ilhan koman papers , mathematical sculpture

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Spiral Developable Sculptures of Ilhan Koman Ahmet Koman Molecular Biology and Genetics Department Bo˜gazic¸i University Istanbul, Turkey [email protected]

Tevfik Akg¨un and Irfan Kaya Faculty of Art and Design Communication Design Department Yildiz Technical University Istanbul, Turkey [email protected] [email protected]

Ergun Akleman Visualization Sciences Program, Department of Architecture Texas A&M University College Station, Texas, USA [email protected]

Abstract In this paper, we present spiral developable sculptural forms invented by Ilhan Koman approximately 25 years ago. We identify the mathematical procedure behind Koman’s spiral developable forms and show that using this procedure, a variety of spiral developable forms can be constructed.

1

Introduction and Motivation

This paper presents the spiral developable forms invented by sculptor Ilhan Koman during the 1970’s. Figure 1 shows Ilhan Koman in front of one of his spiral developable sculptures. For detailed photographs see Figures 2 and 4C.

Figure 1: A Composite of Ilhan Koman in front of one of his Spiral developable sculptures. (The photograph of Ilhan Koman by Tayfun Tunc¸elli, 1980’s). Ilhan Koman has not recorded his method to construct these spiral sculptures. By examining his sculptures, we were able to reconstruct his method. In this paper, we demonstrate Ilhan Koman’s method that is reconstructed by us, hereinafter we simple refer as our method, but, it is, in fact, Koman’s method. To demonstrate that our method is viable, we show that using our method it is possible to construct circular developable sculptures from one piece of rectangular paper. Note that using the same method we can construct a spiral shape by changing the curvature. As shown in Figure 3B, a circular developable sculpture can be constructed using our method from one piece of rectangular paper shown in Figure 3B. 1

Figure 2: A detailed photograph of one of the spiral developable sculptures of Ilhan Koman. Our method can also provide a spiral by using a quadrilateral piece of paper, shown in Figure 4. Figure 4C shows one of Ilhan Koman’s spiral sculptures. Our method allows us to create a comparable form as seen in this comparison. However, Koman’s metal sculpture looks much more elegant than ours as seen in this comparison.

2

Overview

Ilhan Koman is one of the innovative sculptors of the 20th century who frequently used mathematical concepts in creating his sculptures and discovered a wide variety of sculptural forms that can be of interest for the art+math community. Koman was born in 1921, Edirne, Turkey, studied at the Art Academy in Istanbul, opened his first workshop and exhibition in Paris, 1948, moved to Sweden in 1958, where he taught at the Konstfack School of Applied Art in Stockholm until his death in 1986. His works cover a wide spectrum of styles and materials, including 12 public monuments in Sweden and 4 in Turkey. He is represented in several museums including Moderna Museet, Stockholm, Museum of Modern Art, New York, Mus´ee d’Art Moderne de la Ville de Paris. During the last twenty years of his life, he worked on inventing diverging geometrical forms which he developed as prototypes to be realized in large-scale projects. Among the geometrical shapes he developed, tetraflex was a flexible polyhedron which he registered at the Swedish Patent Office in 1971. At the time he had created and proposed these structures as modules for architecture, constructions in space and aviation fuel tanks and published his method in Leonardo [8] For a detailed Ilhan Koman biography see [8, 9, 1]. Ilhan Koman also invented a variety of developable forms, but he did not published his methods. Some of his developable surfaces were the subject of our earlier paper [1]. Developable surfaces are defined as the surfaces on which the Gaussian curvature is 0 everywhere [18]. The developable surfaces are useful since they can be made out of sheets metal or paper by rolling a flat sheet of material without stretching it [14]. Most large-scale objects such as airplanes or ships are constructed using un-stretched sheet metals, since sheet metals are easy to model and they have good stability and vibration properties. Sheet metal is not only excellent for stability, fluid dynamics and vibration, but also one can construct aesthetic buildings and sculptures using sheet metal or paper. Developable surfaces are frequently used by

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Figure 3: Constructing a circular developable sculpture from one piece of rectangular paper shown in (A) using our method. contemporary architects, allowing them to design new forms. However, the design and construction of largescale shapes with developable surfaces requires extensive architectural and civil engineering expertise. Only a few architectural firms such as Gehry Associates can take advantage of the current graphics and modeling technology to construct such revolutionary new forms [11]. Developable surfaces are particularly interesting for sculptural design. It is possible to find new forms by physically constructing developable surfaces. Recently, very interesting developable sculptures, called D-forms, were invented by the London designer Tony Wills and introduced by Sharp, Pottman and Wallner [15, 12]. D-forms are created by joining the edges of a pair of sheet metal or paper with the same perimeter [15, 12]. Pottman and Wallner introduced two open questions involving D-forms [12, 6]. Sharp introduced anti-D-forms that are created by joining holes [16]. Ron Evans invented another related developable form called Plexagons [5]. Paul Bourke has recently constructed computer generated D-forms and plexons [3, 5] using Evolver developed by Ken Brakke [2]. Ilhan Koman’s spiral developable sculptures are the result of a procedure that transforms a single flat developable surface to spiral shapes in 3D. Spirals are one of the most common shapes in nature, mathematics, and art [23]. Behind the beauty of many natural objects such as snail shells, seashells and rams’ horns lies their spiral shapes [22]. The spiral forms exist in almost all cultures as artistic and mystical symbols. This widespread usage of the spiral form may imply that humans innately find it aesthetically pleasing and interesting. Spirals are also popular in mathematical art. Spirals frequently appears Fractal art [21]. Spiral forms can be seen Charles Perry’s mathematically inspired sculptures [13]. Daniel Erdely’s Spidrons also shows spiral structures in 3D [19, 20]. Spirals are among the most studied curves since ancient Greek times. Although spirals are usually represented by parametric equations, there are a wide variety of methods that can be used to construct and represent spirals. There exist a wide variety of spirals such as the Archimedean spiral, the Fermat’s spiral, the Logarithmic spiral and the Fibonacci spiral [23].

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Our Method to Construct Ilhan Koman’s Developable Spiral Forms

In this paper, we present our reconstruction of Ilhan Koman’s method to build developable spiral sculptures. We have developed first a method to build circular sculptures. Spiral sculptures can be obtained by changing the curvature of circular sculptures. The way we change the curvature can directly affect the shape of the spirals.

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Figure 4: Constructing a spiral developable sculpture (B) from one piece of a quadrilateral paper shown in (A) using our method. We traced the boundary with pen to show the the structure better. (C) shows a photograph of a spiral sculpture of Ilhan Koman. Figure 5 shows the the basic procedure to create a circular sculpture. Figure 5A differentiates two different parts of the rectangular paper strip that is used to construct circular sculpture. Dark-blue pieces always stay planar and light-yellow pieces eventually become curved and create the 3D structure of the sculpture.

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Figure 5: Square paper that is used to create circular sculpture. We color coded paper to differentiate the two types of pieces. Blue pieces stays planar and yellow pieces becomes curved. The values of d, h and l can be chosen independently. Our method starts by gluing or stapling the dark-blue piece in the boundary to a planar surface. Once it is flattened, the next neighboring dark-blue piece is flattened by rotating with an angle a and translating such that the second dark-blue piece enters under the now-curved light-yellow piece that is between two dark-blue pieces as shown in Figure 6A. Note that this procedure just makes the yellow piece shorter and forces it to be curved. If we continue this procedure using the same angle a, we start to see a circular arc as shown in Figure 6C. Note that if a = 2π/n where n is an integer, the circular arc is closed as an n sided regular polygon and n becomes the number of blue pieces. The value of a can be uniquely determined by the values of h and d as a = arctan(2d/h) (See Figure 6B). In practice, 2d is chosen much smaller than h as in the example shown in Figure 3B. If 2d is

much smaller than h, then the value of a becomes very small. For very small values of a, the regular polygon closely approximates a circle with a radius r where r≈

nd d d h = = = 2π a 2d/h 2

since tan a ≈ a = 2d/h. Note that r ≈ h/2 is also visible in Figure 3B. For a circle, curvature is constant and can be computed 1/r. Therefore, we can change the curvature by changing either d or h or both.

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Figure 6: Basic procedure. The triangle drawn in (B) shows the relationship between angle a, d and h. The circular structure in (C) is drawn by using exaggerated values: d = 4 and h = 21. Therefore, the angle a is not small, which is approximately 200 . One possible approach is to change only the d value and keep a constant. The value of a can be kept constant by choosing h/d constant. We have created our spiral sculptures this way by changing only the d value and keeping a constant as seen in Figure 4B. In this example, since the d value is changed linearly, our spiral is not a logarithmic spiral, which is more common in nature and considered more aesthetic [22]. We think that Koman’s sculpture is more like a logarithmic spiral and therefore it looks more aesthetic. Although, it is possible to create logarithmic spirals by changing only the d value, for logarithmic spirals d values quickly becomes smaller and it is very hard to cut such a thin slits by hand with scissor. We have not developed a software for this purpose, but, it is possible to automatically create such a drawing and cut the paper or metal using a laser cutter. On the other hand, since laser cutters were not available in 1980, this was not an option for Ilhan Koman. Observing his sculptures it is clear that he did not change d value drastically, instead he made h exponentially smaller. We are sure that he used a simple procedure to change the values of h and d. However, we do not know how he changed the values and what kind of procedure he used to change them.

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Conclusion and Future Work

In this paper, we presented Ilhan Koman’s spiral developable sculptures. We have identified the mathematical ideas behind these spiral forms. We have also introduced a method that can allow the construction of variety of a spiral forms by changing only two parameters. Using the same type of cuts, Ilhan Koman also created more complicated developable surfaces. Our next goal is to understand how he created those more complicated sculptures and disseminate our findings.

References [1] Tevfik Akgun and Ahmet Koman and Ergun Akleman. Developable Sculptural Forms of Ilhan Koman. In Bridges’2006, Mathematical Connections in Art, Music and Science, pages 343–350, 2006. [2] Kenneth A. Brakke, Surface Evolver, http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html [3] Paul Bourke, D-Forms, http://astronomy.swin.edu.au/ pbourke/surfaces/dform/ [4] C. R. Calladine, ”Theory of Shell Structures”, Cambridge University Press, Cambridge, 1983. [5] Ron Evans, Plexons created by Paul Bourke, http://astronomy.swin.edu.au/ pbourke/geometry/plexagon/ [6] Erik D. Demaine and Joseph O’Rourke, “Open Problems from CCCG 2002,” in Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG 2003), pp. 178-181, Halifax, Nova Scotia, Canada, August 11-13, 2003. [7] Ilhan Koman Foundation For Arts & Cultures, ”Ilhan Koman - Retrospective”, Yapi-Kredi Cultural Activities, Arts and Publishing, Istanbul, Turkey, 2005. [8] Ilhan Koman and Franoise Ribeyrolles, ”On My Approach to Making Nonfigurative Static and Kinetic Sculpture”, Leonardo, Vol.12, No 1, pp. 1-4, Pergamon Press Ltd, New York, USA, 1979. [9] Koman Foundation web-site; http://www.koman.org [10] Jun Mitani and Hiromasa Suzuki, “Making Papercraft Toys From Meshes Using Strip-Based Approximate Unfolding. ACM Trans. Graph. 23(3). pp. 259-263, 2004. [11] Frank Gehry, http://www.gehrytechnologies.com/ [12] Helmut Pottmann and Johannes Wallner, “Computational Line Geometry”, Springer-Verlag, 4, p. 418, 2001. [13] Charles Perry. Sculptures. www.charlesperry.com, 2006. [14] http://www.rhino3.de/design/modeling/developable/index.shtml [15] John Sharp, D-forms and Developable Surfaces, Bridges 2005, pp. 121-128, Banff, Canada, 2005. [16] John Sharp, D-forms: Surprising new 3D forms from flat curved shapes,Tarquin 2005. [17] Meng Sun and Eugene Fiume, A technique for constructing developable surfaces, Proceedings of the conference on Graphics interface ’96, Toronto, Ontario, Canada, pp. 176 - 185, 1996. [18] Eric W. Weisstein. ”Developable Surface.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/DevelopableSurface.html. [19] Daniel Erdelyi. Some surprising properties of the spidrons. In Bridges’2005, Mathematical Connections in Art, Music and Science, pages 179–186, 2005. [20] Daniel Erdelyi. Spidron domain: The expending spidron universe. In Bridges’2006, Mathematical Connections in Art, Music and Science, pages 549–550, 2006. [21] B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman and Co., New York, 1980. . [22] D’Arcy Wentworth Thompson. On Growth and Form (Originally Published in 1917). Dover, 1992. [23] Eric W. Weisstein. Wolfram mathworld: Spirals. mathworld.wolfram.com/Spiral.html, 2006.

mathematics Article

Turning Hild’s Sculptures into Single-Sided Surfaces Carlo H. Séquin EECS Computer Science, University of California, Berkeley 94720, CA, USA; [email protected] Received: 15 December 2018; Accepted: 17 January 2019; Published: 25 January 2019







Abstract: Eva Hild uses an intuitive, incremental approach to create fascinating ceramic sculptures representing 2-manifolds with interesting topologies. Typically, these free-form shapes are two-sided and thus orientable. Here I am exploring ways in which similar-looking shapes may be created that are single-sided. Some differences in our two approaches are highlighted and then used to create some complex 2-manifolds that are clearly different from Hild’s repertoire. Keywords: Eva Hild; 2-manifolds; computer-aided design; 3D printing

1. Eva Hild’s Sculptures Eva Hild is a Swedish artist [1], who creates large ceramic sculptures (Figure 1) in an intuitive, incremental process [2,3]. I have been fascinated by her work for several years [4], before I was finally able to meet her in her studio in Sparsör, Sweden in July 2018 (Figure 1b).

Figure 1. Eva Hild: (a) A portrait; (b) working in her studio; and (c) with the sculpture Hollow [1].

Her creations are typically thin surfaces, which may take on the structure of bulbous outgrowths in plant-like assemblies (Figure 2a), or configurations of intricately nested funnels (Figure 2b). These surfaces may be bordered by simple circular openings or by long, sensuously undulating curved rims, which are connected by a system of crisscrossing tunnels (Figure 2c). These sculptures invite mental explorations, raising questions such as: how many tunnels are there? How many separate rims are there? Is this a 1-sided or 2-sided surface? These sculptures also have inspired me to create similar shapes [4]. I do not possess the skills to create large ceramic pieces myself, so I have tried to create computer-aided design (CAD) models of such surfaces, and then have realized the more promising ones as 3D printed models. My students and I found it rather difficult to capture Hild’s sculptures in CAD tools such as Blender [5] or Maya [6]. We had difficulties in defining the proper topologies—often just starting from a few photographs found on the World Wide Web. We also found it difficult to fine-tune these topologically correct models, so that they would reproduce the organic

Mathematics 2019, 7, 125; doi:10.3390/math7020125

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flow and natural elegance of Hild’s creations. We thus ended up developing some additional modeling tools to make the design of such shapes less cumbersome and frustrating.

Figure 2. Eva Hild’s ceramic creations [1]: (a) a bulbous surface with oculus-like openings; (b) various nested funnels; and (c) hyperboloid tunnels and undulating rims.

To place the results of this paper into proper perspective, I start with a brief review of the developments that led me to this current investigation and some of the many other contributions in this domain, made by artists as well as mathematicians. 2. Background and Previous Work For much of my life I have been fascinated by abstract, geometrical sculpture. As a student, some of my heroes were Alexander Calder, Naum Gabo, and Max Bill. In the 1980’s, when I started to teach courses in computer graphics and computer-aided design [7], I employed the help of such techniques to analyze and synthesize topologically interesting sculptural shapes. The key stimulation for my strong involvement in this field came from my interaction with Brent Collins [8]. This connection was prompted by an analytical write-up by George Francis [9] in a special issue on visual mathematics in Leonardo in 1992, which clearly showed the connection between the intuitive sculptural work of Collins and the geometry of minimal surfaces, the topology of interconnected tunnels and handles, and basic knot theory. I started to write a specialized, parametrized computer program, called Sculpture Generator I (Figure 3a) [10] that allowed me not only to capture some inspiring shapes created by Collins, but also enabled me to make several derivative designs that seemed to belong to the “same family”. Collins was fascinated by this prospect and was eager to sculpt additional, more complex shapes by relying heavily on my detailed, computer-generated printouts. Our first two collaborative pieces were Hyperbolic Hexagon II (Figure 3b) [11] and Heptoroid, (Figure 3c) [12], both of which were a generalization of a 6- or 7-story Scherk-Tower [13] bent into a closed, possibly twisted loop. Heptoroid was particularly exciting to us, because it introduced a non-orientable, single-sided surface with a heavily knotted border. Sculpture Generator I was a very specific program that could only generate sculptures based on Scherk’s Second Minimal Surface [13]. To capture any other one of Collins’ inspirational sculptures, such as Pax Mundi (Figure 4a) [14], I needed a new program with different geometrical primitives: in particular, versatile progressive sweeps that would allow me to move an arbitrary cross section along any 3D sweep path. This gradually led to the development of the Berkeley SLIDE program (Figure 4b) [15]. This modeling tool has a variety of parameterized geometric modules that can be combined in many different ways. By using its sweep construct, I was able to capture Pax Mundi and design a large collection of Viae Globi sculptures [16], in which a ribbon travels along the surface of a sphere. With this program we scaled up Pax Mundi, and with the help of Steve Reinmuth [17] realized it as a large Bronze sculpture [18]. The SLIDE program then allowed us to create other Viae Globi sculptures, some of which had knotted sweep paths, such as Music of the Spheres (Figure 4c) [19].

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Figure 3. Sculpture Generator I: (a) The user interface; (b) Hyperbolic Hexagon II; and (c) Heptoroid.

Figure 4. Roads on a Sphere: (a) Pax Mundi; (b) SLIDE user interface; and (c) Music of the Spheres.

Collins created his inspirational sculptures in an intuitive way, driven by the aesthetics of the emerging form, not by thinking about mathematics or topological issues. Other artists start out quite consciously with mathematics on their mind. Helaman Ferguson [20] has created stone and bronze sculptures based on mathematical equations, given knot structures, or topologically intriguing surfaces [21]. George Hart [22] is often driven by some high-order symmetry group and then finds ways to interlink many identical pieces in an intricate manner, while maintaining the envisioned symmetry [23]. He relies on computer-aided tools to define these shapes [24], and even to fabricate them with a computer-driven paper cutter [25]. A similar way of using CAD tools also enables the work of many other artists, such as Bathsheba Grossman [26], Vladimir Bulatov [27], Rinus Roelofs [28], and Hans Schepker [29]. Charles O. Perry [30,31], even though he often starts by defining crucial elements of his sculpture by shaping physical materials such as steel cables, is also quite conscious of the mathematical underpinnings of some of his sculptures [32,33]. For other authors, the key motivation is to generate a visualization of some mathematical concept, rather than creating a piece of art. Jarke van Wijk uses computer graphics to generate the Seifert surfaces supported by mathematical knots [34]. His demonstration tool is readily available on the web [35], and by choosing the right smoothing option, one can produce visualization models that can also stand on their own as pleasing sculptural shapes. Other mathematical concepts, where visualization models can readily turn into art, concern non-orientable surfaces—such as Boy’s surface (Figure 5a) [36,37] or Klein bottles (Figure 5b) [38,39]—and the depiction of regular maps [40–42], regular meshes [43,44], woven surfaces [45,46], knot theory [47–49], or high-genus objects (Figure 5d) [50,51].

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Figure 5. Mathematical visualizations as art: (a) a Boy’s surface [36]; (b) a Klein bottle [38]; (c) a regular map [40]; and (d) a high-genus object [50].

For most of the above mathematical concepts and mentioned artists, Berkeley SLIDE [15] or readily available programs such as Rhino3D [52] or TopMod [53] are adequate tools to capture the corresponding models, because these shapes typically can be depicted with a few clearly defined geometrical primitives. This is true even for the works of artists like Brent Collins [8], Keizo Ushio [54], or Bob Longhurst [55], who intuitively sense some underlying mathematical principles, but do not explicitly use a mathematical formulation or a computer program to define their shapes. Also, the “D-Forms” introduced by Tony Wills [56] or the developable surfaces by Ilhan Koman [57,58] have a clearly defined geometrical structure that can be captured in a procedural specification. Eva Hild’s work [1], on the other hand, presents a new modeling challenge. Her ceramic surfaces are truly free-form [2,3], and if the underlying topology would allow for some regularity or symmetry, Hild would deliberately remove it [4]. Thus, the predefined modules in SLIDE are no longer sufficient, and modeling her work requires some additional capabilities, which will be described in Section 4. Moreover, as discussed in Sections 7 and 8, I faced another novel challenge: how to make a significant topological change to some intuitively conceived geometries, while keeping the visual look-and-feel the same. 3. Classification of 2-Manifolds From a mathematician’s point of view, Eva Hild’s sculptures ( Figures 1 and 2) are smooth 2-manifolds with one or more borders [59]. As a brief reminder: every interior point in a 2-manifold has a neighborhood that is topologically equivalent to a small disk; and every border point has a neighborhood equivalent to a half-disk (Figure 6a,b). Branching of surface sheets, as in the spine of a book, is not allowed. Thus, if the 2-manifold surface is finely tessellated, then every facet edge is used by either one or by two adjacent facets.

Figure 6. Various 2-manifolds: (a) a 2-sided cylinder; (b) a 1-sided Möbius band; (c) a genus-2 handle body; (d) a disk: Euler characteristic χ = 1, genus g = 0; and (e) a disk with 5 holes and 5 cuts.

All possible 2-manifolds can be classified by just three numbers: b, the number of borders; s, its sidedness; and its connectivity described by its genus, g. For a given 2-manifold, the number of borders is easily determined: We start with some border point; we follow along this rim line until we arrive back at the starting point; we mark this rim as ‘counted’; and we repeat this until there are no further unmarked borders. To determine the sidedness, s, we start with an interior point and paint its

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neighborhood with a flood-filling algorithm; we continue this process as far as we can without ever stepping across a border line. If, in the end, all surface points have been painted, then the surface is single-sided, and also non-orientable. If only half of all surface areas have been covered, then the surface is two-sided, and orientable. The connectivity of a 2-manifold is more difficult to determine. Its genus, g, is defined as the maximal number of closed curves that can be drawn on the 2-manifold without dividing its surface into two separate “countries.” But how do we know that we have indeed found the maximal number of such curves? One more practical approach is to close off all punctures in the surface by gluing in disk-like surface patches into any open border loops, as long as this can be done without creating any self-intersections. If this results in a well-behaved handle-body (Figure 6c), we count the number of its handles or tunnels to find its genus. As an alternative, we may make cuts through the 2-manifold from some border point to another border point until the resulting 2-manifold has acquired the topology of a single disk without any holes (Figure 6d). If there are still interior holes in the surface, we need to make additional cuts from the current disk borders (Figure 6e). However, we must not split the surface into separate pieces; everything must remain connected. We know that the Euler characteristic, χ, of a disk is 1, and that every cut adds 1 to χ. Thus, we derive that the original manifold must have had an Euler characteristic of 1 minus the number cuts that we have made. The Euler characteristic and genus are then related according to the following formula: Genus = (2 − χ − b)/s,

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An intriguing discovery is that the 2-manifolds created by Eva Hild [60] show a wide variety in their number of borders and in their genus—but they are all double-sided! Later, I will speculate how this may have come about. Furthermore, I will also try to transform some of these sculptures into single-sided 2-manifolds, while making only small style changes. 4. My Modeling Approach After some experimentation, I found that a good way for capturing the basic geometry of Hild’s sculptures is to locate some defining key features, in particular, “rims”, “funnels”, and “tunnels”. These features have been marked on a few sculptures in Figure 7. These features typically form the starting point in my attempts to model these geometries. They are placed in appropriate locations in 3D space, and the surface is then constructed, facet by facet, between points on these border curves in an interactive manner through a graphical user interface. Our home-brewed modeling system, Berkeley SLIDE [15], developed two decades ago, is well suited to define and place such key features in a parameterized manner. It provides a slider for every parameter, so that complex configurations of such features can be fine-tuned in an interactive manner. Unfortunately, it has no interactive capabilities to form the connecting surface between these features. Moreover, SLIDE cannot handle single-sided surfaces in operations such as surface smoothing and offsetting.

Figure 7. Key features marked on some Hild sculptures: red indicates rims; blue indicates funnels; and green indicates tunnels.

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To overcome these shortcomings, my research students have helped me to create a new CAD environment, called NOME (Non-Orientable Manifold Editor) [61,62]. Just like SLIDE, this program allows the procedural placement of the parameterized key features though a simple text interface. It then provides crucial editing capabilities to form a surface between these features. Moreover, there are pre-programmed generators for funnels and tunnels with all their defining variables. Once these features have been placed reasonably well, the user constructs the connecting mesh in a point-and-click manner. The user may select some number, n, of vertices and then create a n-sided facet (typically irregular and non-planar) spanned by those vertices. The key characteristic is that this facet behaves like a rubber sheet, staying attached to the different tunnels, funnels, or rims, as these elements are repositioned and re-shaped. The parametrization of the key features remains active through the entire design process. Even after subdivision smoothing has been applied and offset surfaces have been generated, the parameters can still be changed, and the user will immediately see the effect that this has on the final geometry to be sent to the 3D printer. This is a crucial capability of NOME that we have found difficult to implement in other CAD tools. I will now describe my current modeling approach with the example of a reconstruction of two relatively simple sculptures by Eva Hild. The first one is Interruption [60]; Figure 7a shows a bottom view of this sculpture, and Figure 8a is a side view. In Figure 8b, I placed a (blue) funnel at the bottom, fitted a (red) free-form B-spline curve along the top rim, and inserted three (greenish) tunnels to define the internal passages. Identifying, defining, and placing these key features is a crucial first step in creating a replica of an existing sculpture. So far, we have not been able to think of an automated process that would extract that information from a few images. Being able to adjust the parameters that define these features through the very end of the design process is thus an important capability for obtaining good-looking results.

Figure 8. Interruption: (a) Hild’s sculpture with key features marked; (b) key features modeled; (c) first quad faces added; (d) the connecting mesh; (e) the subdivision surface; and (f) the resulting 3D print.

The next step is to specify the polyhedral mesh components that properly connect these feature geometries to one another. First, a few connecting quads are introduced, shown in red in Figure 8c; then the various mesh patches are filled in, shown in yellow in Figure 8d. This composite mesh is subjected to three or four steps of Catmull-Clark subdivision [63] to produce a smooth, tangent-continuous surface (Figure 8e). This smoothing process has been robustly implemented in SLIDE as well as in NOME, and it typically produces soap film-like surfaces for polyhedral models with a variety of facets. The best results are obtained when the starting mesh is composed mostly of quadrilaterals that all have about the same edge length. Once we have a pleasing looking mesh, which is tessellated finely enough to display the desired level of smoothness, NOME generates two offset surfaces that lie at a distance t/2 on either side of the subdivision surface. Additionally NOME generates a set of rectangular facets along all border curves to connect the two offset surfaces into a water-tight boundary, creating a physical object of thickness, t, which then can be fabricated on a 3D printer (Figure 8f). Even at this late stage in the design process, the various parameters are all still fully in effect. For instance, by raising the position of the blue funnel, the tubular base of the sculpture can be shortened, or by changing the diameter or length of any of the tunnels, the corresponding hole-geometries can be fine-tuned. This final geometry is then captured as an STL-file [64]. In this

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procedure, the facets of the boundary representation are split into triangles, which are then listed with their three vertices and their face normal. This is a rather wasteful description, and the resulting files are typically tens of Megabytes in size; it is, however, a format that is understood by all 3D printers. Creating good 3D models represents additional challenges. 3D printers come with their own software for handling the tricky task of constructing adequate scaffolding structures for supporting hollow or overhanging geometries as needed. However, print orientation, print speed, and fill-in density must still be decided by the user, and they can strongly affect printing time and the quality of the print. 5. Modularity: Deriving New Topologies The CAD tools described above, allow me to now focus on designing new geometries inspired by Hild’s creations. In a first demonstration, I started from Hild’s Whole (Figure 9a,b). To model this structure, I created a new auxiliary geometrical shape, a cross-tunnel (Figure 10a). Two instances of this new feature were then surrounded with an undulating 3-period Gabo curve [65] to form the complete feature model (Figure 9c) defining my version of this sculpture. Figure 9d shows the resulting 3D print. In contrast to Hild’s sculpture, which has an organically flowing, slightly irregular shape, my model exhibits strict “C2h symmetry” (Schönflies notation), also known as “2*” symmetry (Conway notation), featuring a vertical mirror plane (most easily visible in Figure 9c) and a horizontal C2 rotation axes (Figure 9d).

Figure 9. Modeling Whole: (a,b)two views of Hild’s sculpture; (c) skeleton computer-aided design (CAD) model; (d) 3D printed model.

Figure 10. Wrapped Cross-Tunnels: cross-tunnel sculpture.

(a,b), a one cross-tunnel sculpture; and (c,d) a three

These same basic features can now be readily used in different configurations to make new derivative geometries. Figure 10a,b show what happens when I use just a single cross-tunnel and augment it with a simple 2-period Gabo curve (as found in the seam of a baseball). In yet another variation, I have placed three such cross-tunnels side-by-side and wrapped a 4-period Gabo curve around them (Figure 10c). NOME [62] was used to fill the voids between the Gabo ribbon and the inner core formed by the three cross-tunnels. The resulting 3D printed object is shown in Figure 10d.

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Figure 11 shows a sculpture that results, when three cross-tunnels are placed in a circular arrangement, so that the lower tunnels point in radial directions. This inner core is surrounded by a 3-period Gabo curve, and an additional large funnel is placed at the bottom of this arrangement to produce a stand for the sculpture. Figure 11a shows the key features placed in 3D space. To this I then add one third of the connecting control surface (Figure 11b). Two copies, rotated by ±120◦ , result in the complete topological model. After three levels of Catmull-Clark-subdivision [63] and offset surface generation (Figure 11c), a corresponding STL file is saved and sent to a 3D printer (Figure 11d).

Figure 11. The Lighthouse: (a) the key features; (b) 1/3 of the control surface is added; (c) model after smoothing and offsetting; and (d) the resulting 3D printed object.

This model has overall 6-fold C3v symmetry. Eva Hild would not create such geometries with perfect symmetry. Even if the topology offers some inherent symmetry, she would deliberately deform the geometry to break that symmetry and obtain a more organic, natural look. A drastic example of this can be found in Wholly, a metal sculpture located in the town of Borås, Sweden. 6. Wholly—A More Challenging Modeling Task Free-form surfaces, such as Wholly (Figure 12a), offer a bigger modeling challenge. The difficulties in capturing the shape of this piece demonstrate the need for a well-selected procedural description of the basic geometry. The first step is to figure out the topology and connectivity of the given surface. Here it is captured in the relatively simple model shown in Figure 12b. This is an orientable surface of genus 4 with a single border. Geometrically, it can be seen as a chain of eight side-by-side tunnels separated by seven saddle surfaces. The bottom half shows a simple polyhedral model that yields the proper topology for Wholly. However, it would be overwhelming to ask the user to move all 72 vertices of this model into the appropriate locations to recreate Wholly. Thus, we need a higher level of control!

Figure 12. Hild’s Wholly: (a) the sculpture in Borås; (b) models capturing its topology.

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For this model, I defined nine cross-sectional planes, seven of which go through the walls between pairs of adjacent tunnels, and two more that are located at the two ends of the chain. Each of these sections can now be non-uniformly stretched, rotated, and shifted. A result of such coordinated edits is shown in Figure 13a. All of the nine cross sections remains planar and symmetrical, but their sizes and positions define the shapes and orientations of the tunnels between them. The shape in Figure 13b starts to show the look and feel of Hild’s Wholly sculpture.

Figure 13. Modeling Wholly: (a) a deformed control mesh; (b) the resulting shape after smoothing.

The described approach was not yet satisfactory for modeling this particular sculpture. A key difficulty was keeping all eight tunnels close to circular. So, I looked for a higher-level procedural model that comprises circular tunnels as one of its basic primitives, accompanied by a convenient user interface to appropriately scale each tunnel and to place them snuggly next to one another at the desired rotation and tilt angles. Figure 14 shows such a procedural model, positioning eight partial toroids into a flexible chain. I did not use complete rings, because the walls between adjacent tunnels must be a single shared surface, rather than two almost coinciding surfaces contributed by two adjacent toroids. The missing connectivity between the open gap in one toroid and the ends of the adjacent one were then established in an interactive editing session in NOME.

Figure 14. A procedural model of Hild’s Wholly: (a) eight toroidal elements in appropriate positions; (b) smoothed model after linking the building blocks.

But even this remaining gluing-operation was relatively tedious and did not immediately lead to the smooth saddles found in Hild’s sculptures. Thus, in a further iteration, I focused on these saddle elements. I defined a procedural model for a saddle geometry in which I could vary the two principal curvatures individually (Figure 15a). Now the gluing operations can be done between the four open ends of two adjacent saddles. This operation now takes place in a more open space, and it was thus less difficult to produce nice, smooth transitions (Figure 15b).

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Figure 15. A procedural saddle model: (a) the basic element, (b) a chain of three elements, and (c) the end module.

For added expediency, I also introduced a special end module, consisting of a complete, well-formed toroid with a single saddle connector on one side (Figure 15c). This then allowed me to obtain a reasonably good match with the original sculpture. 7. Introducing a Möbius Rim With a reasonable modeling technique in hand, I will now focus on topological issues in Hild’s sculptures. Most of her sculptures seem to be two-sided. I have not yet found a picture of one that is clearly single-sided; but it is difficult to analyze the more complex ceramic pieces from just one or two images. This prompted me to try to construct some Hild-like 2-manifolds that are single-sided and non-orientable—like a Möbius band or a Klein bottle. A first approach is derived from the Sue-Dan-ese Möbius band (Figure 16a) [66]. It has a single circular border connected to a roughly spherical bag. But it does not look very Hild-like; its rim does not resemble a funnel, and part of it is obscured by the geometry of its bag-shaped body. Thus, I extend the circular rim into a loopy figure-8-shape, letting the border follow the tangential pull of the attached surface (Figure 16b). This leads to an undulating 3D border curve that is more akin to what is found in Hild’s sculptures. In Figure 16c, this new Möbius rim (red) is combined with a bottom funnel (blue) on which the sculpture may rest stably; in addition, part of the connecting mesh is shown in green. Figure 16d shows the resulting smooth, single-sided surface.

Figure 16. (a) The Sudanese Möbius band [66]; (b) relaxing the rim into a 3D figure-8 shape; (c) the key features: Möbius rim plus bottom funnel; (d) the resulting single-sided surface.

In a related experiment, I tried to make a single-sided version of Hild’s Interruption. With this goal in mind, I replaced the bent oval border loop of Interruption with a stretched version of the Möbius rim (Figure 16c), and I used the same internal combination of three tunnels as in Figure 8b. Figure 17a shows the skeleton formed by these key features and a few initial connecting faces (shown in red). NOME made it easy to add all the other facets to form the complete connecting mesh (Figure 17b). Two steps of subdivision already form a nice, smooth surface (Figure 17c). Figure 17d shows the resulting 3D printed model.

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Figure 17. Modeling Single-Sided Interruption: (a) a skeleton with a Möbius ribbon on top; (b) the connecting mesh; (c) the subdivision surface; (d) a resulting 3D printed model.

8. Rings of Dyck’s Disks Another approach to generate a non-orientable surface starts with a single circular disk from which two tubular stubs emerge in opposite directions (Figure 18a); this is known as Dyck’s surface [67]. If these two stubs are joined together with a toroidal loop, one obtains a single-sided 2-manifold (Figure 18b). However, such a prominent toroidal handle is not a typical occurrence in Hild’s artwork. Thus, I replaced the toroidal handle by inserting additional Dyck disks into the loop (Figure 18c,d). Any such loop formed by an odd number of disks will result in a single-sided 2-manifold.

Figure 18. (a) Dyck’s surface; (b) with a toroidal loop added; (c) five disks in a circuit; (e) seven Dyck disks in a circuit.

Eva Hild typically avoids rigid symmetry and makes the tunnels and lobes in a sculpture of somewhat different sizes (Figure 9a,b). Thus, in Figure 19, I scaled subsequent instances of Dyck’s disks by 10% and let this logarithmic spiral sweep through only 300◦ . The remaining 60◦ are then filled with a bulbous element (as found in some of Hild’s sculptures), which in this case, is a convenient way to connect the two tubular stubs of rather different diameters. This sculpture was exhibited at the Bridges 2017 Mathematical Art Gallery [68].

Figure 19. Five Dyck disks of different sizes connected into a loop: (a) the placement of the defining key features and part of the connecting mesh; (b) the resulting 3D printed model [68].

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9. Dyck Clusters of Higher Genus The Dyck disk has only two tubular connectors, so it can form only open chains or simple loops. Therefore, I created a modification with two stubs emerging from either side of the disk. Now we have a valence-4 element that can realize more intricately connected graphs. A single Dyck disk with the stubs emerging from the same side connected to one another, yields a two-sided surface (Figure 20a). If, instead, loops are formed between stubs emerging from opposite sides, a non-orientable surface results (Figure 20b). In Figure 20c, four disks have been connected in a symmetrical manner to form an orientable, two-sided surface. In the following constructions, I have moved in the opposite direction of where Eva Hild would be going; I have aimed to maximize symmetry as well as the connectivity of my 2-manifolds. To obtain a high degree of symmetry, I start with the symmetries of the Platonic solids. In Figure 21a, I have placed six 4-stub Dyck disks at the edge-midpoints of a tetrahedron. I then interconnected groups of six stubs that point towards the same tetrahedron vertex so that they form a 3-branch ring, reminiscent of a truncated corner. In Figure 21b,c, I apply the same process to an octahedron. In these latter figures, we are looking down onto a truncated valence-4 corner. The genus of these surfaces is always one higher than the number of disks, and every rim of a disk represents a topological “puncture” in these 2-manifolds.

Figure 20. Dyck disks with four stubs: (a) local connections resulting in a two-sided surface; (b) local connections forming a one-sided surface; (c) four disks forming a two-sided surface.

Figure 21. Clusters of 4-stub Dyck disks: (a) 6 disks in tetrahedral symmetry; (b) 12 disks in octahedral symmetry: CAD model; (c) octahedral 3D printed model.

In Figure 22a, twelve 4-stub Dyck disks have been placed at the mid-edge points of a cube; here we are looking at one of the original cube faces. Finally, in Figure 22b, 24 disks have been placed perpendicularly to the edges of a rhombic dodecahedron. The result is a single-sided surface of genus 25 with 24 punctures. It has been exhibited at the JMM 2018 Mathematical Art Gallery [69].

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Figure 22. Platonic Dyck clusters: (a) based on cubic symmetry; (b) based on rhombic dodecahedron symmetry [69].

10. Discussion and Conclusions In my modeling efforts assisted by CAD tools, I start with the placement of border curves and other defining features, and then I fill in surface patches between them (Figure 17 and Figure 19). Alternatively, I may place modular surface elements, so that some of their borders coincide and then merge them into a single cohesive 2-manifold (Figures 21 and 22 ). Eva Hild, on the other hand, conveyed to me, in my visit to her studio, that she mostly thinks about spaces. Her ceramic surfaces serve to define those spaces and to separate adjacent spaces. She may start with some bowl-shaped surface or some cylindrical wall and then gradually grow that surface in an organic manner. In this process, the surface may form funnels or bulbous outgrowths. Tunnels may split into two or more branches and then recombine in different ways, and border curves may warp into sensuous undulations. Twisted ribbons are not natural elements for defining spaces, so they are not typically found in Hild’s sculptures. Tubular loops, starting and ending on different sides of the same surface patch, as in Dyck’s disk (Figure 18b), would be another way to produce single-sided surfaces. Such loops rarely show up in her 2-manifold sculptures. Thus, it took some effort to introduce non-orientable surface constructs into my models, while hiding them, so that the result does not immediately stand out as something that one would not find in Hild’s studio. Our emerging NOME modeling environment [62] makes the design process for 2-manifold sculptures in the style of Eva Hild manageable, even though it is still very much in the development stage. It is particularly convenient and powerful for geometries with a high degree of symmetry. On the other hand, it is unlikely that with our current approach I will ever create something approximating the fluid, natural beauty of Hild’s sculptures. Eva explained to me that her pieces grow slowly and organically. She has an initial idea and a starting point, but this will then change and develop gradually during a process, which may take months or even years. She is often surprised when she looks at the resulting final sculpture and wonders: “Where did that shape come from?” In contrast, my own computer-based approach is much more “top-down.” I start with a well-defined plan, and specify the overall symmetry that I want to maintain. The use of symmetry significantly reduces the amount of detailed design work that I have to do. Moreover, the use of computer-aided procedures allows me to create structures of a complexity that would be difficult to achieve in a gradual, bottom-up approach. While my own creations may have a quite different look-and-feel to them, I still would like to thank Eva Hild—and many other “intuitive” artists—for the inspiration they provide.

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Funding: This research received no external funding. Acknowledgments: I would like to thank the staff of the Jacobs Institute for Design Innovation at UC Berkeley for their help in fabricating many of the sculptural models presented. Conflicts of Interest: The author declares no conflict of interest.

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Bridges 2017 Conference Proceedings

Homage to Eva Hild Carlo H. Séquin CS Division, University of California, Berkeley E-mail: [email protected] Abstract Inspired by the work of Eva Hild, I have developed some auxiliary CAD programs and design processes that make it easier to model her undulating free-form sculptures on a computer, so that similar looking maquettes can be fabricated on a 3D printer. A special focus has been placed on trying to make single-sided 2-manifolds in a similar style.

1. Introduction Eva Hild is a Swedish artist, whom I have never met personally. But for several years, I have been fascinated by pictures of her artwork found on the Internet [5]. Her sculptures (Fig.1) are not only a pleasure to look at, but they invite mental exploration of questions such as: How many tunnels are there? How many separate rims are there? Is this a 1-sided or 2-sided surface? Even more importantly, these sculptures inspire me to create similar shapes. I do not possess the skills to create large ceramic pieces myself, so I aim to create CAD models of such surfaces, and the more promising ones I then realize on a 3D printer. It turns out that trying to create such shapes with a typical CAD program is difficult and tedious. Those programs have not been designed to create such smooth, free-flowing shapes. If I wanted an exact replica of one of those sculptures, I could travel to Sweden and capture a few hundred thousand surface points with a Kinect [9] or some other 3D scanning device. However, I am more interested in creating my own new shapes, reflecting some of the key features of Hild’s creations. Thus, I started an effort to develop auxiliary program modules and processes to define the typical elements found in Hild’s sculptures and to compose them in new ways. This paper describes these efforts and shows some initial results.

(a) (b) (c) Figure 1: Eva Hild’s ceramic creations; (a) bulbous surface with oculus-like openings; (b) various nested funnels; (c) hyperboloid tunnels and undulating rims.

2. Key Features in Hild’s Sculptures From a mathematician’s point of view, Hild’s sculptures (Fig.1) are smooth 2-manifolds with one or more borders. Some of these borders may be close to circles or ellipses, while others form smooth, undulating, 3D space curves. Some of her sculptures are characterized by bulbous outgrowths with positive Gaussian curvature (Fig.1a); others by giant funnels (Fig.1b) and saddles, or by tunnels in the shape of single-shell hyperboloids with negative Gaussian curvature (Fig.1c); the latter are particularly interesting to me. 117

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After some studies, I found that a good approach for capturing the essence of such a sculpture is to locate some defining key features, in particular, “rims,” “funnels,” and “tunnels.” These features have been marked on a few sculptures in Figure 2. They should be captured in any computer description, and they then form the basis from which the complete sculpture surface can be developed.

(a) (b) (c) Figure 2: Key features marked on some Hild sculptures: red rims; blue funnels; green tunnels. Rims are highlighted by red lines that follow some 3D border curves; in a CAD model, these 3D space curves are represented conveniently as B-splines. Funnels are ribbons that follow circular or elliptical openings; they can be generated by circular sweeps or by truncated cones. Tunnels are smooth cylindrical segments appearing in a sculpture. They can either represent internal passages (Fig.2b) or the “waists” of some bulbous bellies (Fig.2c). All of these elements can either be defined as 1-manifold curves, or they can be modeled by narrow ribbons, as indicated in Figure 2a. The latter approach is particularly useful for rims and for funnels, because it makes it possible to specify the tangential direction under which the surface takes off from those borders. For this purpose, the ribbon description must have a way to specify the local azimuth angle of its cross-section with respect to the sweep line. Progressive sweeps with individual control of azmuth and twist, as implemented in SLIDE [12], are ideally suited for this purpose.

3. A Computer-Aided Design Process I will now describe my current modeling approach with the example of a reconstruction of two relatively simple sculptures by Eva Hild. The first one is Interruption [7]; Figure 2a gives a view of its bottom, and Figure 3a is a side view.

(a) (b) (c) (d) (e) Figure 3: Modeling “Interruption” in SLIDE: (a) Hild’s sculpture with key features marked; (b) key features modeled; (c) connecting control mesh; (d) subdivision surface; (e) a first 3D print. In Figure 3a, I placed a (blue) funnel at the bottom, fitted a (red) free-form B-spline curve along the top rim, and inserted a (green) tunnel to define the central opening. Figure 3b shows these defining features by themselves, modeled in the Berkeley SLIDE environment [12]. The next step is to specify polyhedral mesh components that properly connect the three feature geometries to one another. For a simple structure, one can specify the necessary faces with a text editor, referring to the proper control vertices that define the key features (Fig.3c). However, this approach is rather tedious and error-prone in any CAD environment that 118

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does not support it with some interactive graphics technique. When the topology of the whole surface has been defined in this manner, the composite mesh is subjected to two or three steps of Catmull-Clark subdivision [2] to produce a smooth, tangent-continuous surface geometry (Fig.3d). Finally, two offset surfaces are generated, lying at distance t/2 above on either side of this subdivision surface, so as to create the geometrical description of a physical object of thickness, t, that can be fabricated on a 3D printer (Fig.3e). To make this construction process more amenable, my research students have helped me to create a new CAD environment, called NOME (Non-Orientable Manifold Editor) [14], which provides these crucial editing capabilities. As in SLIDE, the defining features can be parameterized, allowing the user to finetune the shapes and placement of these features. There are pre-programmed generators for funnels and tunnels with all their defining variables. Once these features have been placed reasonably well, the user can construct the connecting mesh in a point and click manner. The user may select some number, n, of vertices and then create a (typically irregular, non-planar) n-sided facet spanned by those vertices. The key characteristic is that this facet behaves like a rubber sheet, staying attached to the different tunnels, funnels, or rims, as these elements are repositioned and re-shaped. The parametrization of the key features remains active through the whole design process. Even after subdivision smoothing has been applied and offset surfaces have been generated, those parameters can still be changed, and the user will see immediately the effect that this has on the final geometry to be sent to the 3D printer. This is a crucial feature, that distinguishes NOME from most other CAD tools. Proper placement of the three defining elements shown in Figure 3 and interactive fine-tuning of their locations, sizes, and shapes, allowed me to capture quickly the topology and basic geometry of Interruption. However, the resemblance between the original and my model was not very close; the leftward pointing tunnel entrance in the lower half of Figure 3a has a rather elliptical shape in my model, while in Hild’s ceramic it is much more circular. To obtain a better match in geometry, I introduced two additional tunnels (Fig.4a). This makes for a more complicated connecting mesh; but in the NOME environment, it is easy to construct it. First, a few connecting quads are introduced, shown in red in Figure 4b; then the various mesh patches are filled in, shown in yellow in Figure 4c. The rest of the process is almost automatic: a subdivision surface (Fig.4d) and two offset surface (Fig.4e) are produced. Even at this stage, the various parameters are still fully in effect. For instance, by lowering the position of the blue funnel, the tubular base of the sculpture can be elongated. When a satisfactory shape has been achieved, the geometry is saved in .STL format, which can then be sent on a low-end 3D-printer [13] to obtain a physical maquette (Fig.4f).

(a) (b) (c) (d) (e) (f) Figure 4: Modeling “Interruption” in NOME: (a) augmented key features; (b) first quad faces added; (c) connecting mesh; (d) level-2 subdivision surface; (e) offset surface; (f) resulting 3D print; Figure 5 shows this same design process applied to “Hollow” [6], a more complicated sculpture by Eva Hild (Figs.5a-5d). To start with, I had difficulties understanding the complete connectivity of this sculpture. Two years ago, when I first started this quest, I could find dozens of front-view pictures on the Web, but not a single view from the back. The best I could do, was to go into Google Street View [4] and virtually position the viewpoint, so that a reflection of the backside (Fig.5c) could be seen in the plate-glass windows of the building in front of which the sculpture is located. Only recently could I find a direct view of the backside (Fig.5d). To get a clear understanding of the topology of this sculpture, I formed a small, crude clay model (Figs.5e, 5f). With this in hand, I could then specify a whole set of properly sized and spaced 119

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funnels and tunnels that outline the geometry of this sculpture (Fig.5g). NOME then made it easy to fill in the connecting mesh fragments. It offers a utility that allows the user to select two whole border curves and then invoke an automatic zippering routine. This routine fills the space between the two border curves with a ribbon made of quadrilateral and triangular faces, trying to keep the number of these facets small, and the lengths of the connecting edges as short as possible (Fig.5h). After subdivision smoothing (Fig.5i) and surface thickening (Fig.5j), a 3D print can be obtained (Figs.5k, 5l) that has a fairly close resemblance to Hild’s original.

(a)

(e)

(b)

(c)

(f)

(g)

(d)

(h)

(i) (j) (k) (l) Figure 5: “Hollow”: (a),(b) front views; (c) reflected back view; (d) newer back view of sculpture; (e) clay model front and (f) back view; (g) key elements positioned; (h) connecting mesh; (i) a first subdivision step; (j) 3 subdivisions plus offset surfaces; (k) 3D maquette, back and (l) front view.

4. Deriving New Topologies With the NOME editing utilities [14] at my fingertips, I could now focus on creating new geometries inspired by Hild’s creations. In a first experiment, I started from Hild’s “Whole” (Figs.6a, 6b) [8]. A clay model helped me again to understand the connectivity of this sculpture (Fig.6c) and to make a corresponding skeletal CAD model (Fig.6d). To model this structure, I created a new auxiliary geometrical shape, a “cross-tunnel” (Fig.7a). Two instances of this new feature were then surrounded with an undulating 3-period Gabo curve [10] to form the complete feature model (Fig.6d) that defines my version of this sculpture. In contrast to Hild’s sculpture, which has an organically flowing, slightly irregular shape, my model has strict “C2h symmetry” (Schönflies notation), also known as “2*” symmetry (Conway notation), featuring a vertical mirror plane (most easily visible in Fig.6d) and a horizontal C2 rotation axes. Figure 6e shows the resulting 3D print. 120

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(a) (b) (c) (d) (e) Figure 6: (a),(b) Hild’s “Whole”; (c) clay model; (d) skeleton CAD model; (e) 3D print. Figures 7a & 7b show what happens when I use just a single cross-tunnel and augment it with a simple 2period Gabo curve (as found in the seam of a baseball). In yet another variation, I have placed three such cross-tunnels side by side and wrapped a 4-period Gabo curve around them, which is stretched left-to-right to fit more gracefully around the central structure (Fig.7c). NOME was used to fill the voids between the Gabo-ribbon and the inner core formed by the three cross-tunnels, and to obtain a 3D-print (Fig.7d).

(a) (b) (c) (d) Figure 7: “Wrapped Cross-Tunnels”: (a),(b) one cross-tunnel; (c),(d) three cross-tunnels. Figure 8 shows sculptures that result, when three or more cross-tunnels are placed in a circular arrangement. The sculpture shown in Figure 8a starts with 3-fold symmetrical arrangement of three cross-tunnels, where the lower tunnels point in the radial direction. This core is surrounded by a 3-period Gabo curve that smoothly connects to the various tunnel openings. An additional large funnel is placed at the bottom of this arrangement to produce a stand for this sculpture. Figure 8b is an arrangement of four cross-tunnels; but here all upper tunnels point in the same direction (towards the viewer). A 6-period Gabo curve has been wrapped around this core. It naturally connects to half of all the tunnel openings, but it also makes an extra up/down transition in between (Fig.8c). This leads to a somewhat convoluted inner border curve, for which I have not found a good way to let it terminate in one or more additional funnels. Fortunately, this odd border is not readily visible from the outside. This model has overall 8-fold D2d symmetry (“2*2” Conway notation).

(a) (b) (c) Figure 8: (a) 3 cross-tunnels in a 3-fold symmetrical cycle, enhanced with a 3-period Gabo curve; (b) 4 cross-tunnels with alternate orientations, (c) enhanced with a 6-period Gabo curve. 121

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5. Single-Sided 2-Manifolds Most Hild sculptures seem to be two-sided. I have not yet found a picture of one that is clearly singlesided; but it is difficult to analyze the more complex ceramic pieces from just one or two images. This prompted me to try to construct some Hild-like 2-manifolds that are single-sided and non-orientable – like a Möbius band. I pursued three different approaches. And for each conceptual approach, I also asked myself, what might be the simplest possible geometry that could result from it, and which would still exhibit some Hild-like appearance. Möbius Rim A first approach is derived from the “Sue-Dan-ese” Möbius band (Fig.9a) [1]. It has a single circular border connected to a roughly spherical bag. But it does not look very Hild-like; its rim does not resemble a funnel, and part of it is obscured by the geometry of its bag-shaped body. Thus, I let the circular rim be transformed into a loopy figure-8-shape, letting the border follow the tangential pull of the attached surface (Fig.9b). This leads to an undulating 3D border curve that is more akin to what is found in Hild’s sculptures. In Figure 9c, this new Möbius rim (red) is combined with a bottom funnel (blue) on which the sculpture may rest stably; part of the connecting mesh is shown in green. Figure 9d shows the resulting smooth surface.

(a) (b) (c) (d) Figure 9: (a) The Sudanese Möbius band; (b) relaxing the rim into a 3D figure-8 shape, (c) key features: Möbius rim plus bottom funnel; (d) resulting single-sided surface. In a related experiment, I tried to make something like a “Single-Sided Interruption.” With this goal in mind, I replaced the bent oval border loop of Interruption with a stretched version of the Möbius rim (Fig.9c), and used the same internal combination of three tunnels as in Figure 4a. Figure 10a shows the skeleton formed by the key features and a few initial connecting faces (shown in red). NOME made it easy to add all the other facets to form the complete connecting mesh (Fig.10b). Two steps of subdivision already form a nice, smooth surface (Fig.10c). Figure 10d shows the resulting 3D print.

(a) (b) (c) (d) Figure 10: Modeling “Single-Sided Interruption”: (a) Skeleton with Möbius ribbon on top; (b) connecting mesh; (c) subdivision surface; (d) resulting 3D print.

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Tunnels and Tongues My second approach to producing non-orientable Hild-like surfaces starts with an odd number of tunnels in the form of hyperbolic “cooling towers.” A “tongue” extends from the top of one tower to the bottom of the next one, thus connecting the inside of one tower to the outside of the subsequent one (Fig.11a). The minimalistic version of this approach is a single tower with a helical tongue connecting the top to the bottom rim. I experimented with a few different geometries, changing the length, width and steepness of that ribbon (Figs.11b−11e); my favorite geometry is shown in Figure 11e.

(a) (b) (c) (d) (e) Figure 11: (a) 3 connected “cooling towers.” − (b-e) A single tower with a connecting tongue that is: (b) too long, (c) too wide, (d) too small, (e) “just right.” Dyck’s Surface A third approach starts with a single circular disk from which two tubular stubs emerge in opposite directions (Fig.12a); this is known as Dyck’s surface [3]. If these two stubs are joined together with a toroidal loop, one obtains a single-sided 2-manifold (Fig.12b). However, such a prominent toroidal handle is not a typical occurrence in Hild’s artwork. Thus, I replaced the toroidal handle by adding two more Dyck disks to obtain the desired loop closure (Fig.12c). Any such loop formed by an odd number of disks will result in a single-sided manifold; Figure 12d and 12e shows loops with five and seven such disks.

(a) (b) (c) (d) (e) Figure 12: (a) Dyck’s surface: (b) closed with toroidal loop; (c) three elliptical disks in a circuit; (d) five Dyck disks in a circuit; (e) seven Dyck disks in a circuit.

(a) (b) (c) Figure 13: Five Dyck disks of different sizes connected into a loop: (a) the placement of the defining key features and part of the connecting mesh; (b) the smooth subdivision surface; (c) a 3D print. 123

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Eva Hild typically avoids rigid symmetry and makes the tunnels and lobes in a sculpture of somewhat different sizes (Fig.6a). Thus, in Figure 13, I scaled subsequent instances of Dyck’s disks by 10% and let this logarithmic spiral sweep through only 300°. The remaining 60° are then filled in with a bulbous element (as found in Fig.5d); this is a convenient way to connect two tubular stubs of rather different diameters.

6. Summary and Conclusions NOME [14] makes the design process for 2-manifold sculptures in the style of Eva Hild much easier. Still, it is unlikely that with my approach I will ever create something approximating the fluid, natural beauty of her sculptures. In a first e-mail exchange, in which I had sent Eva a draft of this paper, she mentioned in her response that her pieces grow slowly and organically. She has an initial idea and a starting point, but this will then change and develop gradually during a process, which may take months or even years. Often she is surprised when she looks at the resulting final sculpture and wonders, where that shape might have come from. In contrast, my own computer-based approach is much more “top-down.” I start with a well-defined plan, and specify the overall symmetry that I want to maintain. The use of symmetry significantly reduces the amount of detailed design work that I have to do. On the other hand, the use of computer-aided procedures allows me to create structures of a complexity that would be difficult to achieve in a gradual, bottom-up approach. While my own creations may have a quite different look and feel to them, I still would like to thank Eva Hild – and many other “intuitive” artists – for the inspiration they provide.

Acknowledgements I also thank the staff of the Jacobs Institute for Design Innovation at UC Berkeley for their help in fabricating many of the sculptural models presented.

References [1] D. Asimov, D. Lerner, Sudanese Möbius Band, SIGGRAPH '84 Electronic Theater, Issue 17, (1984). [2] E. Catmull and J. Clark, Recursively generated B-spline surfaces on arbitrary topological meshes. ComputerAided Design 10 (1978), pp 350-355. [3] Dyck surface. – http://www.mathcurve.com/surfaces/dyck/dyck.shtml [4] Google, Street View. – https://en.wikipedia.org/wiki/Google_Street_View [5] E. Hild, Homepage, (2017). – http://evahild.com/ [6] E. Hild, “Hollow” (2006). – http://evahild.com/?page_id=366 [7] E. Hild, “Interruption” (2002). – https://www.bukowskis.com/en/auctions/H043/67-an-eva-hild-stonewaresculpture-interruption-2002 [8] E. Hild, “Whole” (2007). – http://evahild.com/?page_id=365 [9] Microsoft, Kinect: Motion sensing input device. – https://en.wikipedia.org/wiki/Kinect [10] C. H. Séquin, Shape Representation: “Gabo curve.” (slide 36). – http://slideplayer.com/slide/9035370/ [11] C. H. Séquin, 2-Manifold Sculptures, Bridges Conf. Proc., pp 17-26, Baltimore, July 29-August 2, (2015). – http://people.eecs.berkeley.edu/~sequin/PAPERS/2015_Bridges_2manifolds.pdf [12] J. Smith, SLIDE design environment, (2003). – http://www.cs.berkeley.edu/~ug/slide/ [13] Type A machines. – https://www.typeamachines.com/ [14] Y. Wang, Robust Geometry Kernel and UI for Handling Non-orientable 2-Mainfolds (EECS-2016-65). – https://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-65.html

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Topological Design of Sculptured Surfaces Helaman Ferguson Supercomputing Research Center Bowie, MD

Alyn Rockwood Computer Science Dept. Arizona State University Tempe, AZ

Topology is primal geometry. Our design philosophy embodies this principle. We report on a new surface &sign perspective based on a “marked” polygon for each object. The marked polygon captures the topology of the object surface. We construct multiply periodic mappings from polygon to sculptured surface. The mappings arise naturally from the topology and other design considerations. Hence we give a single domain global parameteriration for surfaces with handles. Examples demonstrate the design of sculptured objects and their ntanufimture.

Jordan Coxt Dept. of Mech. Eng. Purdue University West Lafayette, I&J

boundaries. The burden of maintaining topological integrity falls to the designer. In these methods, there is no single parameter space, there are many sets of separate coordinate functions. Control of the topology may be simple for some shapes, but it is difficult for topologically complex ones. Some “solid” modelling systems check topology after the design stage, e.g. via the Euler-Poincark formulae mof89]. The check only determines when an invalid operation has occured. but does not participate in the design process. This paper describes a design philosophy that includes surface topology as an integral part. Our interest in this subject came from a desire to automate the sculpture of mathematical concepts such as in Figures 1.1 and 1.2 (see [Ferf39, Fer90, or Roc861).

CR Categories: 1.3.5 [Computer Graphics] Computational Geometry and Object Modeling - Geometric Algorithms; 1.3.6 [Computer Graphics] Methodologies and Techniques - Interactive Techniques. Additional Key Words and Phrases: Computer-aided design, sculptured surfaces, topology, marked polygon, automorphic functions, multiply periodic functions, boundary value problems. 1.

Introduction

Creating shape excites many artistic and scientific minds. It is important to areas as diverse as sculpture, mathematics, cartoon animation, molecular modeling, architecture, mechanical design. It is also one of the most difficult to automate because it demands a comprehensive toolset to handle such needs as topology, geometry, analysis, and manufacture. Traditional design methods (see [Far88, Hof891) define the surface of an object as a quiltwork of many patches; topological issues are treated superficially during the design stage. For example, the methods of Coons and BBzier and the B-rep method of solid modeling define an object as a collection of patches (or faces) that match at tcurrently at the Dept of Mech. Eng., Brigham Young University, Provo, UT. Permission IO copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage. the ACM copyright notice and the title of the publication and its date appear. and notice is given that copying is by permission of the Associati?m for Computing Machinery. To copy otherwise, or to repubhbh. requires a fee and/or specific permission.

Figure 1.1. Automatically sculptured “Umbilic Torus NC.” The continuous NC path was generated by a PeanoHilbert surface filling curve. The torus itself is defined on a single domain.

1” 1992

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ACM-0-X9791-479-1/92100710149

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The most fundamental mappings are C?, these preserve topolosY * Topology is primal geometry. Our design philosophy embodies this principle. This paper is organized as follows. Section 2 introduces the topological concepts mapping, domain, image, homeomorphism, immersion, embedding, and genus. Section 3 gives three examples from this design philosophy: a circle, a torus, and a genus three surface with many Bezier sub-patches over a single twelve-sided polygon. Section 4 reveals how to give a single polygon parameterization of a surface of arbitrary genus by constructing real analytic (smooth) multiply-periodic coordinate functions. Section 5 constructs an embedding of a double torus by solving boundary value problems. Figure 1.2. Milling the “Umbilic Torus NC” on a three axis milling machine, ball end mill tangent to the torus surface defined globally over a single parameuic patch. Our approach decomposes the surface of an object into a polygon, using a series of virtual surface cuts. The virtual cuts are identified with oriented markings that bound the polygon. Such a marked polygon encodes the topology of the object. A mapping from the two dimensional interior of this polygon into three dimensional space is then constructed. The domain of the mapping is a the interior of the polygon; the image of the mapping is the surface of the object. The surface is a single topologically consistent patch. As we investigated the problem, the value of the single polygon patch approach for analysis and manufacture was better appreciated. Even topologically simple objects benefit. The texture on the object in Figure 1 depends on a pattern that is continuous across the marked boundaries of the domain and drove the milling tool. In his famous Erlangen lecture, [Fir82], Felix Klein described geometries by their associated mappings, especially groups of mappings. Objects within the geometries are related and classified by the transformations between them. Rigid body transformations preserve congruence, projective transformations preserve similarity, Cl transformations preserve differentiability, C? preserves topological structure. For example, a cuboid with comers is C” to a sphere but not Cl; a sphere is not Cc to a torus.

2.

Surface Topology

Figuratively, two objects are homeomorphic if one object can be re-formed into the other by a continuous deformation that does not tear the object nor make it selfintersect. For a readable exposition see [Nas831, for rigorous definitions see [Ho&l]. We keep to an intuitive level. Consider a polygon in the plane. Think of the polygon as elastic material to be sewn up along the edges. Give each edge a direction and name each edge with a captial letter. We define, figuratively, a surface to be a ‘sewing up’ of this polygon in space by stitching together the edges with the same capital letters. Assume there are no leftover edges and that the sewing is direct, without twisting or knotting the material. There are two possibilities: the sewn material may self-intersect or not. If there are self-intersections we call the spatial result an immersion. If there are none we call the resulting surface S an embedding. The embeddings bound natural objects. There exists a set of directed virtual cuts on any surface S that induce a one-to-one correspondance from the surface with virtual cuts to its polygon with directed edges (see [Fir82, Sie88]). The polygon together with its directed and labelled edges is called the marked polygon of the surface. Note that a pairwise identification of two edges occurs from a single cut in the surface. This pair of edges is identified by the same letter, but differentiated by the superscripts. Figure 2.1 illustrates two cuts which convert a torus into a marked rectangle. It is also the topology used to create the “Umbilic Torus NC” of Figure 1.1.

Figure 2.1. The torus and its fundamental polygon with formal word ABA-lB-1.

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Move clockwise around the polygon and give the letter a “-1” superscript if its edge runs counter to the movement, otherwise the letter gets no superscript. The “word”that describes the toms in Figure 2.1 is ABA- lB-l. The word ABAB-l, for which the direction of the thirdedge differs from the torus, represents the Klein bottle and is always an immersion. Immersions and embedding are both possible for tori. A surface is orientable if an inside and an outside can be distinguished, i.e. bounds a natural object. The sphere and torus are orientable; the Klein bottle is not. The following theorem classifies all orientable surfaces: A closed, orientable surface

has a marked polygon word either of the form AA-l or AI BIA1-l B1-l ... AgBgAg-l

Bg-l (g> 0). The fmt case is a sphere. The second can be thought of as gluing g tori together. The number g is the genus of the object. It counts the number of “donut holes.” These are called handles. All objects of the same genus are topologically equivalen~ they can be continuously deformed from one to another. A corollary to the above theorem is A closed, orientable suflace (g>O) has a marked polygon with 4g sides. Many cutting schemes are possible for objects of g>l. The textbook cuts [Fu82, Sre88] usually emanate from a central point and emphasize symmetries as in Figure 6. We have devised non-standard cuts that are better for some purposes. Consider the double torus (“Borus”)shown in Figure 2.2. All cuts begin at point A inside on the bottom of the lower handle. The fnt cut divides the bottom loop into two sleeves. The next one slits the sleeve orthogonally to the fust along C. The next pair of cuts run from one sleeve comer to the other throughalong B and then D. The example is easily generalized to a genus g objecq g>2. The object is first deformed into an extended figure “8.” All cuts are made from the same point. The even cuts traverse through each handle and return to the point. The odd set of cuts start from the point and pass through the antipodal point in the handle without intersecting any other cuts. The next step in the marked polygon design process determines vector valued functions that define the surface. Consider the function fl P+S, for P a polygon in R2, surface S in R 3 and f: (u, v) +(x, y, z). Notice that pairwise related edges of the marked polygon must map so theirdirections coincide on the surface. The function ~ is vector-valued with coordinate functions fj: R2 + R, j=l,2,3, where x = ~l(u,v), y = genus

\2(u,v)

and z = ~3(u,v).

These coordinate

polygon as a single common domain.

functions

B-l

B

have the

Figure 2.2. Illustrationof the systemof virtualcuts on a doubletorusas it k openedup to a stop sign shapedmarked polygon with formal word ACBDA-lC-lB-lD-l.

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SIGGRAPH ‘92 Chicago, July 26-31, 1992

3. The Circle, The Torus, and the Embedding by Many Bkzier Patches. 3.1. The Circle. We now apply our design philosophy to the simplest posssible example which has all the essential elements of the general case -- a circle. The cut on a circle corresponds to a single point. The one dimensional polygon domain corresponds to a line segment. We take the line segment to be the interval (-l/2, l/21. Note that integer translations of this interval cover the real line. A polynomial mapping of the interval into the plane will image some kind of curve, which may cross itself (immersion) or be simple without self-intersection points (embedding). Two x and y coordinate polynomials which give a circle-like curve are x(t) = l-16r* and y(t)=f( 14*). But polynomials are not periodic and the curve has a comer. Its tear-drop shape is in Figure 3.1. The coordinate functions should have period 1 since the distance from cut to cut on the (-l/2,1/2] interval is 1.

context: 1) We think of an inner tube shape, mark it with two virtual cuts so that the shape corresponds to a marked square. 2) We pick some qualitative bivariate polynomial guesses to give an embedding. 3) We recognize the torus as doubly periodic and convert our guesses into doubly periodic coordinate functions with a bivariate sum over all pairs of integers, Z x Z. This leads to the following parametric equations for a torus with major radius a and minor radius b: (u,v) -> ((a+bcos(2zv)) cos(2nu). (a+bcos(*xv)) sin(*xu), b sin(2ltv) ). This maps the marked polygon (-l/2,1/2] x (-l/2,1/2] to the torus. This mapping is an embedding if a>b>o. Note that the coordinate functions in the vector are doubly periodic, i.e., invariant under replacing u by u+l or v by v+l. Furthermore these doubly periodic functions are real analytic (smooth) and bounded. 3.3. Traditional method. It is possible to formulate the coordinate functions using traditional design methods, e.g. Bezier cross plots [Far88], which are piecewise polynomial. The single twelve sided polygon for the genus three surface in Figure 3.3 organizes the hundred or so Bt5zier patches in a coherent way. In this case, the domain polygon is tessellated into four-sided regions which are associated with the domains of the polynomial patches; thus inducing a topological aspect into the design paradigm for conventional patch methods.

Figure 3.1. Left: A polynomial embedding of a circle. Note the comer at the cut point of the circle, because polynomials are not periodic. Right: A periodic embedding of a circle using the same polynomials in the periodizing transformation. How does one make numerically computable periodic functions out of polynomials? For any polynomial f(t), the rapidly convergent sum

F(f)= c f(f+k)e-(‘+QZ is periodic: F has period 1, F(z) = F(t+l), because the sum is over all integers Z. Replace f by x in the sum to get a coordinate function X(t) and f by y to get Y(f). Then X and Y both have period 1. Normalizing X and Y appropriately and taking eleven terms in the sum gives the classical periodic cosine and sine functions within eleven decimal places, cf. Figure 3.1. We recapitulate the features of this simple example: 1) we began with an idea of what we wanted to draw, a closed ‘circular’ curve which can be cut open at one point. 2) we picked some qualitative guesses for the coordinate functions which could give an embedding. 3) We recognized the essential periodic nature of the curve and made our guesses periodic to get a smooth curve. Recognizing a period for a circle is the same idea as recognizing the topological genus or system of virtual cuts for a surface. The periods are roughly speaking distances between cuts. 3.2. The Torus. The next simplest case is the torus. The torus is a Cartesian product of two circles, so we could replicate our design philosophy in this similar 152

Figure 3.3. Genus 3 object from many BCzier patches coherently organized over a marked 12-gon.

Computer Graphics, 26, 2, July 1992

4.

Defining

Periodic

the

Embedding

with

Multiply

Functions.

Natural sculptured objects are always bounded by a surface, are closed, and have genus. For a surface to have genus amounts to having multiple periods. A simple closed curve in space is periodic and has periodic coordimte functions. Closed surfaces in space can be defined by coordinate functions that are multiply periodic. A rich class of multiply periodic functions are available in the classical In this literature multiply literature [For51, Sie88]. periodic functions are usually called automo~hic functions because these functions are invarient under a group of automorphisms. A problem with the classical literature is that it tends to be preoccupied with complex analytic functions that do not give bounded embedding. We circumvent this obstruction by constructing enough real analytic functions which are bounded. These real analytic (smooth) functions are the analogs of the classical doubly periodic real analytic functions of variables u and v, the finite Fourier series in Sin(u), Sin(v), Cos(u), Cos(v) encountered for the torus in Section 3.2. Another feature is that the multiple periods of interest for surfaces of genus greater than one are not additive. The periods are elements of a multiplicative and non-commutative group of transformationsof the variables u and v. Multiply periodic functions can h of any arbitrary differentiability class. In what follows, we solve the problem of defining a smooth embedding from the marked polygon to the surface of an object. It allows the embedding to be defined smoothly across the boundary marks of the domain, and thus the virtual cuts of the object. We realize this markedpolygon in the hyperbolic disc with sides which are geodesics in the hyperbolic geometry. The multiple periods will be hyperbolic translations. All these hyperbolic translations form a multiplicative group, The hyperbolic translationstessellate the hyperbolic disc with copies of the markedpolygon. Traversing the polygon from one tesselant to a neighbor is equivalent to recentering the original polygon at another point of the tmundary. A “periodizing” transformation is defined on the tessellated disc that takes functions, say polynomials, over the fundamental domain and blends them at the marks, making them seamless, i.e. infinitely smooth across the virtual cuts. This was done for taking the tear drop to the smooth circle in Section 3.1. This “multiple periodizing” transformation involves a sum over all of the elements of the group of hyperbolic translations. A different group will be defined for each genus. 4.1. The Hyperbolic Disc. The natural geometry for the surfaces with handles which are of interest to us is a hyperbolic geometry. A surface with g handles can be cut with 2g cuts. The comesponding polygon has 4g sides and can be laid down in a hyperbolic geometry, mar83]. One model for this geometry, the Poincan5model, is given by imposing some geometric structure on the unit disc (z I ZZ*< 1, z E C } where C is the usual complex number field and z* is the complex conjugate. The points of this hyperbolic geometry are the points of the unit disc. The lines or geodesics of this hyperbolic geometry are given

by arcs in the unit disc of circles peqwmdicular to the disc boundary at their endpoints. Lengths of arc segments, angles between intersecting arcs, and area of polygons bounded by arcs are all &fincd, [Sie66]. One-to-one, onto, analytic mappings of the disc to itself that pr-e lengths and angles are given by linear fractional transformationsof the form z -> (az+b)/(b*z+a*) where a, b are complex numbers such that aa*-bb*=l. Among these are the hyperbolic rzanslationswe define for the periods of a sphere with finitely many handles. 4.2. The Single Marked Polygon Patch. As described in Section 2, it is possible to mark a surface with exactly g handles in such a way that when the surface can be ‘sewn up’ from the interior of a polygon of exactly 4g sides, where opposite sides are identified and the sewing is done without twisting. We specify a particular such polygon with 4g circular arc sides in the hyperbolic disc. This special set of 4g circular arcs comprises the boundary of the what we will call the canonical 4g -gon. It is symmetric about the origin. Ad@ining arcs are constrained to meet a vertex with interior angles of exactly n/2g radians. Hence, the sum of these 4g angles is exactly 2x. This ensures, by a theorem of Poincar6 ~ir82], that a group of h~rbolic translations, rg which we define Mow, of the 4g-gon exactly covem the hyperbolic disc. Each boundary circular arc is given by the inside arc of a chcle of radius sin(7t/4g) r= ‘e centered at integer multiples of x/2g and radius cg=rg cot(n/4g). For genus two, Q=J%”dc2=J%

The mdhs of the excribed vertices

circle containing

the eight

and the radius of the

of the poly on is 2-1/4

inscribed circle is (21b - 1)12. 4.3. The Group Define lg and mg by

rg

of

Multiple

Periods.

$tan;(l-:) and

21* m=—. g 1+~ Define the hyperbolic translation

z-m Tg:z+—,

1-

m~z

of intinite order, and the single rotation

L Rg:z+e2g

-1 “

z,

I 53

SIGGRAPH ‘92 Chicago, July 26-31. 1992

of order 4s. Note that Tg and Rg do not commute. All of the elements of rg are generated by the 4g fundamental hyperbolic translations Hg,~Rg-k Ts Rsk, kO.1.2 ,..., 4g- 1. 4.4. Enumeration of the Elements of rg .

Every period or element of rg is realized by a “word” in the symbols Hg& of the form Hg&tal . . . H~J,,,~, where the ui are non-negative integers. The length of this word is given by the non-negative integer q+...+u,. These words correspond to fractional linear transformations and there are relations among the words; some words are equal to other words that are spelled differently, thii gives an equivalence class for a given word. A shortest word in a set of equivalent words is called a geodesic word. It is possible to write down finite state automata which generate all geodesic words in increasing order IEps921. It is important to enumerate algorithmically the geodesic words, in order of increasing word length, to efftciently compute a large class of multiply periodic functions. For the case of the double torus and the group r2, there are eight generators. Set H2.t = HL The generators are I-lo. Hl, Hz, H3, Hq. H5. Hg and H7, and the particular word HoH~H~H~H~H~H~H~ is equivalent to the identity word of zero length. The numbers of words of length 0.1, 2, 3,4, 5 ,... are 1, 8, 56, 392. 2736, 19096 ,... respectively. The generating function for this double torus case is ( 1+2t+2r2+2$+r4)/( l-6t-6r2-6t3+r4), [Can84,92].

fjer = (1 - 1~1’ )j Re(zk) a

j L = (l- lzl’)’ Im(zk) for j=2,3 ,... and kO,1,2 ,... . Note that the sum is over rs ; thus the need for the algorithm above to generate geodesic elements of this group in order. These series converge rapidly; for visual accuracy on the double torus fewer than 1+8+56+392=457 terms suffice. See Figure 4.5 for an example basis function. f.

Figure 4.5. A multiply periodic basis function for the

group r2 plotted over the marked octagon in the hyperbolic disc.

Figure 4.6. An infinitely smooth double torus definable

on a single marked octagon. Figure 4.4. Images of the octagon for genus two in the

hyperbolic disc under hyperbolic translations by the geodesic words of the group r2. The center octagon is the identity surrounded by the alphabet of generators Ho, Hl, Hz, H3, H4, H5, H6, H7, then words HiHj, then HiHjHk, . . . . 4.5. Multiply Periodic Functions for the Sphere with g handles, g>l. A set of I’g multply

periodic basis functions is given by q,k(Z) = C fj,k(H * f) Her* Where

154

5. Defining the Embedding Boundary Value Problems.

by Solving

This approach to embedding homeomorphisms poses separate boundary value problems for the x, y, and z coordinate functions. The solution to these 2-dimensional boundary value problems models the specific coordinate function over the entire domain. The 3-tuples are formed by collecting the values of each coordinate function at specific parametric domain points. The parametric domain connects these 3-tuples to give the geometry of the de&d object.. Each boundary value problem is constructed by selecting an appropriate differential operator which models the geometric shape of the coordinate function and then

Computer Graphics, 26, 2. July 1992

applying boundary conditions derived from the appropriate coordinate values of the markings on the actual object. Similar methods have been used to model surface patches (see [ Blo90, Blo9Ocl ). Since the markings represent the boundaries of the domain, see Figure 6, it is appropriate to define the boundary conditions for the boundary value problem to be the coordinates at these markings. The development of the solutions to the boundary value problems can be accomplished through approximation techniques like finite element methods (see Lap82). For genus g > 1 objects the domain is 4g-sided. These 4g sides require that the solution approximation technique include non-rectangles to “mesh” the domain. For this next example of a double torus, domain composition methods were used (see [Cox9la, Cox9lb. Cox9lcl). These methods allow overlaps in the finite elements. Thus the 4g-sided domain can be meshed with overlaps. The selection of the differential operator in the double torus example is based on a traditional potential problem. Potential problems (i.e. heat transfer, pressure, etc.) are modeled using Laplace or Poisson operators. The differential operator choice should be dictated by the shape of the object and the periodicity of the coordinate functions. The selection of the differential operator is important and further work on this topic is forthcoming. For this example the Laplace operator generally produces the required smoothness in the coordinate function. Most existing finite element packages provide Laplace or Poisson operators.

internal constraints can be applied using penalty functions. Figure 7 shows the z-coordinate function for a specific embedding and decomposition of the double torus. For example, Figure 7 shows a cross shaped flat area above the coordinate function indicating the location of the internal constraints. These constraints force the solution to be zero along the corresponding curve on the surface. These constraints prevent self-intersections in the double torus as the octagonal domain is embedded in space. The ftrst subfigure of Figure 7 shows the double torus. The second subfigure shows the z-coordinate function as height over the octagonal domain. Both tigures are contoured and rendered with a height related color map A 3-tuple for the embedding is constructed from the x, y, and z coordinate values at the same domain point. All the 3-tuples of the embedding thus come from the three boundary value solutions representing the coordinate functions. If each of the coordinate functions are generated using equivalent finite element meshes in the domain, then the solutions at the nodes can be used directly as the 3-tuples that model the object. The connectivity of the elements in the domain will translate into appropriate connectivity in the object space. The double torus of Figure 7 is modeled by combining each of the elements in the three coordinate maps into elements covering the object. This is the image of the embedding from the octagonal domain into three

Boundary conditions specified along cuts

Extrema of torus Figure 6. Topological form of a double torus where the embedding is constructed by solving three boundary value problems. Boundary conditions are determined from the coordinates along the four cycles or virtual cuts as marked. Internal constraints to force an embedding arise from the outer and inner radii. Once the partial differential equation, the finite element mesh, and the complete set of boundary conditions are prescribed, there will exist a unique solution. However, it may not produce the desired coordinate function. Internal constraint conditions are added to the specification of the problem to achieve the desired results. These constraints specify some of the coordinate x,y, or z values in the interior of the solution surface. These can correspond to extrema of the object (i.e. the outer radius or inner radius of the double torus example.) Each of the coordinate boundary value problems will have different internal constraints. The

Figure 7. The double torus showing views of the zcoordinate function.

SIGGRAPH ’92 Chicago, July 26-31, 1992

The surface of the double torus of Figure 7 has anomalous bumps and ridges. Changes in the partial differential operator affect bumpiness. For example, the “stiffness” of the surface could be increased, the order or form of the differential operator can be modi.tkl, or a body force applied to smooth the resulting object. Further investigation into these methods is ongoing. Another problem is the distortion of the grid on the octagonal domain to the image grid on the double torus. More research on partial differential operators and more &velopment of methods of selecting internalconstraints is invited to conmol the modeling of coordinate functions. It is significant to note that the genus of the object is preserved throughout while them is great flexibility in the shape and locations of the handles. This gives a spatial richness needed for geometric design. 6.

Conclusion

Our approach is unique in its incorporation of topology as well as geometry into the design process. This leads to more comprehensive models for scientific visualization and manufacturingneeds and is well-suited to a variety of representations of complex objects which arise naturally from tie abstrtwtionprocess in scientitlc research. A basic outstanding problem is to provide a good set of sufficient conditions (on the coordinate functions) to give an The techniques embedding of a sphere with handles. introducedhere work in principle for any dimension. Future directions includes applying these ideas for volume domains and volume images with attached functions.

Acknowledgements We appreciate Sareddy Madhukar of Spatial Technologies, Boulder, CO for help on Figure 3.3 and Dave Smittley of the Super computing Research Center Bowie, MD for help with Figure 4.5. References llW90]

@lW89]

[can84]

[Can92]

Bloor, M. I. G., Wilson, M. J., “Using Partial Differential Equations To Generate Free-Form Surfaces”, Computer-Ai&d Design, vol 22, number 1, May 1990. Bkmr, M. I. G., Wilson, M. J., “Generating Nsided Patches with Partial Differential Equations”, Computer Graphics International, Wyvil, B. (Editor), Springer-Verlag 1989. Cannon, J., “The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups,n Geometrt”ae Dedicata, VOI16,1984,123-148.

Cannon, J,, Wagmich, P., “Growth Functions of Surface Groups”, to appear in Math. Annalen, 1992. [Cox91a] Cox, J., “Domain Composition Methods For Combining Geometric And Continuum Field Models”, PhD Thesis, Purdue University, West Lafayette, IN, December 1991.

156

[Cox91b] Cox, J., Anderson, D., C., “Single Model Formulations That Link Engineering Analysis With Geometric Modeling”, Product Modeling for Computer-Ai&d Design and Man#actura”ng, J, Turner, J. Pegna and M. Wozny (Editors), Elsiver Science Publishers B. V., North-Holland, New York 1991. [COX91C] Cox, J., Charlesworth, W., W., Anderson, D. c., “Domain Composition Methods For Associating Geometric Modeling With Finite Element Modeling”, Proceedings of the ACMISIGGRAPH Symposium on Solid Modeing Foundations and CADICAM Applications, Austin, TX, June 5-7,1991. [COX88] Cox, J., Ferguson, H. R. P., Kohkonen, K., “Single Domain Methods For Modeling Objects In The Round For Engineering And Manufacturing Applications”, Advances in Design Automation, ASME Design Automation Conference, Orlando, FL, Sept. 25-28, 1988. Epstein, D.B.A., J.W.Cannon, D.F.Holt, F.V.F. rEPs921 Levi, M. S.Paterson, W. P.Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992. Farin, G. E., Curves and sur$aces for Computer rFar88] Aided Geometric Design, Academic Press Inc., Boston, 1988. Ferguson, H., “Umbilic l-Fe&9] Torus NC,” SIGGRAPH89 Art Show, Boston, Mass., Leonardo, Journalof the Intematiomd Society for the Arts, Sciences and Technology, Supplemental Issue, August 1989, page 117, 122. Ferguson, H., Two Theorems, Two Sculptures, l-Fer90] Two Posters, American Mathematical Monthly, Volume 97, Number 7, August-September 1990, pages 589-610. Ferguson, H., “Algorithms for Scientific rFer92] Visualization,” Supmmmputing Research Center Tech. qort SRC-92-XXX,1992. Firby, P. and C. Gardiner, Surface Topology, Fir82] John Wylie & Sons, New York, 1982. Ford, L R,, Automorphic Functions, Chelsea, FO151] New York 1951. Hoffmann, C., Geometric Modeling, Morgen rHof90] Kaufman, New York, 1990. [~2] Lapidus,L., Finder,G. F., Munericai Solutions of Partial Differential Equatwns in Science and Engineering, John Wiley & Sons, New York, 1982. Martin, G. E., The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, New York, Second Rinting, 1986. Rockwood, Alyn, “The (5,3) Toroidal Knot,” B-1 SIGGRAPH86 Art Show Catalog~ugust,1986. Siegel, C.L., Topics in Complex Function [Sie88] Theory, vol.1: “Elliptic Functions and Uniformization Theory”; VO1.2,“Automorphic Functions and Abelian Integrals”, Wylie Interscience,New York 1988

Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture

2-Manifold Sculptures Carlo H. Séquin CS Division, University of California, Berkeley E-mail: [email protected] Abstract Many abstract geometrical sculptures have the shape of a (thickened) 2D surface embedded in 3D space. A fundamental theorem about such surfaces states that their topology is captured with just three parameters: orientability, genus, and number of borders. When trying to apply this classification to interesting sculptures of famous artists depicted on the Web, a first non-trivial task is to establish an unambiguous 3D model based on which the three topological parameters can be determined. This paper describes some successful, practical approaches and gives the results for sculptures by M. Bill, C. Perry, E. Hild, and others. It also discusses the surprising topological equivalences that arise from such an analysis.

1. Introduction Many abstract geometrical sculptures have the shape of a (thickened) 2D surface embedded in 3D space. Examples of such 2-manifold sculptures range from Max Bill’s Endless Ribbon [1] which is just a Möbius band cut from stone (Fig.1a), to Brent Collins’ wood sculpture Heptoroid [3], which is a toroidal composition of seven 4th-order saddles into a surface of genus 22 (Fig.1d).

(a) (b) (c) (d) Figure 1: 2-manifold sculptures: (a) “Endless Ribbon,” Max Bill [1], (b) “Dual Universe,” Charles Perry [13], (c) “Loop_785,” Eva Hild [10], (d) “Heptoroid,” Brent Collins [3]. To gain a deeper understanding of these pieces of art, we will analyze the topology of these sculptures. We make use of the Surface Classification Theorem [8], which states that all 2-manifolds (thin surfaces with borders) can be uniquely classified by their orientability (double-sided or single-sided), by the number b of their borders (loops of 1-dimensional rim lines), and their genus g (the number of independent closedloop cuts that can be made on such a surface, leaving all its pieces still connected to one another). This analysis partitions the world of all 2-manifold sculptures into discrete families. For members of the same family that agree in all three of the above topological parameters, there exists a homeomorphism, i.e. a bijective mapping that smoothly transforms the surface of one member into the surface of the other. This reveals some surprising relationships between sculptures that may look quite different.

17

S´equin

2. Understanding Complex Surface Sculptures Many of the sculptures that I want to analyze I know only from images in books or on the Web. A first task then is to construct a mental 3D image of the sculpture. Often the available images do not show the sculpture from all sides. Sometimes recognizing an inherent symmetry allows one to infer what is going on on the invisible “back side.” In other cases no such obvious symmetries exist. Moreover the sculpture may have a generally recognized “good front side,” and all the pictures were taken from that direction. In the case of Eva Hild’s patinated bronze sculpture Hollow (Fig.10c) located on the Varberg Campus in Sweeden, I resorted to Google Maps and its associated Street View to virtually drive by the sculpture with the goal of obtaining additional images. However, the road from which these images were taken, passed only “in front” of this sculpture. On the other hand, luckily, this sculpture is positioned in front of a café with large plateglass windows. By choosing my street-view position carefully, I could see a faint reflection of the backside of this sculpture in those windows. Some enhancements done with Photoshop gave a clear enough image to let me extract the geometry of the back side of this sculpture. It can still be a puzzling experience to look at some of the complex sculpture images of artists celebrated in Figure 1. One may try to trace out the sharp rim lines to see how long it takes to come back to the starting point. One may also make a conscious effort to see whether one can move along the surface from “one side” to “the other” without ever stepping across any rim-line. I found that a good way of gaining an understanding of such sculptures is to try to make rough, physical models from paper, clay, pipe-cleaners, or styrofoam. These models then serve as the basis for developing a parameterized computer graphics model. Paper-strip Model of Charles Perry’s Tetra Sculpture For the analysis of Perry’s Tetra sculpture [13] [14] I had to rely on just a small set of photographs (Fig.2). As a first step I tried to identify corresponding features in various views. In particular, I labeled the six “branches” with (red) numbers 0 through 5. I then discovered that exactly three such ribbon-branches join together in each of four “vertices”, which I then labeled with (green) letters A through D. Now the name Tetra suddenly made a lot of sense.

(a) (b) (c) (d) Figure 2: Charles Perry’s “Tetra” sculpture [13] seen from four different directions. To make a rough model of this sculpture, I represented the six edges with narrow paper strips and glued together three of them at every vertex, twisting each strip in accordance with the sculpture. The first such model looked rather messy (Fig.3a), but it still was good enough to determine the sidedness of the surface and the number of its border loops. Subsequent, cleaned-up paper models depict more clearly an untwisted tetrahedral frame (Fig.3b) and a better, tailor-made model of the Perry sculpture (Fig.3c). 18

2-Manifold Sculptures

(a) (b) (c) (d) Figure 3: Paper models: (a) initial model of Perry’s “Tetra;” (b) untwisted tetrahedral ribbon frame; (c) capturing the twistedness of Perry’s “Tetra;” (d) initial model of Perry’s “Continuum”[13]. A more complicated sculpture by Charles Perry is Continuum, located in front of the Aerospace Museum in Washington D.C. [13]. Several images were collected from the Web showing this sculpture from different directions (Fig.4). Many of these images had to be processed by Photoshop to enhance contrast within the gray-level domain of the actual sculpture in order to bring out details concerning the curvature of different areas of this sculpture and to reveal the way in which some of the ribbons were twisting (Fig.4b,c). A view crucial for the understanding of this structure was Figure 4d; it clearly showed that this sculpture has an axis with 6-fold D3 symmetry. By combining this insight with the view in Figure 4a, we see that this sculpture actually has 12-fold symmetry of type D3d (Conway notation: 2*3), since it also has rotational glide symmetry. Again a paper model was made (Fig.3d) to gain clarity how the connecting ribbons twist through space. It turns out, this is a single-sided, non-orientable surface of genus 6 with a single contiguous rim curve. Some curve segments comprising 1/6 of the whole geometry are highlighted in blue and green in the paper model.

(a) (b) (c) (d) Figure 4: Charles Perry’s “Continuum” [13] seen from 4 different directions. A Parameterized Topological Computer Model Once the connectivity and the amount of twisting has been figured out, it is not too difficult to create an approximate computer model that captures the topology of this surface, but which also makes it possible to introduce some variations and to study their impact on the classification of the surface. The model shown in Figure 5 is based on four hexagonal plates placed at four tetrahedral corners of a reference cube. They lie perpendicular to the space diagonals of this cube and have been rotated around this axis so that three hexagon sides are perpendicular to the face diagonals along which we want to form connections to other junction plates. The ribbons that connect to three of the edges of these hexagonal plates are modeled as sweeps along Bézier curves that join the hex-plates with tangent continuity. The basic, untwisted configuration of the complete tetrahedral frame is shown in Figure 5a. To make it possible to study the effects of different amounts of twisting of the ribbons, the model allows the twist of the ribbons to be

19

S´equin

adjusted in increments of 180°. To emulate Perry’s Tetra sculpture, the three pairs of opposite edges have to be given twist values of +360°, 0 , and −360°, respectively (Fig. 5b).

(a) (b) (c) (d) Figure 5: Virtual model of Perry’s “Tetra” sculpture: (a) untwisted tetrahedral frame, (b) 4 twisted branches as they occur in Perry’s “Tetra,” (c) two branches twisted through −180°, leading to a single-sided surface, (d) two ribbons with +180° twist and two branches twisted through −720°. It is interesting to note that, in spite of all the twisting, Perry’s Tetra is a double-sided (orientable) surface. But in our generator we can produce a single sided surface by twisting one pair of branches through only ±180° (Fig.5c). We can also increase the twist in some branches. In Figure 5d one pair of branches is twisting through −720°; but this is still a single-sided surface because of the +180° twist in the two branches crossing the z-axis. Models Based on Rims and Tunnels and Connecting Surfaces Eva Hild’s ceramic creations [10] cannot be easily decomposed into a collection of connected ribbons. Some of her sculptures are characterized by bulbous outgrowths (Fig.6a), others by giant funnels (Fig.6b) and saddles, or by tunnels in the shape of single-shell hyperboloids (Fig.6c). For all her sculptures, key defining features are the free-flowing boundary curves that border the smooth surfaces. Some of the earlier Hild structures have mostly circular rim lines. In some of her sculptures these rims define the borders of “funnels” (Fig.6b); in others they open inwards like in an “oculus” or inverted funnel (Fig.6a). We need different paradigms to produce parameterized computer descriptions of such shapes.

Figure 6: Eva Hild’s ceramic creations; (a) bulbous surface with oculus-like openings; (b) various nested funnels; (c) hyperboloid tunnels; (d) a complicated combination of features. Since the borders of these surfaces constitute crucial, defining features, they should be a central element to be captured in the computer description. But it is not just the geometry of the rim curve that matters, but also the tangential direction under which the surface takes off from this rim. Thus I found that an appropriate modeling primitive is a ribbon that can be controlled in shape (by default I use cubic B-splines) as well as in its orientation in space (the local azimuth angle of the cross-section), so that we can create funnel-like shapes (Fig.6b) as well as bulbs with a hole at the apex (Fig.6a).

20

2-Manifold Sculptures

In addition, we need a way to place internal tunnels that do not involve any boundary curves. A short cylinder with some controls of the tangent directions at its two open ends readily serves that purpose. If this cylinder is given a large diameter and inward-sloping surface seams, it can also serve to define the girth of a large bulb or worm-like protrusion. As an example, Figure 7 shows how these defining elements have been used in the Berkeley SLIDE environment [22] to re-create a simple ceramic Hild sculpture called Interruption (Fig.7a). I placed a (blue) funnel at the bottom, draped a (red) free-form B-spline rim over the top, and inserted a (green) tunnel to define a corresponding opening (Fig.7b). Next, I specified a mesh of quads connecting the control vertices of the three defining elements. This is a rather tedious and error-prone task in any CAD environment that does not allow to do this with some interactive graphics technique, and which requires the user to type in sequences of vertex identifiers, while the display does not even show any identifying labels. This is the weak spot in the current approach where, a good procedural front end could make a big difference!

(a) (b) (c) (d) Figure 7: (a) Eva Hild’s sculpture “Interruption”; (b) placing some defining elements; (c) creating a subdivision surface connecting the rims; (d) a first 3D-printed maquette. Proper placement of the three defining elements and interactive fine-tuning of their locations, sizes, and/or shapes allowed me to capture the topology and rough form of this sculpture. However, the geometric resemblance between the original and my model is definitely lacking. In particular, the left-ward pointing tunnel entrance in the lower half of Figure 7a has a rather elliptical shape in my model, while in Hild’s ceramic it is much more circular. To obtain a better emulation of the original geometry, we need additional defining features on this surface. Work along this route is in progress. Capturing the Geometry of Some Perry Sculptures The above rim-based modeling approach also works well to emulate Perry’s ribbon sculptures. The topological model developed above (Fig.5) does not capture appropriately the actual geometry of Perry’s Tetra sculpture. In order to obtain a truer model of the geometry, I started out with four rings lying on the faces of a regular tetrahedron. By adjusting the tilt angle of the circles and shifting their positions in the zdirection, I ended up with the interlinked position shown in Figure 8a. Now, using the 12 vertices of the duo-decimal control polygons of the rings, I could define a coarse polyhedral surface that could then be used as the starting shape for a Catmull-Clark subdivision surface Error! Reference source not found.. To match the geometry of the actual sculpture even more closely, I needed to control the principal curvatures of the ribbons and junction areas in both directions. Thus, I also introduced control points along the medial center lines of the six ribbons in this sculpture. These control points were displayed with small colored balls (Fig.8b), and their positions could be fine-tuned with some extra sliders. To reduce the amount of tedious work that had to be done, I made use of the symmetry inherent in this sculpture and defined only the control polyhedron for the minimal amount of unique geometry (Fig.8a). The complete control polyhedron was then generated by instantiating four suitably rotated copies of this master geometry. I generated two surfaces separately for both orientations of all the quad facets and rendered them with different colors, so that it was easy to distinguish the front- and the back-sides of this 21

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surface (Fig.8b). By interactively adjusting the positions of the six balls, while the surface was displayed with three levels of recursive subdivision, good visual agreement could be obtained with the curvature observed in the various photographs of this sculpture (Fig.2). Finally I picked a suitable orientation and view point for the model for a side-by-side comparison of model and sculpture (Fig.8c,d).

(a) (b) (c) (d) Figure 8: (a) Placing four circles in a “Tetra” configuration, connected with a sparse set of polygons; (b) turning this set of polygons into a subdivision surface; (c) adjusting the ring placement to match Perry’s Tetra sculpture; (d) Perry’s “Tetra” sculpture with circular border curves highlighted.

3. Analysis and Classification of 2-Manifolds Once we have a clear and unambiguous 3D model of a sculpture, we can proceed with the analysis of its topology. As mentioned at the beginning, we need to determine three topological parameters: its orientability (number of sides σ), its genus g, and the number of borders b. The focus of this paper is on surfaces embedded in Euclidean 3-space (R3), i.e., surfaces without any self-intersections. Two-sided closed surfaces are easy to deal with; for each handle-body defined by such a surface of genus g, there is only one topological variant, since there is only one type of torus without selfintersections [9],[18]. Thus, no matter how different they may look, for any pair of handle-bodies of the same genus g there is always a homeomorphism (a smooth one-on-one mapping) from the surface of one of the bodies to the other one [15]. On the other hand, closed single-sided surfaces cannot be embedded in R3. To avoid self-intersections an artist may introduce one or more punctures, i.e., openings in the surface through which another surfacebranch may pass. (For the classical “inverted sock” Klein bottle [19] one could cut an opening large enough to let the narrowed-down end of the tube pass sideways into the larger end without any intersections). From this it follows that any embedded non-orientable surface must have some border that cannot be capped off with a topological disk without intersecting the surface in some way. Among the three parameters that we need to establish, the simplest one is the number of borders: We just start moving along one of the rims until we come back to the starting point, and we count the number of individual loops that we find in this way, until all border segments have been traced out. To determine the orientability or “sidedness” of the surface, we conceptually start painting the surface from any single spot and spread the painted area without ever moving over any of the sharp border lines. If the whole surface gets painted eventually, it is single-sided and non-orientable; if only half of it gets painted, it is double-sided and thus orientable. If less than half of it gets painted in this manner, then it is not really a (thickened) 2-manifold, but a more complicated geometrical object in which three or more surface regions are attached to the same 1-dimensional “spine”, or a volumetric object such as a polyhedron. In many cases, though, it is sufficient just to look for one path that leads from some surface point to its antipodal point on the “other side” without stepping over any border, to determine that this is a single-sided, nonorientable surface.

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2-Manifold Sculptures

Determining the genus of a surface is by far the most difficult of the three tasks. But rather than trying to find directly the genus of a given surface, one may determine instead its Euler Characteristic χ. To find χ, one covers the surface with a mesh of vertices, edges, and polygonal facets. The Euler Characteristic is then given by the expression χ = V – E + F. This is an equally good invariant to classify all 2-manifolds. However, given one of the above complex pieces of art, it may be tedious and error prone to try to sketch a mesh that covers the complete 2-manifold represented by the sculpture. There is a different approach that often proves to be much simpler. The Euler Characteristic of a set of n separate disk is χ = n. Whenever we form a ribbon-shaped bridge-connection between the rims of two disks, the Euler Characteristic is reduced by 1, because the added ribbon has two edges, but only one face. This applies whenever we make an additional connection to an independent disk or between disks that are already part of our connected network. Thus we can apply this process in reverse and ask how many independent cuts c we have to make in order to “open up” the given surface until it assumes the topology of a single disk. The Euler Characteristic is then 1 – c. Thus the Euler Characteristic of a closed ribbon (Fig.1a), whether twisted or not, is zero, and for a tetrahedral frame (Fig.3b) χ = –2. With the Euler Characteristic established, the genus of a surface is then derived with the expressions: g=2–χ–b

for non-orientable surfaces, and

g = (2 – χ – b)/2

for double-sided surfaces.

It is a non-trivial task to demonstrate that these definitions are equivalent to the number of independent closed-loop cuts that can be made without cutting a surface into disconnected pieces. Figures 9 through 13 show the results of applying the above analysis to various 2-manifold sculptures. For the sculptures depicted and discussed above: these are the results: Perry’s Tetra (Fig.2):

σ = 2, b = 4, χ = –2, g = 0;

Perry’s Continuum (Fig.4):

σ = 1, b = 1, χ = –5, g = 6;

Hild’s Interruption (Fig.7):

σ = 2, b = 2, χ = –2, g = 1;

Here are some more examples: in pictures with appended analysis of the topological parameters:

(a) “Shell” Brent Collins: σ = 1, b = 2, χ = –1, g = 1,

(b) “Dual Universe” (c) “Antichron” Charles Perry: B. Grossman: σ = 2, b = 4, σ = 1, b = 2, χ = –2, g = 0, χ = –4, g = 4, Figure 9: Analysis of several sculptures.

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(d) “Totem 1” Carlo Séquin: σ = 2, b = 2, χ = –8, g = 4,

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(a) “Heptoroid” Brent Collins: σ = 1, b = 1, χ = –21, g = 22,

(b) “Arabesque-XLV” (c) “Hollow” Robert Longhurst: Eva Hild: σ = 2, b = 2, σ = 2, b = 1, χ = 0, g = 0, χ = –3, g = 2, Figure 10: Analysis of a few more sculptures.

(d) “Costa in Cube” Carlo Séquin: σ = 2, b = 3, χ = –5, g = 2,

For 2-manifolds that are not primarily understandable as networks of ribbons (as were the sculptures in Figure 9), and which are dominated by large contiguous surface areas (Figs. 7, 10c, 10d), it is often easier to derive the genus directly, rather than first determining the Euler characteristic. By definition, the genus does not change if border circuits are closed by inserting topological disks. If one can seal off all rims without obtaining any intersections, one has produced the orientable surface of a handle-body, and its genus can then readily be determined by counting the number of handles and/or tunnels. Specifically in Figure 7a one can close off the red and blue openings, and the resulting shape can then readily be recognized as a simple torus. This torus would open into an annulus if cut along the green path. Similarly, for the Costa surface shown in Figure 10d, one can easily close off the 3-segment top and bottom openings. To close off the remaining equatorial rim, consisting of six segments, one should add a large bulging balloon that does not interfere with any other parts of the sculpture. If this balloon is bulging out in the bottom/backwards direction, then the resulting shape will have three tunnel entrances near the top/front, which internally join in a Y-shaped valence-3 junction. This is the hallmark of a genus-2 handle-body.

4. Surprising Topological Equivalencies The surprising topological equivalence between Max Bill’s Tripartite Unity [2] (Fig.11a) and a sketch (Fig.11b) appearing in George Francis’ Topological Picture Book [7] has already been discussed [12][20]. Both are single-sided surfaces of genus 3 (Dyck’s surface [6]) with a single puncture. Such a surface can be formed in a generic way by grafting a Sudanese Mӧbius band [11] onto a torus (Fig.11c).

(a) (b) (c) (d) (e) Figure 11: Surfaces with the same classification: σ=1, b=1, χ = –2, g=3: (a) Bill’s Tripartite Unity [2], (b) sketch by George Francis [7], (c) connected sum of a torus and a Sudanese Mӧbius Band [11], (d) monkey-saddle toroid [17], (e) suitably twisted Tetra-frame (Fig.5).

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2-Manifold Sculptures

In a completely different approach, Sculpture Generator I [17] can also be used to create a surface with the same topological parameters (Fig.11d). An odd number of “stories” (i.e., a saddle-tunnel combination; here there is just one.) has to be used to obtain a single-sided surface. A minimal amount of twist that closes the toroid smoothly will typically lead to a single border curve of maximal length; the branch factor for the saddles is set so as to obtain the desired Euler Characteristic. Alternatively, Perry’s Tetra can also be turned into the same topological object if the twisting of the ribbons is chosen in the right combination: exactly three of them need to make an odd number of half-flips (Fig.11e). Next I want to explore what surfaces are single-sided and single-bordered but have a genus g = 4. The Minimal Trefoil (Fig.12a) belongs into this family; and again the Sculpture Generator I can be employed to create a corresponding surface with just a single saddle (Fig.12b). This type of surface is topologically equivalent to a connected sum of 2 Klein bottles; we can create an embedding of this surface by letting both cross handles pass through the one opening created by the single border (Fig.12c). Rinus Roelofs has created an elegant embedding of this topology class by cutting open a torus and appropriately twisting the remaining connecting ribbons so that a single border curve is formed (Fig.12d).

(a) (b) (c) (d) Figure 12: Surfaces with classification: σ=1, b=1, χ = –3, g=4: (a) Séquin’s Minimal Trefoil [17]; (b) 4th-order-saddle toroid [17]; (c) connected sum of 2 Klein bottles; (d) Roelofs’ Mӧbius Torus [16]. Now let’s look at a family of sculptures representing an orientable surface. All of the objects in Figure 13 are topologically equivalent to a sphere with four punctures! Perry’s Tetra and also his sculpture D2d [13] belong into this family. The bamboo basket [21] by Shoryu (Fig.13b) is more difficult to analyze when just looking at pictures. When looking at the physical 3D basket, it becomes readily apparent, because of the symmetry in this object, that the result of closing the four openings will be a severely deformed sphere. Again, Sculpture Generator 1 can be used to tailor-make an object with the right topology; now we use two stories (an even number) to obtain an orientable surface, and the two saddles need to be of the ordinary biped type to result in the right genus (Fig.13c). The simplest version of a surface of this type is a disk with 3 holes; but the 3 struts between each pair of holes can be looped through the opposite hole to make a more intriguing structure (Fig.13d).

(a) (b) (c) (d) Figure 13: Surfaces with the same classification: σ=2, b=4, χ = –2, g=0: (a) Perry’s “D2d” [13]; (b) bamboo basket by Shoryu [21]; (c) 2-story twisted toroid; (d) 3-hole button − entangled version.

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5. Conclusions Trying to figure out the topological classification of many intriguing 2-manifold sculptures forced me to take a much more focused look at whatever images I could get access to. A major task was to form a mental 3D image of these shapes and then capture these geometries in some parameterized description that would allow me to fine-tune them to get as close a match as possible − yet, at the same time, to maintain the freedom to change the sculpture in some significant way. In the case of Hild’s geometries, this may be mostly a change of proportions between different funnels and tunnels, or a modification in the voluptuousness by which a dominant border would flare out. In the more ribbon-like surfaces created by Perry, I could modify the twist of individual ribbons, an operation that may affect the number of borders as well as the orientability of the surface. The topological classification that this paper is focused on, is the coarsest way in which 2-manifolds can be analyzed. By this measure Figures 5a and 5b both belong into the same topological family. One way to distinguish them would be to analyze the interlinking of the various border loops; 5a shows no interlinking at all, while in 5b four of the possible six border pairs are linked. In general we find a strong tendency to exhibit linked border curves in Perry’s sculptures. On the other hand, Hild’s creations may have very long and complicated borders, but they are rarely interlinked. Focusing on the border loops, another interesting question to ask is whether, in any particular sculpture with more than one border, all of them are “equivalent” in the sense that for each pair of borders there exists a homeomorphism that maps the surface back onto itself and also maps the selected two borders onto one another [5]. These issues are ongoing work and will be the subject of a future publication.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

M. Bill, Endless Ribbon (1953-56). – Middelheim Open Air Museum for Sculpture, Antwerp. M. Bill, Tripartite Unity (1948-49). – http://www.lacma.org/beyondgeometry/artworks1.html B. Collins, Heptoroid (1998). – http://bridgesmathart.org/bcollins/gallery6.html E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. ComputerAided Design 10 (1978), pp 350-355. W. Cavendish and J. H. Conway, Symmetrically Bordered Surfaces. Math Monthly 117 (2010). W. Dyck, Beiträge zur Analysis situs. I, Math. Ann. 37 (1888) no.2, pp 457–512. G. K. Francis, A Topological Picturebook. Springer, New York, 1987, p 101, fig. 33. G. K. Francis and J. R. Weeks, Conway’s zip proof, Amer. Math. Monthly 106 (1999) pp 393–399. J. Hass and J. Hughes, Immersions of Surfaces in 3-Manifolds. Topology 24 (1985) no.1, pp 97-112. E. Hild, Homepage. – http://evahild.com/# D. Lerner and D. Asimov, The Sudanese Mobius Band. SIGGRAPH Electronic Theatre, 1984. T. Marar, Projective planes and Tripartite Unity. – http://www.mi.sanu.ac.rs/vismath/marar/ton_marar.html C. Perry, Topological sculpture – http://www.charlesperry.com/sculpture/style/topological/ C. Perry, Sculpture List. – http://www.charlesperry.com/sculpture/list/ U. Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985) pp 421–434. R. Roelofs, Mobius Torus. – http://www.rinusroelofs.nl/rhinoceros/rhinoceros-m13.html C. H. Séquin, Sculpture Generator I. – http://www.cs.berkeley.edu/~sequin/GEN/Sculpture%20Generator/bin/ C. H. Séquin, Tori Story. Bridges Conf. Proc., pp 121-130, Coimbra, Portugal, July 27-31, 2011. C. H. Séquin, From Moebius Bands to Klein-Knottles. Bridges Conf. Proc., pp 93-102, Towson, July 2012. C. H. Séquin, Cross-Caps – Boy Caps – Boy Cups. Bridges Conf. Proc., pp 207-216, Enschede, the Netherlands, July 26-31, 2013. H. Shoryu, Galaxy; Bamboo Basket (2001). Museum of Asian Art, San Francisco (Exhibit 2014). J. Smith, SLIDE design environment. (2003). – http://www.cs.berkeley.edu/~ug/slide/

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Nexus Netw J (2015) 17:981–1005 DOI 10.1007/s00004-015-0261-9 DIDACTICS

Schematizing Basic Design in Ilhan Koman’s ‘‘Embryonic’’ Approach ¨ zkar2 Benay Gu¨rsoy1 • Mine O

Published online: 1 August 2015  Kim Williams Books, Turin 2015

Abstract With an outwardly professed interest in mathematics, Ilhan Koman has produced his nonfigurative, abstract sculptures mostly as various series of forms. The difference and similarity between the works in any series is achieved through repetitions and variations of certain relations between parts. This corresponds to creating a relational system and it requires having control over the underlying principles of that system much as basic design students are encouraged to do. In order to substantiate the implications of work such as Koman’s in learning about design thinking, we first delineate the mathematical concepts in Koman’s ‘‘embryonic’’ approach through visual schemas. These visual schemas are then supplied to first-year design students as guides and design constraints as well as tools to formalize their design thinking. We observe that introducing Koman’s schemas to students helps them grasp how they establish relations between parts in their own design processes. Keywords Ilhan Koman  Design computing  Basic Design  Shape grammars  Visual schemas  Architectural education  Design algebras

& Benay Gu¨rsoy [email protected] ¨ zkar Mine O [email protected] 1

Mimarlık Faku¨ltesi, Istanbul Bilgi Universitesi, KD-210, Santral Kampu¨su¨, Eyu¨p, 34060 Istanbul, Turkey

2

¨ niversitesi, Taksim, Mimarlık Bo¨lu¨mu¨ Tas¸ kıs¸ la, Mimarlık Faku¨ltesi, Istanbul Teknik U 34437 Istanbul, Turkey

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¨ zkar B. Gu¨rsoy, M. O

Unity is a criterion habitually sought in design. Learning to unify various design input are among the main objectives of a modernist architectural education. A basic design curriculum, the core of the foundational year in many architecture schools, serves towards this objective. It engages students with various abstract tasks that are detached from real life architectural problems. This way, students find the chance to directly reflect on their reasoning over a simpler and formal vocabulary. Although liberated from the complexity of real world problems, the problem of relating abstract shapes is itself complex under the microscope and so provides a rich material for learning about multiple aspects and the relational nature of design. Unity in a basic design work is achieved through establishing consistent relations between ‘‘the elements of a composition in an orderly manner’’ guided by concepts such as ‘‘balance, contrast, harmony, repetition, dominance and hierarchy’’ (Aytac¸Dural 2012: 117–118). The elements are often two- and three-dimensional shapes, and their features such as size, orientation, position, texture, and color. Using the similarities and differences between the elements to establish relations, students are expected to create perceivable repetitions and their variations in unity. Creating repetitions and variations constitutes not all but some of the design knowledge conveyed in foundation studios—significantly, a computable part. Previous studies have already dwelt on the possibility of inquiry into design through its visually computational aspects (Stiny 2006) and of ‘‘learn[ing] design by visual ¨ zkar 2007). A designer’s shape rules, as instructions computation’’ (Knight 1999; O in visual computations—i.e., designs—are tools not only to understand, communicate, and control the relations between shapes but also to learn the know-how. Visual schemas, as more generalized versions of shape rules (Stiny 2011), reflect overarching relations and are even more suitable than rules to the level of beginning ¨ zkar 2011). design students’ formal studio talk (O This paper provides a case for guiding entry-level design students with visual schemas that particularly formalize repetitions and variations. The visual schemas that are introduced to the students are developed based on the nonfigurative mathematical art of the Turkish sculptor Ilhan Koman (1921–1986) whose work is characteristically very repetitive. We observe that introducing Koman’s schemas to students consequently helps them grasp how they establish relations between parts in their own design processes. Koman’s work provides ideal examples of compositions where elements come together with perceivable repetitions and their variations. With an outwardly professed interest in mathematics and mathematical relations of forms, Ilhan Koman has produced his art mostly as various series of forms. The difference and similarity between the works in any series is achieved through repetitions and variations of certain relations between parts. In (Koman and Ribeyrolles 1979: 1) this approach is called ‘‘embryonic in the sense that each series embodies new ideas and the need of different know-how that could be exploited in making further works of the same type’’. Koman’s ‘‘embryonic’’ approach in which he develops a relational system and creates variations in it by controlling the underlying principles of that system is similar to what basic design students are encouraged to do in learning to achieve unity in design.

Schematizing Basic Design in Ilhan Koman’s ‘‘Embryonic’’…

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Below, we first investigate Koman’s Infinity-1 series using visual schemas and shape rules. Later, in order to substantiate the implications of such works in learning about design thinking and achieving unity, we present a study realized with basic design students where the underlying visual schemas of Infinity-1 series serve as guides and design constraints as well as tools to formalize students’ design thinking. The analysis uncovers the types and the extent of variations in shape relations. Reflecting on the analysis, our text presents the schemas of the case, a pedagogical proposal, students’ processes and a discussion of the outcomes.

Repetition, Variation and Computability in Koman’s Work There is a current and growing interest in Koman’s work in the field of computational design, where his approach is deemed digitally and parametrically reproducible (Bes¸ liog˘lu 2011). There is also an interest in identifying the mathematical concepts behind the creation of form in Koman’s sculptures (Akgu¨n et al. 2006, 2007). Although within the same context of computational design, we present a pedagogical incentive to discuss Koman’s work and focus on its systematic shape relations towards a basic design methodology. Polyhedra and Derivatives, Hyperforms, 3D Moebius, Pi Series and Infinity-1 are some of the sculpture series conceived and produced by Koman. There is often an apparent formal unity between the works in the same series and it is achieved through the repetition of varying formal relations. We claim that what lies beneath this ‘‘embryonic’’ approach is the use of shape rules and their unifying visual schemas in the generation of works in a series. We illustrate this through a comprehensive analysis of the Infinity-1 series.

Schematizing Infinity-1 Koman divides either a single sheet of a material such as aluminum, titanium, and wood, or a prismatic block of a material such as wood into connected parts to create the sculptural works of the Infinity-1 series (Figs. 1, 2, 3). Although repetitive, the connected parts display gradual transformations in space such as rotation or translation, and, hence, produce a variation while still maintaining an unfragmented whole. Repetition of certain relations is perceivable at first sight both in the works themselves and across the body of works in the series. We identify these perceivable shape relations across all the works in the Infinity-1 series as a set of rules that we have retrospectively categorized under two schemas: X x! prtð xÞ ðSchema1Þ and x ! x þ tðxÞ

ðSchema2Þ

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¨ zkar B. Gu¨rsoy, M. O

Fig. 1 Rule 1 and works from Koman’s Infinity-1 series. From top to bottom, left to right: ‘‘Whirlpool’’, ‘‘Untitled 2’’, ‘‘To Infinity…’’, ‘‘Untitled 1’’, ‘‘Untitled 3’’. Photos: Yıldırım Arıcı

Schematizing Basic Design in Ilhan Koman’s ‘‘Embryonic’’…

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Fig. 2 Rule 2 and ‘‘Shell’’ from Koman’s Infinity-1 series. Photo: Yıldırım Arıcı

Fig. 3 Rule 3 and works from Infinity-1 series: ‘‘Untitled 4’’, ‘‘Untitled 5’’. Photos: Yıldırım Arıcı

Coincidentally, both schemas are categories defined by Stiny (2011). Schema 1 very generally divides a whole into its parts. A number of rules from Koman’s Infinity-1 works, to be explained below, fall under this schema in algebras U12 and U23. Algebras, as introduced to shape computing by Stiny (2006: 180–196) indicate the spatial dimension of a shape in the left index and the spatial dimension of its environment in the right index. The different algebras here correspond to physical operations of dividing either the planar sheet material or the solid block material. The physical parts obtained after this division are subsequently transformed, through rotation and translation in Euclidean space, using Schema 2. The first two shape rules, Rule 1 (Fig. 1) and Rule 2 (Fig. 2) under Schema 1, are in algebra U12, as they correspond to line drawings on a planar surface to generate the cut patterns which divide a sheet material into strips. With Rule 1, Koman divides a shape into equally sized sub-shapes to obtain a continuous meandering strip of steady width and length from a sheet material (Fig. 1).

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Rule 2 divides a shape into sub-shapes in a similar manner but this time subshapes sequentially change in size to create an overall trapezoidal shape. In short, Koman applies Rule 2 to divide a sheet of material into a continuous meandering strip with gradually scaled width and length (Fig. 2). Although we will mostly focus on shapes in algebra U12 from this point onwards, the division rule Rule 3 under Schema 1 provides a notable and expected exception for a solid. It divides a three-dimensional shape into sub-shapes of equal size in U23. Koman uses this rule to divide a rectangular prismatic block into connected sprouts (Fig. 3) which he later opens up to create various forms. We create variations of Rules 1 and 2 to change the number and size of the subshapes obtained in the right side of the rule. Rule 1 is varied as shown in Fig. 4. Length l is kept constant while width w and depth d are changed resulting in different outcomes of the subdivision. These variations of Rule 1 directly determine the number and the width of the generated sub-shapes as well as the depth of the connection part in between sub-shapes. All variations in this example are purposefully done so that the sheet material is always divided with no leftovers. As seen in the works generated with Rule 1 in Fig. 1, the width of the strips and the depth of the connection part between strips are of equal length. We create variations for Rule 2 as well. A previous study by (Akgu¨n et al. 2007) on the mathematical relations of the Shell, the only work Koman produced using Rule 2, sets the basis for understanding the parametric variations for this rule. We alter two variables: the sequential scaling of the length l and of the width w of subshapes, resulting in different number and sizes of the generated sub-shapes. Rule 2b specifies the relation between l’s as increments smaller than in Rule 2a. This diminishes the number of sub-shapes. These are also longer in average when compared to those in Rule 2a. Rule 2c specifies values for w higher than in Rule 2a. The result is even fewer sub-shapes (Fig. 5). Rule 2 is also a parametric variation of Rule 1. However, in the context of this paper, we will still be denoting them separately as Rule 1 and Rule 2. Once the sub-shapes are obtained with the rules under Schema 1, Koman rearranges them by applying three rules under the schema x ? x ? t(x) (Schema 2) repetitively to all the sub-shapes. This schema suggests a spatial relation between a shape and its secondary instance with a transformation—in this case, a Euclidean translation in space—and therefore denotes an additive and repetitive process. In other words, x corresponds to a sub-shape that connects to a second sub-shape, which is its transformed similitude. The three rules we have identified under Schema 2 across the body of works in Infinity-1 are shown in Fig. 6. It should be noted that the shapes in these rules are abstract representations for the material strips obtained by dividing the sheet material with Rule 1 or Rule 2, and rules themselves represent how the strips are physically aligned in space. Rule 4 shows how to connect x and t(x) where t(x) is a transformation along the yaxis of the first shape x. The rule indicates the distance of the drift along the y-axis. This distance is the same size as the width of the sub-shapes in Rule 1. In the application of this rule, a strip is drifted on top of another strip in one axis only and of the same size as its width. Rule 5 sets a shape apart from the transformation of that shape so that the two shapes remain detached. Koman uses different types of

Schematizing Basic Design in Ilhan Koman’s ‘‘Embryonic’’…

Fig. 4 a Legend for how dimensions are named on the sheet, and b three variations of Rule 1

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¨ zkar B. Gu¨rsoy, M. O

Fig. 5 Three variations of Rule 2

specifications to control their relative position in space. This will be elaborated further below. Rule 6 shows how to connect x and t(x) where the transformation t(x) is in both y and x axes of the first shape x. The rule indicates the distance of the drift in both axes. The drift is half the depth of the strip along the y-axis and the width of the strip along the x-axis. We claim that the final three-dimensional forms of the sculptural works emerge through the repetitive and hands-on application of the rules under Schema 2 and are not conceived in advance through mathematical calculations as a previous study (Akgu¨n et al. 2007) suggests. What differentiates the works are the choice of the rule or rule set, the sequence of the rules in the chosen rule set, the customized variables in the rules, the total number of repetitions in the application of the rule or rule set (iterations), and the specifications to be introduced to Rule 5 to indicate the relative positions of the detached parts in space. Table 1 illustrates these factors excluding the specific values for variables.

Schematizing Basic Design in Ilhan Koman’s ‘‘Embryonic’’…

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Fig. 6 The three shape rules under Schema 2 and their repetitive physical applications a Rule 4, b Rule 5, and c Rule 6

Specifications that Koman introduces in Rule 5 control the relative position of spatially detached strip parts in space and can uniquely modify the overall form of the sculptures. In the Shell, Koman fixes all the strip edges onto one planar surface applying Rules 4 and 5. While applying the rules in an alternating order, Koman keeps every other strip flat on the plane while inevitably curving the rest as he goes along. This gives the overall curved character to the piece. How much Koman

990 Table 1 Comparison of works in the Infinity-1 series

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Fig. 7 References of ‘‘Shell’’ shown on photograph. Photo: Yıldırım Arıcı

Fig. 8 References of ‘‘To Infinity…’’ shown on photograph. Photo: Yıldırım Arıcı

curves each strip is defined by his application of Rules 4 and 5. The mean depth of each strip is the point of reference for the translation drift in Rule 4 while a small rotation angle inevitably emerges (Fig. 7). Accordingly, in the application of Rule 5, each flat strip edge (shown as red lines in Fig. 7) is laid next to the previous one as far as it reaches and produces the geometric sequence of spiraling edges. In To Infinity…, Rule 5 is applied twice with two different specifications (Fig. 8). In its lower part, the sculpture is fixed on a planar surface. Koman creates an axis on this surface and prepares horizontal slits parallel and at equal distances from one another. These slits correspond to where the strip edges are fixed in the every other application of Rule 5 in the rule sequence. In short, Koman fixes the distances between them in this manner. This is one specification. In the upper part of the

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sculpture, Koman creates notches near the edge of each strip to interlock another strip to it. In corresponding strips (those that will interlock) these notches appear to be in mirrored but exact locations, at double the depth and half the width of a strip. This is the second specification for Rule 5 that designates a particular spatial relationship for x and t(x). Both specifications give the sculpture its overall form. Lastly, the number of rules in a rule sequence, as identified in the table for each sculpture, has an impact on the overall spatial form of the sculptures. Whereas in Untitled (1), the sequence has two alternating rules, resulting in a linear form development around a central axis, in Whirlpool, the sequence has three rules resulting in a volumetric development of a triangular cross-section and one that curves onto itself. On the other hand, sculptures such as To Infinity… and Untitled (2) have four rules in a sequence and produce different volumetric structures. Untitled (3) has five rules in a sequence and produces a more complex volume.

Redefining Koman’s ‘‘Embryonic’’ Approach As shown above in the use of rules that specify size parameters, it is possible to discuss parametric variations in Koman’s works. However, our analysis of the Infinity-1 series makes evident that what lies beneath Koman’s ‘‘embryonic’’ approach goes beyond numeric specifications and includes rather spatial ones. Spatial specifications are mostly revealed through the process of making and this carries the approach further away from being a top-down mathematical formula. Koman makes visual calculations instead and these lead to the emergence of the final forms. Still, the mathematical analysis of the Shell in (Akgu¨n et al. 2007) has guided us in writing a code in RhinoScript to generate cut-sheets of the strips and the spiral template on which these strips are fixed onto a plane. The code enables us to generate parametric cut-sheets to digitally manufacture works with great precision by modifying the parameters in Rule 2. Below are some of the parametrically produced works built with the cut sheets generated using this code and cut in laser cutter (Fig. 9). The variety is endless and is deemed useful pedagogically. These

Fig. 9 Parametric Shells produced using the code in RhinoScript

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cardboard shells comprise a part of the study realized with the basic design students, presented in the second part of this paper. Nevertheless, the variations Koman achieved in and across different works in Infinity-1 series are more than just the parametric variations as we exemplify with the Shells. Koman sets a relational system through certain spatial rules as illustrated in Table 1. Within this relational system, an infinite number of variations can be generated. While trying to uncover the existing relations within the works in Infinity-1, we have built several mock-up models and by misinterpreting the set of rules in the original works, we accidentally obtained different, yet similar, ‘‘embryonic’’ Infinity-1 forms. These mock-ups are illustrated in Table 2. The outcome of the detailed analysis of the Infinity-1 series therefore might serve not only the reproduction of the existing works but also future experimentation with various materials towards additional Infinity-1 sculptures at various scales. Sculptures in the Infinity-1 series also strongly demonstrate how perceived complexity can be achieved through repetition of simple rules. The underlying relations are best understood and conveyed spatially in mock-ups. What we have further discovered through the mock-ups is that the making of these sculptures also necessitate a systematic approach because the connected parts of one meandering strip tend to easily jumble up. We believe that Koman also developed a know-how concerning the making processes behind these sculptures. Differentiating each side of the sheet material with a different color, a trick we made use of while making our models, is previously developed by Koman as in Untitled (3) and is helpful in the making process. The second part of this paper discusses the ways in which Koman’s ‘‘embryonic’’ approach relates to basic design and how understanding the visual schemas of perceivable repetitions and their variations in Koman’s works can help basic design students to understand and establish a relational system for formal unity.

Koman’s ‘‘Embryonic’’ Approach in Basic Design Basic design students are often encouraged to create variations in a relational system and have control over the underlying principles of that system, much as Koman did with the Infinity-1 series. However most of the time, entry-level design students struggle in defining a relational system, tend to intuitively seek unity between parts, and mostly fail. Hence, one of the objectives of basic design is to make students aware of the emergent relations they discover intuitively while trying to relate shapes, and to help them in formalizing their discoveries in order to translate them to design decisions. Therefore, showing students their works as visual computations may help them observe what they are doing and develop it further. However, conveying the computable aspects of design through the formalism of visual rules to students already struggling to adapt to the abstract visual language is not an easy task. Visual schemas, as ‘‘more general ways to understand design decisions’’ than visual rules, may be more suitable in making students aware of their ¨ zkar 2011: 115). design process (O

994 Table 2 Paper mock-ups by the authors placed in the Infinity-1 table

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Below we present the outcomes of a basic design exercise where the underlying visual schemas of Koman’s Infinity-1 series are set as guides and design constraints as well as tools to formalize design thinking. The study was realized with fifteen volunteering basic design students at Istanbul Bilgi University, Faculty of Architecture. The volunteers had recently started their design education and had been engaged with basic design exercises for 3 weeks at the time of the study. The exercises they had dealt with so far required two-dimensional compositions by physically cutting and gluing shapes on specifically sized canvases. As part of the broader basic design curriculum at that university, students were especially asked to design systematic wholes where seeking unity between shapes in both the figure and the ground, as well as establishing consistent relations are of the utmost importance. At the same time, they were cautioned to avoid symmetrical arrangements and guided to consider emerging shapes as design elements. The behavioral properties of the materials had not been acknowledged yet as design input. The study in connection to our paper was realized in two consecutive stages. Students were given the task of designing a two-dimensional composition by cutting and gluing shapes obtained from tangram squares of three different colors on a given canvas area. The task was repeated in the second stage. In between the two stages students were introduced to Koman’s works under Infinity-1 series and were asked to discover the underlying principles (i.e., visual schemas) and to observe similarities and differences in these works. In the second stage, they were asked to focus on two variations of Koman’s Shell and translate the underlying spatial relations in these works as guides to their own two-dimensional designs. Whereas the students mostly failed in successfully designing a systematic whole in the first stage, there is visible evidence that in the second, they came up with more conscious, consistent and systematic arrangements of tangram shapes that we could retrospectively denote with shape rules. Below, we present the tangram exercise in detail and selectively analyze the design processes of students.

Infinity-1 and Tangram Exercise Students were given a verbal brief to design a systematic whole. The brief described the task as gluing on a defined canvas some cardboard tangram shapes of specific size but in three different colors. For our notes not shared with the students, the description of the exercise included visual rules that fall under Schema 1 and Schema 3: X x! prtð xÞ ðSchema1Þ and prtð xÞ !

X

tðprtðxÞÞ

ðSchema3Þ

The first shape rule under Schema 1 divides a square into seven tangram subshapes (Fig. 10). This corresponds to physically cutting the cardboards. The subshapes obtained through this division are of three different geometries. The square

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Fig. 10 Shape rule which generates a tangram square

and the parallelogram are unique but the triangles come in three sizes. The shapes already have certain defined relations in that there are inherent similarities of size and geometry between the elements. For instance, the parallelogram and the square can also be constructed using P two of the smallest triangles. The schema prt(x) ? t(prt(x)) (Schema 3) is a summation schema and is consequently used to make Euclidean transformations in U22 on the tangram parts obtained to re-design a whole. In order to test the impact of Koman’s visual schemas, the assignment is first given without referring to Koman’s works. The majority of the students were not able to define a relational system between different kinds of possible transformations that can be applied to shapes under the Schema 3. Rather, they sought for different possible relations between shapes. Three students’ works are redrawn here in Fig. 11, where the left column shows the boundaries of the shapes and the right column shows the shapes with their assigned colors. In these designs, new shapes have emerged in addition to those given in the tangram when shapes with the same colors share boundaries. This does not accord with Schema 3 as the requirement of the assignment but exemplify the potential richness of the exercise. At the end of this first go, students were introduced to Koman’s Infinity-1 series and provided with a code that generates the parametric cut-sheets of the Shell to further investigate the extents of variation within the repetitive system of the series. They produced, in groups of two, the parametric Shells shown in Fig. 9. Among the parametric Shells, one group chose to generate a circular Shell, the rest generated spirally developed Shells as Koman did (the former will further be referred to as the circle, the latter as spirals). They were also asked to verbalize the relations they discovered, as if they were telling them to a friend. One student wrote: ‘‘In the spirals, a shape is related with its rotated and scaled versions in a successive order, whereas in the circle, the same shape is related with its rotated version without difference in scale’’. This verbal definition corresponds to Koman’s visual schema that we identified in the first part of this paper: x ? x ? t(x) (Schema 2). For the spirals, the schema denotes a transformation of translation, and scale, whereas in the circle it denotes translation only. Students were then asked to choose one of the two systems for the spiral and the circle to redo the tangram exercise. This meant changing P the more general schema given with the definition of the exercise, prt(x) ? t(prt(x)) (Schema 3), to a

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Fig. 11 Studio work by three students before being exposed to Koman’s schemas (redrawn on computer)

more specific one, x ? x ? t(x) (Schema 2) and applying it to every tangram part separately. Seeing the applications of Schema 2 in Koman’s sculptures helped students define consistent spatial relations they had previously had difficulties with. The final works of six students are presented in Fig. 12. The arrangements are much more consciously done when compared to versions prior to Schema 2 (Fig. 11). The two students who have chosen to work on the relations they discovered through the circle (C1 and C2 in Fig. 12a), sustained Koman’s x ? x ? t(x) (Schema 2) where the only transformation is translation. They both created a unit using discrete parts. Retrospectively, we formally present in Fig. 13 the process to obtain the student work C1 (in Fig. 12a) with shape rules. The student in C1 defined the total height of the unit in relation to the height of the canvas so that the leftover parts can also relate with the unit (Fig. 13a). Two shape rules (both befitting Schema 2, x ? x ? t(x), and shown in Fig. 13b, c) control in parallel the respective relations between the given elements and between the leftover parts on the canvas. Applying both rules recursively for three times, we obtain the compositions in Fig. 13d, e. When the results are combined, the figureground distinction disrupts and there is no longer a leftover and unconsidered area (Fig. 13f), with the exception of the thin linear strips between the units which the student seemed to ignore. When the overall composition is placed within the boundary of the canvas, there are leftover areas between the canvas borders and the composition (Fig. 13g). In the final design, the student embedded tangram parts, namely a triangle and two parallelograms, in these leftover areas (Fig. 13h). These

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Fig. 12 Workshop results after discussing Koman’s schemas a from the circle, and b from the spirals (redrawn on computer)

added elements are not only tangram shapes but also parts of the unit the student used earlier. We show the recognition of these shapes in the rules shown in Fig. 13i. These rules fall under a new schema: x ? prt(x) (Schema 4). After setting these relations between shapes, the student assigned these shapes their colors. For color assignments, no defined set of rules is visible. However, the student seems to have made variations to avoid symmetry within the repetitive order by assigning the same color to adjacent shapes and creating new larger shapes. The student work C2 (in Fig. 12a) establishes relations that can be denoted by the shape rule in Fig. 14a. While variable a controls the translation on the central axis of x, variable b determines the difference in scale. The significant step that this student takes is in identifying vocabulary shapes in wholes that she creates with other vocabulary shapes. The larger triangles that are a combination of smaller shapes of the same color become the left side of the main rule in more than one

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Fig. 13 The process to obtain the student work C1: a the repeating unit of the student work C1, b shape rule relating the given elements, c shape rule relating the leftover parts, d first rule applied three times, e second rule applied three times, f the overall composition as shown with line drawings only, g the overall composition placed within the boundary of the canvas, h the overall composition on canvas with visually embedded shapes shown in shading. i Student recognizes tangram shapes as parts of the repeating unit

instance (Fig. 14b). This yields a richer variation in how the shape rule in Fig. 14a is applied. The students who have chosen to work on the relations of the spiral made good use of the three different sizes of the tangram triangle. The shape rules we have identified for this comply with the schema x ? x ? t(x) (Schema 2) used in the spirals where the transformation denotes ‘‘translate, and scale’’.

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Fig. 14 Rules identified in the student work C2: a shape rule with variables a and b, and b rule indicating the identification of larger vocabulary shapes in clusters of smaller vocabulary shapes

The process to obtain the student work S2 (in Fig. 12b) is presented in Fig. 15. The first set of rules establishes relations with scale changes between shapes (Fig. 15a). The rules in each column could be parametrically linked. The rules in the bottom row are in fact similar to the ones in the top row. The size of the added triangle t(x) can be specified. Our analysis with the rules shows the student’s conscious decisions in relating the rules to one another. Since the variation in the rules is just this difference in size, for the case of this student, we did not refrain from identifying each rule separately. The second set of rules is similar to the first set of rules already given in Fig. 15a, with the exception of the spatial relation between x and t(x). This relation can be defined by assigning values to variables. We differentiated them as shown in Fig. 15b. The application of the first and second set of rules is highlighted by assigning colors to the shapes in Fig. 15c, d. The student then repeated both sequences to combine in a rotational symmetry on the canvas in a final composition (Fig. 15e). This can be represented with another spatial rule that translates and rotates x, still compliant with Schema 2. It is possible to represent this student’s process with entirely different rule sets. The spatial relation between each grey and black triangle couplet (in Fig. 15e) also reveals a repetitive but varied use of yet another rule. The variation in the rule can be specified in terms of the size of each triangle and their relative position to one another within an imaginary rectangular box. In S3 (Fig. 12b), the student created a unit where the colors are assigned beforehand (Fig. 16a). This unit is used at three different scales. We have identified five shape rules within the schema x ? x ? t(x) (Schema 2). These rules are presented below, in Fig. 16b–f. However, regarding the diversity of the shape rules and noticing that there is no recursion of any rule, except the one illustrated in Fig. 16b, we can say that the student did not define a relational system beyond the similarity due to the repetition of the unit. Instead, he might have sought for alternative transformations within the schema x ? x ? t(x) (Schema 2) to fit within the boundary of the canvas. Nevertheless, by using the same unit at different scales with the same colors and thus acknowledging the corners that emerge in the unit in Fig. 16a, he somehow achieved a unity (Fig. 16g). In S4, we could not identify the use of any consistent shape rule. However, while working on the composition, the student verbally expressed that he had defined certain relations between different parts that he repeated at different scales. Only

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Fig. 15 The process to obtain the student work S2: a the first set of rules, b the second set of rules, c the first set of rules applied with weighted shapes, d the second set of rules applied with weighted shapes. e The student rotationally mirrors one group of shapes on the opposite edge of the canvas

perceivable in the line drawings, there is a repetition of certain shape relations between squares and parallelograms whether they are original tangram pieces or composed of two triangles. The student had defined spatial relations between different shapes (x and y) but created new shapes while assigning color. Hence, in Fig. 17, in order to make these relations more legible, we assigned new colors to the shapes. This way, a different schema x ? x ? y (Schema 5) in addition to Schema 2 is better exposed.

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b Fig. 16 The rules to obtain the student work S3: a The basic unit in S3. Right triangles are in a spatial relation that emphasizes a right corner emerging to the eye between the grey and red ones. b–f The rules we have identified. g Corners emerge as shapes that repeat in the overall composition (color figure online)

Fig. 17 Left S4 with colors assigned by student. Right S4 with colors assigned by the authors

Conclusions and Discussion Making unified wholes with shapes is often the goal in basic design exercises where abstract organizational concepts such as repetition guide students as they develop a relational system towards creating a whole. Our basic premise is that designs where repetitions and variations are perceivable may reciprocally serve as effective tools in conveying to beginning design students the underlying relational nature of design. We analyzed Ilhan Koman’s non-figurative sculpture series, all designs with strongly repetitive features, and illustrated his ‘‘embryonic’’ approach in visual schemas which we then incorporated into a basic design studio exercise to demonstrate students’ learning processes. Exposure to the knowledge of Koman’s Infinity-1 series, particularly the spiral and the circle, visibly helped the students in grasping how the repetition of certain relations enabled the artist to create complex sculptures, and how variations generated different, yet similar outcomes. Our shape computation analysis of students’ proposals for the same exercise before and after being exposed to this knowledge shows the apparent progress in the quality of the works in terms of creating a relational system and achieving unity. The general schema x ? x ? t(x) (Schema 2) of the Infinity-1 series denotes an additive process where repetitions and variations are perceivable. The variations in question can be of features such as size, orientation, position, and color. The adoption of Pthis schema by basic design students instead of the more general schema prt(x) ? t(prt(x)) (Schema 3) given with the definition of the exercise, encouraged them to establish relations among shapes and to compute with them without even talking explicitly about individual shape rules. While, within Schema 2, Koman generates his works within a linear iterative process, in basic design works, repetitions are not necessarily linear, and in fact are rarely so. Basic design works are mostly spatial and processes behind them are frequently obstructed through seeing emergent shapes. This is a key aspect that distinguishes basic design works from linear generative processes and provides a richer playground to incorporate perception into P a hands-on process. The schema prt(x) ? t(prt(x)) (Schema 3) which illustrates the description of the exercise given in the studio remains too general for entry-level design students

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to handle. When asked to design a whole with Schema 3, students mostly concentrated on fitting the shapes at hand within the boundary of the canvas in a way to achieve a unity between figure and ground, which corresponds to a unity between the leftover parts on the canvas and the discrete shapes being used. However, except for this relation, they mostly failed in defining a relational system to organize the whole. When the general schema is further specified as x ? x ? t(x) (Schema 2) after analyzing Koman, students realized that the repetition of certain relations with a degree of variation could help them achieve unity, as in the Infinity-1 series. The variations can be of scale, orientation, and position. They also tested the consequences of assigning colors to shapes to create variations within a repetitive system, as exemplified here in what one student did in C1. In addition to the Koman Infinity-1 schemas, students’ processes present several other schemas: x ? x, prt(x) ? prt(x), b(x) ? b(x), b(prt(x)) ? b(prt(x)), etc. Students keep recognizing shapes that they did not see before as they tackle figureground, solid-void relations, and relations between discrete shapes in additive schemas. After seeing emergent shapes, they can go back and change their rules to cope with emergence. Going back and forth between the schemas and shape rules in basic design makes it a more complicated task than a linear iteration. A relational system, in this case triggered by the exposure to Koman’s ‘‘embryonic’’ approach, helps students to interpret and utilize emergent shapes within a larger whole. Further investigations into the students’ interface with visual schema formalisms are required to discuss how students can define visual schemas for their own work towards a computational understanding of design in their curriculum. Finally, processes similar to those seen in Koman’s artistic production, in which material properties and behaviors are considered as design input and where material manipulation shapes the design outcome, necessitate extended visual formalisms to represent the feedback from materials. There is potential for future research not only on how visual schemas convey key aspects of relational and reflective thinking to basic design students, but also to elucidate the computational aspects of various art and design work, as was done for an example of Koman’s ‘‘embryonic’’ approach in this paper. This will in turn serve to bridge the tacit knowledge of creative processes and contemporary design pedagogy. Acknowledgments All images are by authors, unless otherwise noted. We thank Yıldırım Arıcı for permission to reproduce the photos that appear here. The tangram exercise is credited to the instructors of the 2013 Basic Design Studio at Istanbul Bilgi University - Faculty of Architecture, including Benay Gu¨rsoy. The student work in Fig. 11 is from the studio. All other student work is from the workshop carried out separately from the studio as part of this study. The rules and schemas that describe the exercise and the student work are retrospective and by the authors.

References Akgu¨n, T., ˙I. Kaya, A. Koman, and E. Akleman. 2007. Spiral Developable Sculptures of Ilhan Koman. In: Proceedings of the 2007 Bridges Conference in Art, Music, and Science, eds. Reza Sarhangi and Javier Barallo, 47–52. London, UK: Tarquin Books.

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Akgu¨n, T., Koman, A., and E. Akleman. 2006. Developable Sculptural Forms of Ilhan Koman. In: Proceedings of the 2006 Bridges Conference in Art, Music, and Science, eds. Reza Sarhangi and John Sharp, 343–350. London, UK: Tarquin Books. Aytac¸-Dural, T. 2012. Beginning design education as a process of transformation. In: Shaping design ¨ zkar, 101–128. Aalborg: Aalborg University Press. teaching, eds. N. Steino and M. O Bes¸ liog˘lu, B. 2011. An Inquiry into the Computational Design Culture in Turkey: A Re-Interpretation of the Generative Works of Sedat Hakkı Eldem and Ilhan Koman. Intercultural Understanding, 1: 9–15. Knight, T. 1999. Shape Grammars in Education and Practice: History and Prospects. International Journal of Design Computing 2. Koman, ˙I. and F. Ribeyrolles. 1979. On My Approach to Making Non-figurative Static and Kinetic Sculpture. Leonardo 12(1): 1–4. ¨ zkar, M. 2007. Learning computing by design; learning design by computing. In: Proceedings of the O Designtrain Congress Trailer I—Guidance in/for Design Training, 102–111. Amsterdam. ¨ zkar, M. 2011. Visual schemas: pragmatics of design learning in foundations studios. Nexus Network O Journal 13(1), 113–130. Stiny, G. 2006. Shape: Talking about Seeing and Doing. Cambridge, Massachusetts: The MIT Press. Stiny, G. 2011. What Rule(s) Should I Use? Nexus Network Journal 13(1), 15-47.

Benay Gu¨rsoy is a PhD candidate at Istanbul Technical University, in the Program in Computational Design and a fulltime instructor at Istanbul Bilgi University, Faculty of Architecture where she is teaching Basic Design Studio, Architectural Geometry and Design Computing courses. Her current interests and teaching focuses on foundational design education, material computation, material tectonics, and computational and generative design methodologies. Mine Ozkar is an associate professor of architecture at Istanbul Technical University, where she is the coordinator of the Program in Computational Design. She teaches graduate level design research methods, a computational design studio, several computational theory courses and an undergraduate architectural design studio. She earned her MS in design inquiry and PhD in design and computation from MIT. In some of her previous work, she has interpreted the history and theory of progressive pedagogy in art and design from a computational perspective. Her current research focuses on shape representation for creative computing, visual/spatial computation, and design methods as well as the integration of foundational design education and computational knowledge. She also publishes on the on-going global and local curriculum reforms in architectural education. She recently co-edited a book entitled Shaping Design Teaching and was a Visiting Professor in the MIT Department of Architecture Computation Group during Spring term of 2013.

Leonardo

On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art. With an Artist's Statement by Brent Collins Author(s): George K. Francis and Brent Collins Source: Leonardo, Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue (1992), pp. 313-320 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1575857 . Accessed: 17/06/2014 09:06 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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TOPOLOGY

On

An Knot-SpanningSurfaces: Illustrated on Essay Topological

Art

With an Artist'sStatement by Brent Collins GeorgeK Franciswith BrentCollins

I n this essay I shall set forth basic vocabulary for a dialogue between artist and mathematician concerning certain kinds of mathematical art-in particular, the topological sculptures of Brent Collins. First, I establish some coordinateswithin aesthetic philosophy that will allow us to locate our subject relative to other mathematical art. Second, I introduce the language of topology and use it to analyze in some detail an early sculpture by Collins. In the course of this discussion, we will discover the true topological identity of the surface of this sculpture and its nonobvious affinity to another well-known piece of mathematical art. As with all philosophy, it is appropriate to begin with some distinctions. There is 'implicitly' mathematical art and 'explicitly' mathematical art. In the former, the mathematical content or significance is chiefly in the 'eye of the beholder',

Fig. 1. Brent Collins, One-Sided Surfacewith Opposed Oeimrlities, oiled cedar, 30 x

12 x 4 in, 1984. This image shows a right-handed Haken surface spanning Listing's knot and with a disc removed.

for the artist did not intentionally express mathematical ideas in an aesthetically informed manner. This does not mean ABSTRACT that mathematical aspects in an implicitly mathematical artThe authordiscussesa type of visualmathematics knownas work must be obscure, hidden topologicalsculptures,firstby definor accidental. However, the ingtermsandideas,andthenby mere use of spheres, regular discussingthe workof artistBrent polyhedra or spatial symmetry Collins.Anartist'sstatementby Collinsis also included. does not necessarily mean that the artist intended to impress some mathematical sense into his or her work. Distinctions are best studied at their 'border'. Collins's sculpture is just on the explicitly mathematical side of it. Consider an artist's use of a natural or artificial form that has a long tradition in mathematical illustration-for example, a halved nautilus shell or a stellated dodecahedron. This is implicitly mathematical art if the artist intends to evoke a memory of mathematical cerebration, a flash of recognition. The same objects would be explicitly mathematical art if logarithmic self-similarity, in the first case, or Euclidean stereometry, in the second case, were the true subjects of the artwork. Compare Dfirer's use of mathematical artifacts in Melancholia1 with his lesson on prospective geometry in his ArtistDrawinga Lute. Central to explicitly mathematical art is a second distinction that places such work along the spectrum of mathematical sophistication. At one end of this spectrum are subjects that are quite traditional, such as symmetry, Platonic solids and optical illusions, and subjects that are rather conventional, such as cubes, spheres, M6bius bands and the like. At the other end of this spectrum are mathematical diagrams, frequently called schematicdrawings.The beauty of such explanatory illustrations is often in the eye of the beholder. Precision, information and lack of ambiguity are important here. I also exclude artworks that merely appear George K. Francis (descriptive topologist), Mathematics Department, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, U.S.A. Brent Collins (artist), RR #1, Box 146, Oakland, IL 61943, U.S.A. Received 23 May 1991.

? 1992 ISAST PergamonPressLtd.PrintedinGreatBritain. 0024-094X/92$5.00+0.00

LEONARDO, Vol. 25, No. 3/4, pp. 313-320, 1992

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313

a

b

Fig. 2. Two ways of drawing an unframed link consisting of a trefoil knot and an unknot. (a) The 'interrupted underpass' convention is used for a left-handed trefoil simply linked to an unknot resembling a left-handed figure-8. (b) The 'thickened curve' convention is used for the right-handed trefoil. a

b

a century ago, that still grace the glass cases of a few mathematics departments in this world, as documented in Gerd Fischer's handsome volumes [1]. At the other extreme of this distinction, the artist, deeply involved with a profound mathematical theorem, fashions an impressionisticvisual poem of undeniable aesthetic force. The mathematical initiate recognizes the theorem and is moved by the recollection of its significance. The lay observer senses the beauty but is puzzled by the meaning. This art form is rare-however, Anatoly T. Fomenko's imaginative chapter-title illustrations in the textbook Introduction to Topology[2] fit this category very well. The gamut of Fomenko's entire mathematical art is so vast that it defies easy classification. In his MathematicalImpressions[3] are representatives of all kinds of explicit mathematical art, from models to visual poems. The work of Collins is somewhere orthogonalto Fomenko's, closer to implicitly mathematical art. It is intentionally mathematical at the level of precision, symmetryand definition, in terms of simpler constituent forms. In the process, Collins accidentally created beautiful mathematical models that serve more than one geometrical discipline. I next turn to his works, concentrating on their topological qualities. The metrical qualities will be considered in a subsequent study.

AN EXAMPLE OF MATHEMATICAL ART

Fig. 3. (a) The figure-8 knot in a symmetric but peculiarly twodimensional form. (b) A right-handed one-sided Haken surface spanning this knot. (c) An isotopic deformation of the surface shown in Fig. 3b. (d) A left-handed Haken surface spanning the knot. (e) The orientable surface spanning the figure-8 knot studied by Seifert [28]. (f) Half-twists of short ribbons illustrating graphic techniques.

mathematical to the observer-artworks in which the artist did not consciously impress a mathematical form into the work. Let me describe the poles along this axis. An artificial object may be crafted in such a way that the abstract mathematical principle that it portrays can be unambiguously

read

from it. The artist makes the object appear to represent a phenomenon from which the principle could have been discovered empirically rather than through mathematical insight. I am referring to mathematical models, so popular

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Francis with (ollins, On Knot-Spanning

Collins is a sculptor living in Oakland, a rural town in east central Illinois. He uses common shop tools, such as a hand-held router and a small, electric chainsaw, to fashion abstract shapes from single blocks of cedar wood. Many of his pieces are invariant under discrete spatial symmetry groups, and all of them involve continuously varying sectional curvatures. As a result, the precision and fidelity with which corresponding parts match, together with the subtle permutation of elementary constituent forms, are most attractive to mathematically educated viewers. Collins is not a mathematician. The sculpture shown in Fig. 1 is representative of one of Collins's series of motifs. Apart from the base, it is clearly invariant under 180? rotations about all three principal axes-in a kind of three-dimensional (3D) playing-card symmetry. (For another example of this kind of symmetry, see Color Plate A No. 2.) Such symmetries are very helpful for inferring details of shape from only one picture. Naturally, it would be much more satisfactory to view the piece from all angles or, even better, to view it while rotating it manually. The object is of uniform 1/4-inthickness, and so may be regarded as a material realization of an abstract surface whose mathematical nature I shall explore presently. This abstraction does immediate violence to two further features of mathematical interest, the wood grain and the edge of the sculpture. The former has more aesthetic than technical value. The wood grain frequently suggests the illustrations one finds in treatises on Morse Theory [4]. When the concentric cylinders of dicotyledonous wood are cut transversely by a plane, they form the familiar tree rings. Cutting by domeshaped or saddle-shaped surfaces brings out the characteristic markings of a Morse function in the vicinity of its singularities.

Surfaces

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The second feature has a technically highly interesting aspect that we shall ignore after this paragraph. From the sharp, uniformly /4-in-wideedges of the artwork,we abstract closed, knotted and linked ribbons curving through space. The mathematical surface depicted by the artworkmay thus be regarded as a spanning surface of aframed link. Rolfsen's Knotsand Linksis an authoritative treatment of this subject [5]. We shall ignore the question of how this ribbon is twisted along itself and consider the abstract surface ending along an unframed link (Fig. 2) as its border. Collins's surface (see Fig. 1) has two border curves that are not linked. The little one in the center, shaped like the numeral 8, plays a special role. Imagine that the artist had not yet penetrated the center panel, but left a smooth sheet. This way, the surface spans a single, knotted curve. If it is carefully traced, it can be seen to be shaped like the knot shown in Fig. 3a. (As it will become necessary to compare the shapes in Fig. 3, they appear together in the same figure.) Collins's surface shown in Fig. 1, without the central detail, corresponds the surface shown in Fig. 3b. I shall discuss the significance of this surface later. The knot spanned by Collins's surface shown in Fig. 1 goes by many names:figure-8 knot,four-knotand Listing'sknot, whose namesake coined the term topologyto describe the study of shapes and spatial transformations of abstract objects. This knot is a respectable character of an area of mathematics known, appropriately enough, as knot theory. Rolfsen's Knots and Links [6] is an ample portal to this fascinating kind of mathematics and its useful bibliography points to the remaining important references. However, a small subset of what there is to say about this knot can also be found in A TopologicalPicturebook[7].

A BRIEF TOPOLOGY LESSON Wood is a rigid substance, so much so that it is difficult to imagine a surface carved from wood as having the quality of stretchable rubber. It also seems wasteful to forget the sensuous curvature Collins carves into his sculptures. Yet these operations are necessary for studying an object in topology. Two objects are topologically equivalent, or homeomorphic,if one can be distorted so as to correspond point for point with the other. Actually, this notion is more correctly a common description of an isotopy.The notion of topological equivalence is more generous, allowing for a temporary dismemberment of the object as long as corresponding 'rips' are faithfully 'sewn' together again. Moreover, an isotopy must be realizable in its entirety in space. A homeomorphism may be described piecemeal and unpictorially. In deforming one topological object into an equivalent one, we do not allow knots to come apart, or surfaces to pass through each another. Within these tolerable restrictions, topologists have been able to solve the classification problem for knots and for the surfaces spanning them. In order to understand just what it means to say that there are only six different surfaces spanning Listing's knot, we need to establish a few more notions from the topology of surfaces [8]. A surface is said to be closedifit has no boundary and fits into a finite portion of space. Thus a sphere is closed, as is a torus (Fig. 4a). There are infinite surfaces, with or without borders, such as a plane with or without a disc removed. Each portion extending to infinity is called an end of the surface. Among finite surfaces with borders, some are two-sided, such as a disc or a torus with a disc removed

(Fig. 4c). Some are one-sided [9], such as a M6bius band (Fig. 4b) or a Klein bottle (Fig. 4d). (In addition to Fig. 1, Collins's piece in Fig. 7 is one-sided; the rest of his sculptures shown here are two-sided. The border curves of Fig. 8 naturally extend to infinity. Thus this may be regarded as a part of an infinite surface with three ends. The border curves of Color Plate A No. 2 and Fig. 9 look like the rims of discs that have been removed from two closed surfaces of higher genus.) Compact surfaces, that is, finite surfaces that are either closed or bordered, are classified by three qualities: (1) by the number of component curves contained in their boundary, (2) by whether or not they are two-sided and (3) by an integer presently defined as the Euler characteristic.This means that any two compact surfaces are topologically equivalentor homeomorphicunless they differ in one or more of these qualities. This is the fundamental theorem of the theory of surfaces. To determine the orientability is a matter of tracing your mind's finger along the curve. The Euler characteristic is more difficult to define. I shall first give an operational definition. Two other definitions follow. The disc, naturally, shall have Euler characteristic 1 [10]. Sewing two surfaces together along the entirety of a border circle on each produces a surface whose Euler number shall be the sum of the constituents. Hence, a sphere has Euler characteristic 2 (for the two hemispheres sewn on the equator), and the Euler characteristic of a surface decreases by one for each disc removed. Consistent with this is the rule that cutting across a ribbon increases the Euler characteristic by 1. Thus, the characteristic of a Mobius band is 0 (one less than that of the disc) [11 ]. Two surfaces can always be connected in the following way to produce a third surface. Replace two discs, one on each surface, by a cylindrical tube that connects their border circles. The result of such an operation is called the connected sum of the constituent surfaces. The Euler characteristic of the connected sum of two surfaces is two less than the sum Fig. 4. (a) A torus, (b) a Mobius band, (c) another torus with a circular window removed and (d) a Klein bottle. The torus and the Klein bottle are closed surfaces; the Mobius band and holey torus have a single border curve. The Klein bottle and Mobius band are one-sided surfaces.

a

b

c7~

d

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a

b

Fig. 5. Two further isotopes of Listing's knot. Two of its aliases, 'four-knot' and 'figure-8 knot' derive from its shape in (a). Its three-dimensional symmetries may be seen in (b), which is drawn in strong perspective.

of their Euler characteristics. Considering the flexibility of topological surfaces, it is easy to agree that the connected sum of a sphere with any surface does not produce a new surface at all. The sphere S acts like zero under this topological addition. Among closed surfaces the arithmetic is quite simple. One must be careful to avoid an ambiguity when one connects a surface F to itself. The connection of two copies of the same surface differs from connecting a surface to itself. The latter surgery is equivalent to replacing two discs on a surface by a tubular handle. It is also equivalent to connecting F to a torus T (see Fig. 4a). The classification theorem for closed, two-sided surfaces states that every such surface is the connected sum of a certain number of tori. This number is called the genusof the surface. (For example, Collins's surfaces shown in Color Plate A No. 2 and Fig. 9 are homeomorphic to the connected sum of six, respectively four tori, each with six discs removed.) There are closed, one-sided surfaces, but they are more difficult to imagine because they cannot be assembled in the 3D space of our experience. The simplest of these is called a projectiveplane, P, for historical reasons. Topologically speaking, this surface is a Mobius band sewn to a disc along the common boundary curve. Everygood topology textbook will have instructions for visualizing this object [12,13]. We thus have a second elementary constituent surface, namely P. The connected sum of two copies of P yields a famous one-sided surface, the Klein bottleK (see Fig. 4d) [14]. A common way of visualizing the connected sum of a Fig. 6. The double cover of a Mobius band is a two-sided ribbon with a full twist: (a) shown retaining the shape of the Mobius band, and (b) shown farther away from the (invisible) Mobius band so that its twists can be more easily counted.

b

a

316

projective plane to a given surface is to remove a disc and sew a Mobius band back into the surface. Note how this operation subtracts 1 from the Euler characteristic. If this is too hard to imagine, try visualizing a structure like the one in the very middle of Collins's surface shown in Fig. 1 and then imagine a disc sewn in along the figure-8-like border. Either way, this feat of haberdashery cannot be completed in three-space unless the to-be-added patch is allowed to penetrate the already-existing surface. In the latter case, the resulting figure is called a crosscap. Thus, sewing in a cross cap invariably renders 'i surface one-sided, while attaching a handle does not alter this condition. All one-sided, closed surfaces are obtained as connected sums of projective planes [15]. Finally, there is a most curious and remarkable theorem connecting the sums of tori T and projective planes P, such that F+T+P=F+P+P+P=F+K+P.

This theorem states that sewing three cross caps into any surface F is equivalent to sewing in a handle and a cross cap, or a Klein bottle and a cross cap. The simplest case is for F to be a sphere. Then F can be canceled from all three sums. This remarkable surface (which is at once a torus with a cross cap, a Klein bottle with a cross cap, and a projective plane with two cross caps) was discovered by Walther von Dyck [ 16] but is rarely called by his name. Dyck's surface has most recently been celebrated in marble and onyx by sculptor and mathematician Helaman Ferguson [ 17]. Remove two discs, and it becomes Brent Collins's surface shown in Fig. 1. That these two so artisticallydifferent conceptions are of the same topological realitywas a strong motivation for mywriting this article.

HAKEN SURFACES

FOR THE FIGURE-8 KNOT We say that a surface spansa set of one or more closed curves in space-a knot or link, respectively--simply when the curve(s) form the border of the surface. It is always a nuisance to work informally with collective terms. A link consisting of a single curve is a knot. A knot that is not knotted, is called the unknot by topologists. We say 'the' unknot because all unknots are isotopic. Unfortunately, it is not easy to think inclusively. Recall that all squares are rectangles, and rectangles are trapezoids, yet the word 'trapezoid' never conjures up a picture of a square in our minds. At issue here is the fact that the same link may be spanned by different surfaces. One can form the connected sum of one spanning surface with any number of tori or projective planes. So, we seek, in some sense, the simplest surfaces spanning a given knot [ 18], for example, those surfaces that are not obtained from another one by means of connected summation. Thus there are exactly six Haken surfacesspanning the figure-8 knot. One of them is not a bordered surface-it merely manages to pass for one on technical grounds. It is a tube that runs along the knot. Again, think of the knot as thickened (as it must be to correspond to a real object), and then think of the toroidal tube corresponding to the surface of this thickened knot (Fig. 3a). So far, the original knot is not even on this surface, it runs along the center, inside of the tube. Now push the knot out to the surface. It still cannot be said

Franciswith Collins,On Knot-Spanning Surfaces

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Fig. 7. Brent Collins, One-SidedSurfacewith EfflorescentCenterpiece,wood, 50 x 14 x 4 in, 1989. Both front and back views of this companion piece to Fig. 1 are shown to emphasize the pure symmetry expected of topologically significant art. Aesthetically, it is related to Collins's earlier piece by a variation on central connectedness. Topologically, it is quite different. This nonorientable surface spans a nontrivial link of two unknots. With a Euler characteristic of -6, it is too complex to be a Haken surface.

:-.. .-....:

kSg^y

to be the boundary of the knotted torus (tori are closed surfaces) but the torus does qualify as spanning the knot under a more technical (and therefore more useful) definition of the term. There is no single way of pushing the

original knot out to the surface of the tube. There are as many waysof doing this as there are framings for the figure-8 knot. Three further Haken surfaces for the figure-8 knot can be readily drawn. These are the right-handed (Fig. 3b) and the left-handed Klein bottles with a disc removed (Fig. 3d), and the torus with a disc removed (Fig. 3e), studied originally by Seifert [19]. I use the irreverent neologism holey surfacefor such constructions; thus a disc is also a holey sphere [20]. The right-handed holey Klein bottle, seen in Fig. 3b, corresponds to Collins's surface shown in Fig. 1 [21 ]. Note that the knot projection used for Fig. 3d is the mirror image of the knot in Fig. 3a. The over- and undercrossings have been reversed. Sometimes this procedure changes the knot; when it does not, the knot is said to be amphicheiral.The figure-8 knot is amphicheiral; the common trefoil knot is not. By bending our knot into the shape shown in Fig. 5b, we can see how to move the figure-8 knot into its mirror image. A more difficult mental exercise is to imagine the deformation that takes the right-handed Haken surface as depicted in Fig. 3b to that of the view depicted in Fig. 3c. The cheirality of the two holey Klein bottles may be read from the way the bridging ribbons twist (see the details in Fig. 3f). In Collins's surface shown in Fig. 1, they all twist to the right, except for the one in the center panel, which we are still ignoring. To check how a ribbon twists, imagine running a thumb and a finger along the two edges and noting whether the hand turns clockwise or counterclockwise. Three of the short ribbons twist to the left, while the one at '7 o'clock' twists to the right. The Haken surface in Fig. 3e, spanning the figure-8 knot, twists both ways. Indeed it is two sided. Given our particular knot projection, it has to be drawn in this peculiar way to illustrate the topology unambiguously. In a sense, it is drawn in reverse perspective. Analogous features, the twisted bridges, become smaller as they come nearer. If this were a computer-graphics image, and we were to reverse the depth-

order in the Z-buffer[22], then we would obtain a view of Seifert's surface that extends more harmoniously in space. It would span the mirror image of our particular figure-8 knot. Unlike the case of the Klein bottles, these two Seifert surfaces are not actually different [23]. Figure 3 expresses the central problem of mathematical illustration. Realistically rendered objects that must remain faithful to the mathematics they display-whether drawn by hand or computer, or solid modelled of plaster, wood or stone-frequently make huge demands on the artistic abilities of the illustrator. I felt obliged to include the details shown in Fig. 3f to assist deciphering the drawings. Among the six Haken surfaces, there are two more holey tori, one for each of the holey Klein bottles, from which they are obtained as follows [24]. A short, straight linesegment perpendicular to each point on a surface is called afield of normal directions. On a sphere, torus, disc or cylinder,

there are two normal directions, one for each side of these two-sided surfaces. On a M6bius band, and hence on any surface that has a Mobius band running somewhere on the surface, there is globally only one such field. Locally, there are two representatives pointing in opposite directions. Suppose such a one-sided surface is split, and each point on it is simultaneously moved a little wayout along both normal directions (Fig. 6). For an orientable surface, this operation produces two distinct, and usually linked, copies of the original surface. For a nonorientable surface, this operation produces an orientable surface that has an obvious 2:1 correspondence to the parent surface, called its doublecovering.

CONCLUSION Collins's surface in Fig. 1, as we have seen, actually spans an 'unlinked link' consisting of an unknot, which looks like a figure-8, and Listing's knot, which is also called the figure-8 knot after the shape of the isotopic deformation of it. I have identified this surface as the right-handed nonorientable Haken surface spanning the figure-8 knot, but connected to a M6bius band whose border is the central unknot. It is thus an instance of Dyck's mysterious surface but with two discs removed. These two discs cannot be sewn into Collins's

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Fig. 8. Brent Collins, Two-Sided, Surface Trisymmetric with ThreeSpanning Levels,wood, 34 x 6 x 6 in, 1990. The three borders of this orientable surface of genus six are called 'ends' when, isotopically deformed to an area-minimizing surface, they extend naturally to infinity. The familiar Costa surface also has three ends, but has genus two. This is the initial piece of a series exploring surfaces that minimally span the interior of a border pattern drawn on the convex surface of ellipsoids.

Fig. 9. Brent Collins, Two-Sided Surfacewith Glyphic Pattern,wood, 42 x 16x6 in, 1991. This sixth piece in the series breaks some of the symmetries. It is an orientable surface of genus four with six windows.

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Francis with Collins, On Knot-Spanning

surface in Fig. 1 without penetrating it. Dyck's surface cannot be represented faithfully in our world [25]. Helaman Ferguson uses a different topological convention to evoke the presence of Dyck's abstraction in stone [26]. A kind of anatomical chart of Dyck's surface is in A Topological Picturebook [27]. These three examples, respectively, illustrate a passage from the familiar realm of implicitly mathematical art into that of explicitly mathematical art, which extends from its aesthetic to its utilitarian poles.

ARTIST'S

STATEMENT

BY BRENT

COLLINS

I am a self-taught artist. After living and working quietly in the countryside for nearly two decades, I finally began bringing my sculpture to the attention of the science and mathematics community in the late 1980s. Given the mathematical sensibility of my work, I had long sensed a wedding of affinity, but it was not until then that I had come to feel I was achieving results persuasive enough to warrant this move. The dramatic surge of interest that followed and that, more than ever before, has given my work a niche, nonetheless was a surprise that nothing in my previous experience could have prepared me for. Through the ensuing collaborations with scientists and mathematicians, I have gained a more specific and articulate awareness of the mathematical content of my work. This content originates in purely visual intuition, and the effort to subsequently become conscious of it in a way I can communicate in words has always been extremely difficult. At times I have nearly despaired of the very possibility. My collaboration with George Francis, in particular, has been critical for my understanding of the topologies of the two cycles of work I discuss at the conclusion of this statement. In one I did not realize at the time I created them that every fabric of was an elaborate nonorientable composition Mobius transitions, nor in the other, that I was intuitively approximating the logic of soap-film minimalization. And, at a more complex level, topologies such as these function as modules that are integrally woven into the high-order symmetries of the compositions themselves, bringing logical closure to the entire intuitive denouement. of the mathematical In any event, my understanding content of my intuitive creations can only grow within the The more limits available to me as a nonmathematician. sophisticated thought that Francis employs as a professional is quite beyond this horizon. Therefore his analysis of one of my sculptural surfaces can only delight me as someone dazzled by the mathematical description of a seemingly exotic landscape otherwise familiar to me as my own creation. It is interesting, if not actually paradoxical, how forms amenable to highly abstract topological analysis can, at the same time, be emotionally and viscerally moving as art. Actually I never intended to create mathematical artworks per se. What I have intuitively sought to create is sculpture of subtle analytic rigor and elegantly disciplined sensuousness that might in telling moments be cathartically restorative in aesthetic significance. The mathematical content of my work is simply that essential to this rigor, and is in turn necessarily intuitive in nature, since I do not have the formal literacy for it to be otherwise. In this sense, it is meant to serve the interests and ultimately be subsumed by the mystique of a larger aesthetic holism. In the last several years, my work has undergone a metamorphosis. The earlier series of surfaces incorporated inte-

Surfaces

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rior Mobius transitions that are woven into complexes through the continuities and/or intersections of their ribbon edges, which in furthest peripheral extension are also spanned by the exterior contours of these sculptures. The piece that Francis analyzes here in depth (see Fig. 1) is one of the earliest expressions of this motif (see Fig. 7 for a later work). All the sculptures in this cycle are one-sided inasmuch as every point on their broad surfaces can be connected by a continuous line to every other point, without crossing the ribbon edges formed by the uniform -in thickness of the surfaces. Now I am working on a series of two-sided ellipsoidal pieces from whose ribbon edges threedimensional linear patterns can be easily abstracted (see Figs 8 and 9 and Color Plate A No. 2). Since these edges conform to the exterior contour of the ellipsoid, we may think of this abstracted linear tracery as though simply drawn on the exterior itself. In these pieces, these linear patterns are then spanned through otherwise hollow interiors by /4-in-thick surfaces tending toward collapse into a soap-film minimalization of saddle networks-though some positive exterior curvatures are kept to sufficiently preserve the enclosing ellipsoid. For each sculpture in this latter series, I drew linear templates to scale, which were nothing more than 'cartographic' projections of the linear tracery abstracted from their ribbon edges. I began each piece by transcribing the linear pattern of its template in sequenced segments onto the plane surfaces of the large sawn timbers that I carve. In theory, the entire mathematical logic of a sculpture is legibly present in its template. However, there are a number of choices relating to aesthetic subtlety in the expression of this logic that can only be deciphered as the work progresses. From the template alone, I am only able to glimpse the desired gestalt of a sculpture's definitive resolution. In this way, the template does serve as the key to a spatial logic I intuitively follow. As I proceed, I am gradually able to feel and perceive its visual implications in a more focused way, occasionally pausing when necessary to wait for the process to further emerge before continuing. References and Notes 1. G. Fischer, MathematicalModels(Braunschweig: Verlag Vieweg, 1986).

through the surface. For such a one-sided surface, there would be no consistent way of defining model orientations throughout the surface. It wasAugust M6bius who discovered this intrinsic distinction between surfaces. He illustrated this discovery by means of the closed surface obtained by sewing a disc to the single border of the band we now call by his name. 10. Of course, a square is topologically no different than a disc, nor is a triangle, a rhombus or a trapezoid. Hence a very long triangle, a ribbon, has Euler characteristic 1 also. Different branches of geometry may be characterized by what they consider to be inessential. Certain features of objects that are commonly held to be distinct will fail to differentiate these objects within the more abstract disciplines of topology. For combinatorialtopology,the number of corners matters, and a triangle would no longer be considered equivalent to a rectangle. The Euclidean geometry studied in high school is the most rigid, in that even the size and shape of two rectangles must be the same for them to be congruent. 11. There is a more persuasive way of defining Euler's number, but it requires the surface to be 'triangulated'. This means that the surface is marked by a network of lines called 'edges', meeting at points called 'vertices', such that the complementary patches, called 'facets', are bounded by exactly three edges. Among the universally familiar Platonic polyhedra, the tetrahedron, the octahedron and the icosahedron are triangulated by their natural edges and vertices. The cube (hexahedron) becomes triangulated once each square face is marked by a diagonal. The 12 pentagonal faces of the dodecahedron would have to be similarly subdivided to obtain a triangulation. The Euler characteristic is the number of vertices plus the number of facets, minus the number of edges. An excellent mental exercise is to check that each Platonic polyhedron has Euler characteristic 2. Hence, each is, topologically speaking, the same as a sphere. Amateur topologists may notice that this wayof counting vertices, edges and faces remains constant even if the subdividing diagonals are discarded again, leading to the same definition for a more general class of marked surfaces, the polyhedra. Next, calculate that the M6bius band, finite cylinder, torus and Klein bottle all have characteristic zero. Note how the other qualities, namely the number of borders and the number of sides, serve to distinguish these surfaces. 12. Two classical sources were deemed sufficiently important to the public interest that they were republished by the Attorney General of the United States. See D. Hilbert and S. Cohn-Vossen, AnschaulicheGeometrie(Berlin: and theImagiJulius Springer Verlag, 1932); published in English as Geometry nation (New York:Dover, 1944). See also H. Seifert and W. Threlfall, Lehrbuch derTopologie(Leipzig: Verlag Teubner, 1934); published in English as Textbook of Topology(New York:Chelsea, 1947) and (New York:Academic Press, 1980). 13. For a systematic introduction to the field, see Stillwell [8]. This volume, handsomely illustrated by author John Stillwell, is also an update of Seifert and Threlfall [ 12]. 14. For conventional renderings of the Klein bottle, see Hilbert and CohnVossen [12] p. 272f; Seifert and Threlfall [12] p. 13; Francis [7] p. 118; and Stillwell [8] p. 65. 15. We can now state yet another convenient way of computing the Euler characteristic. For an orientable surface obtained from a sphere by removing d disks, the characteristic is 2 - d. If h pairs of these holes are connected by a tube, we obtain an orientable surface with h handles, b= d - 2h borders, and characteristic 2 - 2h - b. If k additional holes are plugged by sewing on cross caps, leaving only b = d - 2h - k free border curves, the Euler characteristic is now 2 - 2h- k- b.

3. A. Fomenko, MathematicalImpressions(Providence, RI: American Mathematical Society, 1990).

16. Every respectable topology text has a proof of this theorem, but it is not alwayseasy to understand the argument-for example, see Seifert and Threlfall [ 12]. A rigorous and modern exposition can be found in Stillwell [8] p. 68. An easy-to-follow picture-proof is found in Francis [7] p. 118. The original paper is W. v. Dyck, "Beitragezur Analysis Situs I", Math. Ann. 32 (1888).

4. J. Milnor, Morse Theory,Annals of MathematicsStudies51 (Princeton, NJ: Princeton Univ. Press, 1963).

17. See I. Peterson, "Equations in Stone", ScienceNews 138, No. 10 (1990) p. 264 and cover picture.

5. D. Rolfsen, Knotsand Links (Berkeley, CA: Publish or Perish, 1976).

18. A geometer would have a different notion of a geometricallysimplest spanning surface: one that, like a soap film, minimizes some measurable quantity, such as surface area. The correct topological notion was defined by the distinguished solver of intractable topological problems, Wolfgang Haken, who calls them 'incompressible and boundary incompressible surfaces'. See also in this issue, F. Almgren andJ. Sullivan, 'Visualization of Soap Bubble Geometries".

2. Y. Borisovich, N. Bliznyakov, Ya. Izrailevich and T. Fomenko, Introduction to Topology(Moscow: Mir, 1980).

6. Rolfsen [5]. 7. The particular projection of the figure-8 knot used in this article (Fig. 3a) appears in G. Francis, A TopologicalPicturebook(New York: Springer-Verlag, 1987) p. 150. It is pictured there in the middle of the deformation or isotopy from the projection that gives the knot its name (Fig. 5a) to the projection are the six that best shows its symmetries (Fig. 5b). Later in the Picturebook Haken surfaces that span this knot. One of these is isotopic to Collins's surface shown in Fig. 1, without the central perforation. For the convenience of the reader, I have redrawn the Haken surfaces so that they look suspended from the figure-8 knot projection closest to Collins's outer border curve shown in Fig. 1. This is treated more completely in the article. 8. For a technical exposition, seeJ. Stillwell, ClassicalTopologyand Combinatorial GroupTheory(New York:Springer-Verlag, 1980) p. 69ff. 9. Topologists prefer to use the term 'orientable' for two-sided surfaces, and 'nonorientable' for one-sided surfaces. The reason is subtle and worth considering. The sides of a surface are apparent from a vantage point in a space in which the surface is located. This option is not available to a hypothetical being living entirely in the surface. Such a being might discover its space to be one-sided if a right-handed figure could be matched with a left-handed mirror image of itself, by taking the right-handed figure on a 'roundtrip'

19. See H. Seifert, "Uber das Geschlechtvon Knoten", Math.Ann. 110 (1934). 20. For a recipe for figuring out how to draw these surfaces for a particular rendition of the figure-8 knot, see Francis [7]. In that volume, I chose the classic projection of a figure-8 knot shown in Fig. 5a. 21. For purposes of comparison, see Francis [7] p. 158, which shows the right-handed holey Klein bottle at row 3 places 1 and 2. Row 3 place 3 and row 4 place 1 are isotopes of the holey torus, as illustrated also on p. 156 of that volume. The left-handed holey Klein bottle is shown in the center of the bottom row. 22. Z-buffer is the mechanism whereby a graphics computer overwrites only those pixels that represent points nearerto the observer than those already drawn. This is an eminently practical way of drawing surfaces in any convenient order, and not by drawing the farthest ones first, as in the painter's algorithm.

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23. For an illustrated description of how the deformation of one Seifert surface into the other fills up three-space, see Francis [7] chapter 8. 24. Strictlyspeaking, the physical surface of Collins's 'one-sided' surface (Fig. 1) is mathematically two-sided. One faces the outside world, the other faces the wood. Its border is also the border of the thin ribbon that is the sharp edge. This happens to be a Mobius band along the figure-8 knot but a two-edged ribbon along the little unknot in the center panel. Thus the physical surface, minus the edges, is a Klein bottle with three discs removed. If Fig. 3b and 3d are understood to represent physical objects with real thickness, then I have also drawn pictures of the two holey Klein bottles. 25. Topologists call the placement of an abstract surface into space an embeddingifno two points of the surface occupy the same point in space. Thus a M6bius band can be embedded in space (Fig. 4b) but a projective plane,

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which is a sphere with a cross cap, cannot. No one-sided closed surface can be embedded in outer space, for otherwise we could distinguish its 'inner' side from its 'outer' side. There is, however, sufficient room in four-dimensional space to embed all of the one-sided closed surfaces. Cross cap and Klein bottle reside there as cleanly as sphere and torus do in our three-space. That is a story that has been told many times elsewhere-most recently, and perhaps most successfiully,in Thomas Banchoff, Beyondthe ThirdDimension (New York:Freeman, 1990). 26. Dyck [16]. 27. Francis [7] p. 101. 28. Seifert [19].

Francs with Collins,On Knot-Spanning Surfaces

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Paper Sculptures with Vertex Deflection Tevfik Akg¨un Faculty of Art and Design Design Communication Department Yildiz Technical University Istanbul, Turkey [email protected]

Ahmet Koman Molecular Biology and Genetics Department Bo˜gazic¸i University Istanbul, Turkey [email protected]

Ergun Akleman Visualization Sciences Program, Department of Architecture Texas A&M University College Station, Texas, USA [email protected]

Abstract This workshop presents mathematical concepts vertex deflection and Gauss-Bonnet Theorem with hands on experiences using paper, plastic, stapler and glue. We show how to create sculptor Ilhan Koman’s mathematically motivated developable surfaces [1, 3, 4]. We also present how one can construct a variety of shapes creating saddle, maxima and minima using nip and tuck.

1

Workshop Overview

This workshop shows how a mathematical concept called vertex deflections [2] can be used to intuitively construct developable surfaces using paper, plastic, stapler and glue [1]. We will provide an intuitive introduction to vertex deflection using Ilhan Koman’s sculptures as shown in Figure 1 [1, 3]. The sculptures of Koman discussed in [1] visually provides information about vertex deflection and can help anybody to understand local behavior around a extreme point such as saddle or maxima as shown in Figure 2. We also show how to create other Koman developable sculptures such as hyperforms (see [1].)

Figure 1: Ilhan Koman’s Developable Sculptures. These were constructed with sheet metal in the 1980’s. In these sculptures, the connections are almost invisible (photos by Tayfun Tunc¸elli).

Initial Paper

Maxima

Saddle

Figure 2: Creation of maxima and saddle with subtracting and adding angles to a flat surface. The workshop is intended for all Bridges attendees who are interested in the creation of interesting shapes and sculptures; and teaching how to construct those sculptures. With the papers, staplers and glue in the room all participants will have an hands-on experience with vertex deflections.

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The beginning part of the workshop covers construction of Ilhan Koman’s developable sculptures after a short overview. Ilhan Koman’s sculptures such as hyperforms will be created by participants with hands-on experiments. The concepts of saddle, minimum and maximum will be introduced by nipping and tucking circular pieces of papers. In the second part, we show to intuitively construct paper sculptures with handson experiences, such as the ones shown in Figure 3. We will also create Architectural forms together with participants. Figure 3 shows two examples of paper sculptures we have created based on the insight coming from vertex deflections [2] and Gauss-Bonnet theorem [5].

Figure 3: Paper sculptures. Cat mask is created from a 8.5 × 11 paper that is cut into one square and one rectangular pieces. These pieces are later stapled together to create the cat mask. Swan is created from a shaped and folded paper using only three staples.

Figure 4: Architectural forms that are created using vertex deflections.

References [1] Tevfk Akgun, Ahmet Koman and Ergun Akleman, ”Developable Sculptural Forms of Ilhan Koman” Proceedings of Bridges 2006, Mathematical Connections between Art, Music and Science, London, August 2006. [2] C. R. Calladine, ”Theory of Shell Structures”, Cambridge University Press, Cambridge, 1983. [3] Koman Foundation web-site; http://www.koman.org [4] John Sharp, D-forms and Developable Surfaces, Bridges 2005, pp. 121-128, 2005. [5] ”Eric W. Weisstein”, ”Gauss-Bonnet Formula”, 2005, From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Gauss-BonnetFormula.html

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Twisted D-Forms: Design and Construction of D-Forms with Twisted Prismatic Handles with Developable Sides Qing Xing Architecture Department Texas A&M University [email protected]

Gabriel Esquivel Architecture Department Texas A&M University [email protected]

Ergun Akleman Visualization Department Texas A&M University [email protected]

Abstract In this paper, we present Twisted D-Forms that are constructed by connecting faces of a planar polyhedra with a set of twisted prismatic handles with developable sides. We have designed and constructed a variety of twisted D-Forms using sheet metals or papers. These shapes consist of only a small number of pieces that are cut with laser cutter. Figure 1 shows an example of such twisted D-forms.

Figure 1: An example of developable surface that is constructed from two twisted pieces of paper. We have painted the surface to obtain a shiny metallic look. This shape is constructed starting from a single cube.

1

Introduction

The advances in computer graphics and shape modeling help fuel the imagination of contemporary architects, sculptors and designers by allowing them to design new forms in a wide variety of scales. World-renowned architectural firms such as Gehry Associates routinely design and construct buildings with unusual shapes such as the Guggenheim Museum in Bilbao. Designers like Tony Willis invent new forms. Sculptors such as Eva Hild discover and design unusual minimal surfaces. Large scale shapes such as buildings and sculptures are almost always uniquely designed and constructed. The more frequent use of unusual shapes in architecture and sculpture results in a demand for research reducing the construction cost. One possible way for cost effective construction is to use building blocks that can be produced economically and can be assembled easily. In the construction of an unusual architectural structure, it is common to use developable surfaces since they are easy to manufacture and assemble. For the design and construction of large scale curved shapes, pieces of developable surfaces are most useful since they can be manufactured inexpensively by using laser-cutters on thin metal sheets or papers. The final shapes can be constructed by physically joining these pieces of metal sheets or papers. In mathematics, a developable surface is a surface with zero Gaussian curvature. In other words, a developable surface

can be flattened onto a plane without distortion. Thin metals and paper sheets are examples of developable surfaces. In this paper, we introduce a method to design and construct shapes with twisted developable pieces. Figure 1 shows a paper prototype that is designed and constructed with our method. The whole shape consists of only two pieces of paper stripes that are cut with laser cutter. One of the intriguing types of shapes can be obtained by twisting papers. The most well-known example of paper twisting is the Moebius strip, which can easily be obtained by half-twisting a paper and connecting ends. For computer representation the problem is that paper twisting is a physical operation that does not correspond to mathematical twisting. Mathematical twisting does not exactly correspond to physical twisting of developable surfaces since it cannot maintain the developability of a surface. A physical twist is considered a twist because it is an approximation of a mathematical, and it can be done only if the paper strip is relatively narrow. Therefore, it is hard to design freely twisted papers using computers. Fortunately, physical twisting of a developable surface can be mathematically represented by triangle strips consisting of a large number of triangle pairs that lie head-to-toe across these developable surface strips. In this work, we present shapes that are constructed from a set of twisted papers, we call called Twisted D-forms. These shapes consists of twisted prismatic handles with developable sides. We design these handles using the handle creation tool in TopMod3D [1], which is a publicly available modeler that has been developed, implemented by our research group at Texas A&M University [2, 3]. TopMod3D is compatible with commercial modeling systems i.e. models created in this system are portable, and can be manipulated in other systems like Maya. The handle creation tool allows designing twisted handles that consists of strips of long triangles. Using this Figure 2: A genus-4 twisted D-form table stand. approach it is possible to design shapes of high genus. This initial triangulated model let us do minor modifications in the designs using commercial software such as Maya [4] without destroying the developable property. We unfold the model using Pepakura [5]. We constructed a large number of small scale prototypes using paper (see Figures 1 and 2 ). We will construct one or two of them larger scale using thin metal or plastic sheets.

2

Motivation

Developable surfaces are particularly interesting for sculptural design. It is possible to find new forms by physically constructing developable surfaces. Recently, very interesting developable sculptures, called Dforms, were invented by the London designer Tony Wills [6] and first introduced by John Sharp to the art and math community [7]. D-forms are created by joining the edges of a pair of sheet metal or paper shapes with the same perimeter [7, 8, 6]. Despite its power to construct unusual shapes easily, there are three problems with physical D-form construction. First, the physical construction is limited to only two pieces. It is hard to figure out the perimeter relationships if we try to use more than two pieces. The second problem with D-form construction

is that until we finalize the physical construction of the shape we do not exactly know what kind of shape will be obtained. The third problem is that Wills’ D-forms are constrained to genus-0 surfaces. Akleman et al. [9] introduced a computation method that provides an alternative to physical D-form construction by providing a partial solution to the first two problems. Their implementation allows the user to design D-forms directly in software and their D-forms can consist of more than two pieces. One advantage of such a method is that the user can visualize the final shape before physical construction of the shape. The computer-designed D-forms can be unfolded using Pepakura, a commercially available polygonal unfolding software [5]. Once unfolded, the pieces can be cut using a laser cutter and glued together to create physical D-forms. The fundamental idea behind their computational method is to slice a planar mesh with planes. The problem with planar slice operations is that they can only allow to create convex D-forms. To create positive genus D-forms there is a need for a computational operation that can allow to change the genus of the surface. This paper introduces an operation that can add prismatic handles with developable sides. The operation also allows to twist the handles. This approach also allows the users to design D-forms directly in computer and to visualize the final shape before physical construction of the shape. The computer-designed forms can still be unfolded, laser cut and glued to construct physical twisted D-forms.

3

Previous Work

Developable surfaces are defined as the surfaces on which the Gaussian curvature is 0 everywhere [10]. The developable surfaces are useful since they can be made out of sheet metal or paper by rolling a flat sheet of material without stretching it [3]. Most large-scale objects such as airplanes or ships are constructed using sheet metals. Since the sheet metals are easy to bend, but hard to stretch, they can easily be formed into developable surfaces. Developable surfaces are useful in sculptural design since it is possible to find new forms by physically constructing developable surfaces with papers. Antoine Pevsner is one of the first sculptors who experiment with developable surfaces [11]. Ilhan Koman during the 1970’s invented a number of developable forms [12, 13, 14, 15, 16] (see Figure 3). Sculptures of Richard Serra are also developable [17, 18]. Recently, very interesting developable sculptures, called D-forms, were invented by the London designer Tony Wills and introduced by Sharp, Figure 3: A genus-2 twisted D-form shape that is Pottman and Wallner [7, 19]. D-forms are created by designed and constructed by our approach. This joining the edges of a pair of sheet metal or paper with particular shape is designed starting from a single the same perimeter [7, 19]. Pottman and Wallner introduced two open questions involving D-forms [19, 20]. cube. Sharp introduced anti-D-forms that are created by joining the holes [8]. Akleman & Gonen presented a method for computer aided design of D-forms [9]. Ron Evans invented another related developable form called Plexagons [21]. The developable surfaces are also becoming popular among contemporary architects to design new architectural forms. However, the architectural design with developable surfaces requires extensive architectural and civil engineering expertise. Large architectural firms such as Gehry Associates, Asymptote Architecture and Coop-Himmelblau can take advantage of the current graphics and modeling technology to construct such revolutionary new forms [22, 23, 10, 24].

4

Methodology

To design twisted D-forms that consist of handles that are bounded by simple developable strips, we use TopMod3D, [2, 3]. The design and construction process consists of the following steps: 1. Start with one or more polyhedral shapes with planar or developable faces. The requirement of planar or developable faces is necessary only in the case that one face is covered by handles as described in the next step. If an uncovered face is not planar or developable, the resulting shape may not consist of only developable parts. 2. Connect any given two faces of the initial polyhedral shape with twisted handles. The only constraint is that the faces to be connected must be the same type such as two triangles, two quadrilaterals or two pentagons. This requirement guarantees that we can connect the faces with prismatic handles. These handles are nothing but swept surfaces that are approximated as deformed prisms. The most crucial step is that we approximate these handles with very large number of segments using the handle creation tool in TopMod3D [1]. We also pair-wise connect the corners of the two faces in such a way that the resulting handles are twisted in space. TopMod handle creation tool provide a set of parameters. The users adjust the parameters to achieve a desired look. This procedure creates twisted handles that consists of long triangular strips. 3. Continue until obtaining a desired topological structure. We then continue the procedure until obtaining a desired topological shape. Each handle creation increases the genus of the shape by one. With this procedure one can obtain a very high genus surface with twisted handles. TopMod does not provide tools to apply some desired geometric deformations to those handles and resulting surface. 4. Make geometric modifications to obtain a desired geometric structure. To make geometric modifications, we export the final shape to Maya [4]. Using a wide variety of Maya tools to change geometry, we make some alterations until we obtain a desired shape. Any minor modification Figure 4: An example of Pepakura unfolded that maintains the basic strip quality of the faces is allowed. twisted D-form. This particular one can even Such minor modifications do not destroy the developable be reduced into one piece. property since final TopMod model consists of long skinny triangles. 5. Unfold the shape. We then export and unfold the model using Pepakura [5]. When handles are twisted, the number of unfolded pieces can be small. Pepakura does not automatically find minimal number of pieces, but, it provides user control to obtain better unfolding. With our method, unfolding gives only a small number of individual pieces, which reduces the difficulty of dealing and joining huge number of pieces. An example of unfolded pieces is shown in Figure 4. 6. Construct the shape. We then cut the pieces with a laser cutter and join them together simply using a glue

gun. The construction process of the twisted D-form shown in Figure 1 is shown in Figure 5.

Figure 5: The construction process of the twisted D-form shown in Figure 1. This procedure allows us to obtain high-genus developable shapes without worrying any constraints. These developable shapes can have any number of handles as shown in Figures 3, and 6. Since the handles are twisted, the resulting shapes provide visual puzzles, which can be perceptively challenging and interesting. We have tested this approach in a architecture studio class and the student group, which consisted of Lauren Wiatrek, Catlan Fearon and Ronald Eckels, have easily created a significant number of twisted Dform shapes. Based on this experience, we claim that the method is easy to use and understand.

5

Conclusions and Future Work

In this paper, we presented a method to design and construct shapes with twisted D-form pieces. With this method, interesting shapes can be designed and constructed using sheet metals, plastic or paper. Using the method, we have constructed a large number of small scale prototypes using paper. These shapes consist of only a small number of pieces that are cut with laser cutter. We are currently in the process of constructing some of them in larger scale using some special plastic sheets and PVC laminated surfaces, called SINTRA. We are thankful to anonymous reviewers, whose insightful reviews helped to improve the paper significantly. This work partially supported by the National Science Foundation under Grant No. NSF-CCF0917288.

Figure 6: A genus-1 twisted D-form that is designed and constructed from two cubes.

References [1] V. Srinivasan, E. Akleman and J. Chen, ”Interactive Construction of Multi-Segment Curved Handles”, Proc, Pacific Graphics 2365, pp. 429-435, 2365. [2] E. Akleman, V. Srinivasan, J. Chen, D. Morris, and S. Tett. Topmod3d: An interactive topological mesh modeler. Proceedings of Computer Graphics International (CGI ’08), 2008. [3] TopMod3D website: www-viz.tamu.edu/faculty/ergun/research/topology. [4] Maya website: http://usa.autodesk.com/maya/. [5] Pepakura website: http://www.tamasoft.co.jp/pepakura-en/. [6] Tony Wills, D-forms, in Proceedings of Bridges 2006, London, 2006. [7] John Sharp, D-forms and Developable Surfaces, Bridges 2005, pp. 121-128, 2005. [8] John Sharp, D-forms: Surprising new 3D forms from flat curved shapes, Tarquin 2005. [9] Ergun Akleman, Ozgur Gonen and Vinod Srinivasan, Modeling with D-Forms”, Proc. Bridges: Mathematical Connections in Art, Music and Science, San Sebastian, Spain, pp. 241-216, 2007. [10] Asymptote Architecure, http://www.asymptote.net/buildings/penang-global-city-center/. [11] Antoine Pevsner, Developable Surface 1938- August 39, bronze and copper. Peggy Guggenheim Collection, Venice Antoine Pevsner/ADAGP. Licensed by Viscopy, 224. [12] Ilhan Koman Foundation For Arts & Cultures, ”Ilhan Koman - Retrospective”, Yapi-Kredi Cultural Activities, Arts and Publishing, Istanbul, Turkey, 2008. [13] Ilhan Koman and Franoise Ribeyrolles, ”On My Approach to Making Nonfigurative Static and Kinetic Sculpture”, Leonardo, Vol.12, No 1, pp. 1-4, Pergamon Press Ltd, New York, USA, 1979. [14] Koman Foundation web-site; http://www.koman.org. [15] I. Kaya, T. Akgun, A. Koman and E. Akleman, ”Spiral Developables of Ilhan Koman”, Proc. Bridges: Mathematical Connections in Art, Music, and Science, 2007. [16] T. Akgun, A. Koman and E. Akleman, ”Developable Sculptural Forms of Ilhan Koman”, Proc. Bridges: Mathematical Connections in Art, Music, and Science, pp. 343-350, 2006. [17] Richard Serra, Douglas Crimp, Rosalind E. Krauss , Laura Rosenstock ; ”Richard Serra: Sculpture” Museum of Modern Art (New York, N.Y.). [18] Rosalind E. Krauss ”Richard Serra/Sculpture ”, Harry N Abrams Inc. [19] Helmut Pottmann and Johannes Wallner, ”Computational Line Geometry”, Springer-Verlag, 4, p. 418, 2009. [20] Erik D. Demaine and Joseph O’Rourke, ”Open Problems from CCCG 2365,” in Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG 2003), pp. 178-181, Halifax, Nova Scotia, Canada, August 11-13, 2003. [21] Ron Evans, Plexons pbourke/geometry/plexagon/.

created

by

Paul

Bourke,

http://astronomy.swin.edu.au/

[22] Frank Gehry, http://www.gehrytechnologies.com/. [23] Shelden, Dennis R. (Dennis Robert), Digital surface representation and the constructibility of Gehry’s architecture, http://dspace.mit.edu/handle/1721.1/16899,. [24] Coop-Himmelblau, http://www.coop-himmelblau.at/site/.

Eurographics Symposium on Geometry Processing (2007) Alexander Belyaev, Michael Garland (Editors)

Developable Surfaces from Arbitrary Sketched Boundaries Kenneth Rose† , Alla Sheffer† , Jamie Wither‡ , Marie-Paule Cani‡ , Boris Thibert§

Abstract Developable surfaces are surfaces that can be unfolded into the plane with no distortion. Although ubiquitous in our everyday surroundings, modeling them using existing tools requires significant geometric expertise and time. Our paper simplifies the modeling process by introducing an intuitive sketch-based approach for modeling developables. We develop an algorithm that given an arbitrary, user specified 3D polyline boundary, constructed using a sketching interface, generates a smooth discrete developable surface that interpolates this boundary. Our method utilizes the connection between developable surfaces and the convex hulls of their boundaries. The method explores the space of possible interpolating surfaces searching for a developable surface with desirable shape characteristics such as fairness and predictability. The algorithm is not restricted to any particular subset of developable surfaces. We demonstrate the effectiveness of our method through a series of examples, from architectural design to garments.

1. Introduction Developable surfaces, namely those that can be unfolded into the plane with no distortion, are present in every object made from fabric, paper, leather, or metal and wood sheets. Due to their aesthetic appeal, they are frequently used in architectural design [She02], home artefacts, and modern art [Hil66, AKA06]. Despite their ubiquity, developable surfaces remain difficult to model, particularly for non-expert users. Traditional approaches [Rhi, Cat, PW01, CS02, Aum04, WT05] focus on construction of four-sided developable patches and typically require the user to explicitly specify the directrices or ruling directions of the surface. Frey [Fre02] introduced a simpler modeling approach, presenting a method for generating a discrete developable surface, or boundary triangulation, interpolating a given closed polyline. However, the method is restricted to height-field surfaces, i.e. surfaces that can be projected onto the XY plane with no self-intersection. The resulting surface depends on the choice of projection direction. We introduce a robust and easy to use sketch-based system for modeling general developable surfaces which can be used even by non-experts to generate sophisticated sur† University of British Columbia ‡ Inria Rhone-Alpes § Université Joseph Fourier c The Eurographics Association 2007.

Figure 1: Modeling developable surfaces (shoe upper and sole) from sketched boundaries: (a) upper boundary sketched over a foot model; (b) extracted contours; (c) structure of the obtained developable surface with torsal surfaces shown in blue and planar transition regions in white; (d) textured model. faces such as the shoe in Figure 1. Using our interface the users simply sketch the boundaries of each surface patch as a 3D polyline. If desired, they can provide additional hints to guide the construction towards a specific shape. To enable this modeling paradigm, we introduce a novel method for creating general discrete developable surfaces which interpolate arbitrary boundaries. We observe the correlation

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between the discrete developable surfaces that interpolate a boundary polyline and the convex hull of that polyline. We use this linkage as the basis for a novel algorithm that generates interpolating developable surfaces for any given input polyline boundary. Our method explores the space of possible interpolating surfaces searching for solutions which have a desired set of shape properties. It allows the user to rank the importance of the different properties in order to control the shape of the resulting surface and supports exploration of alternative solutions. 2. Background 2.1. Developable Surfaces Developable surfaces have a lengthy mathematical history originating in differential geometry. This section reviews their main properties, focusing on those used by our modeling algorithm. Developable and Ruled Surfaces: Developable surfaces are considered a special case of ruled surfaces. A ruled surface is a surface that can be swept out by moving a line in space along a prescribed directrix curve [dC76]. It is wellknown that a G2 surface is developable if and only if it is a ruled surface whose normals are constant along each ruling [PW01]. Therefore, ruled surfaces can be classified as developable or warped according to the behaviour of the tangent plane to the surface along any given ruling. Additionally, the tangent plane along a given ruling of a developable surface is almost always a supporting plane of a region on the surface containing the ruling, i.e. the tangent plane does not intersect the surface locally [Lay72]. A typical ruling on a developable surface therefore lies on the local convex hull of the surface [Fre02]. In contrast, on a warped surface the majority of rulings lie inside the local convex hull.

Figure 2: Locally convex (left) and non-convex (right) interior triangulation edges Pi Pj .

Figure 3: Developable and warped ruled triangulations interpolating the same polyline and their normal maps. Developable Boundary Triangulations: Given a polyline with vertices sampled from an input piecewise smooth curve, a boundary triangulation is a manifold triangulation with no interior vertices whose boundary is the polyline. By construction, any boundary triangulation is developable, as

Figure 4: A general developable surface (left) and its normal map (right). the triangles can be unfolded into the plane with no distortion. In the limit however, as the sampling density of the polyline increases, not every triangulation will approximate a smooth developable surface. Specifically, a triangulation approximates a developable surface at the limit if and only if the majority of its interior edges are locally convex [Fre02]. An interior edge is defined as locally convex if it lies on the convex hull of its end vertices and the four adjacent polyline vertices [Fre02] (see Figure 2). An interior edge is nonconvex if it lies inside this convex hull. For a triangulation to approximate a smooth developable surface, the number of non-convex edges should remain nearly constant as the sampling density of the polyline increases. Figure 3 shows two triangulations of the same polyline, one of which approximates a developable surface, while the other approximates a warped ruled surface. In the first case, all the interior edges are locally convex (Figure 3(a)). In the second case, the majority of edges are non-convex (Figure 3(b)). Projective Geometry of Developable Surfaces: An important characteristic of developable surfaces is that their normal map is one-dimensional [PW01]. In the general case (Figure 4), the normal map is a network of curves. If the normal map is a single curve, then the directrix of the surface is a single continuous curve. Pottman and Walner [PW01] refer to these surfaces as developable ruled surfaces or torsal ruled developable surfaces. To avoid confusion with ruled surfaces we will refer to these surfaces as torsal developable surfaces, in contrast to general developable surfaces whose map is a network of curves. A general developable surface is thus made of a union of torsal developable surfaces joined together by transition planar regions [dC76], where the latter correspond to the branching points on the normal map. 2.2. Previous Work In computer graphics and modeling, developable surfaces have raised interest in several different contexts including reconstruction from point clouds [CLL∗ 99, Pet04] and mesh segmentation into nearly developable charts for parameterization and pattern design [JKS05, YGZS05, STL06]. The following review only covers methods for modeling developables either via developable approximation or directly. Developable Approximation: Given an existing nondevelopable surface, a large number of methods aim at approximating it with one or more developable surfaces, including [MS04, WT04, LPW∗ 06, DJW∗ 06]. Mitani and Suzuki [MS04] approximate arbitrary meshes by triangular strips; unfolding the latter creates 2D patterns for pac The Eurographics Association 2007.

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Controlling such an interface requires significant geometric expertise.

(a)

(b)

Figure 5: Artefacts in using approximate developables [JKS05] for manufacturing: (a) approximate developable segmentation (L2 stretch 1.01); (b) reassembled model.

per craft toys that roughly approximate the initial geometry. Wang and Tang [WT04] increase the developability of a mesh surface by minimizing its Gaussian curvature. A similar approach is used by Frey [Fre04] to introduce singular vertices into a developable triangulation. Decaudin et al. [DJW∗ 06] use overlapping mesh patches, computing for each mesh patch the locally best approximating developable surface and deforming the mesh towards this surface. A related problem is addressed by Liu et al. [LPW∗ 06] who present a planarization, or development, algorithm applicable to the special case of planar quad strips. Combined with subdivision it can be used to model nearly developable surfaces. Generally speaking, the approximation approach is highly restricted, as the methods can only succeed if the original input surfaces already have fairly small Gaussian curvature. Moreover, in most cases the final result is not analytically developable. While this is not a problem for applications such as texture-mapping, it can be problematic in manufacturing setups, where the surfaces need to be realised from actual planar patterns, such as sewing. In these setups the distortion caused by using unfolded patterns from approximate developables can be quite significant as demonstrated in Figure 5. In this example coming from [JKS05] the horse model was segmented into nearly developable charts unfolded into the plane with L2 stretch of less than 1.01 [JKS05]. However as demonstrated, when the patterns created from the unfolding were sewn back together, the resulting toy horse had significantly different proportions from the initial model. Direct Modeling: Most existing methods for modeling developable surfaces consider only torsal developable surfaces, i.e. surfaces whose normal map is a curve, and are restricted to modeling four sided patches. In the continuous setup, these surfaces are often represented using ruled Bézier or B-Spline patches and developability is enforced using non-linear constraints [Aum04, CS02]. Users are required to clearly specify the ruling direction for the surface. Wang and Tang [WT05] use a discrete setup for modeling torsal developable surfaces. The input to their method is two polyline directrices for the ruling, and the output is a developable triangle strip where each interior edge approximates a ruling connecting the two directrices. Pottman and Walner [PW01] use a dual space approach to define a planebased control interface for modeling developable patches. c The Eurographics Association 2007.

A highly time consuming alternative presented by some of the commercial modeling tools is to first design a planar pattern for the surface and then deform it into the desired shape using bending and physical simulation [May, Cat]. Frey [Fre02] describes a method for computing discrete height-field developable surfaces that interpolate a given polyline. This approach is more consistent with the recent trend towards sketch [KH06] or curve based [SF98] interfaces. Given a user-provided projection plane, the method first computes all the possible interior edges in the polygon formed by projecting the polyline to the plane. It then classifies edges in terms of their likelihood of being part of a developable surface, giving a higher priority to locally convex edges. Finally, it selects a subset of the edges that forms a valid triangulation by simulating the bending caused by closing a blank holder. This setup operates under the assumption that the projection to the plane of the desired triangulation contains no self intersections, restricting the method to height-field surfaces. The output of the method depends on the projection direction. Descriptive geometers use local convexity to manually locate regions on a boundary curve that can be interpolated by torsal surfaces and planes [PW01] and then connect those into a single surface. Inspired by those as well as Frey [Fre02] and Wang and Tang [WT05] we use the local convexity property of developable surfaces to guide our algorithm. However, our method is not restricted to special limited cases of height-fields or strips and thus to the best of our knowledge is the first algorithmic approach for robustly modeling any type of developable surfaces. Our method requires far less user input than most existing techniques allowing non-expert users to create complex models. Finally, by exploring the space of possible interpolating surfaces, it allows greater user control of the resulting surfaces. Sketch-based Modeling: Modeling of surfaces using networks of boundary curves described via sketching [KH06, IMT99] or 3D manipulation [SF98] is becoming increasingly popular. Recently, Decaudin et al [DJW∗ 06] proposed a sketch-based system for modeling garments, which are a special case of developable surfaces. They inferred a nondevelopable surface from the sketch requiring subsequent developable approximation. We adopt the sketching framework for modeling of developable surfaces and use it to obtain the 3D boundaries of the modeled surfaces (Section 6).

3. Toward Locally Convex Triangulations Section 2.1 discussed the potential of boundary triangulations to represent developable surfaces that interpolate a given boundary polyline. As explained, triangulations that approximate smooth developable surfaces have the property that the majority of their edges are locally convex. We now

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secutive triangles on the polyline’s hull having edges on the polyline (Figure 7(b,c)). Formally we define such charts as sequences of hull triangles, such that: (a)

(b)

(c)

Figure 6: Envelope triangulations for a polyline that lies on its convex hull: (a) polyline; (b) convex hull with envelopes; (c) the two envelope triangulations, the framed (right) one is the one selected by the algorithm in Section 5.

1. each triangle shares at least one edge with another triangle in the same chart; 2. each triangle shares at least one edge with the input polyline; 3. all the triangles are oriented consistently with respect to the polyline. The second requirement implies that charts are separated from each other by interior triangles, i.e. triangles of the convex hull with no edges on the polyline (shown in black in Figure 7(b)). The last requirement ensures that the triangulation constructed by the algorithm is manifold and orientable.

Figure 7: Extracting a locally convex triangulation: (a) boundary; (b) convex hull with extracted charts (interior triangle shown in black) (c) individual charts and remaining subloops after subtraction; (d) recursing on the subloop formed by removing the purple chart; (e) resulting triangulations (the framed triangulation is the one returned by the algorithm in Section 5); (f) two of the triangulations created with different chart choices. describe a general method for obtaining such triangulations. Section 5 extends this method to search for triangulations which satisfy additional requirements. The following observation forms the basis for our method: Since most edges of a desirable triangulation must be locally convex, a natural place to identify developable regions interpolating a boundary polyline is the convex hull of the boundary, where every edge is locally convex. We rely on this observation to narrow the search space when looking for desirable triangulations. The convex hull of a closed polyline is a triangular mesh, containing a subset of the polyline’s vertices. If the polyline lies entirely on its convex hull, it separates the hull into two triangulations, the left and right hull envelopes, defined with respect to the boundary orientation (Figure 6). If the polyline is planar, then these envelopes are identical. By construction, both triangulations interpolate the polyline. Moreover, as desired, every interior edge in each triangulation is locally convex since it is an edge on the convex hull. If the polyline does not lie entirely on the convex hull, it will not separate the hull into two envelopes and a more sophisticated modeling strategy is required. The convex hull of almost every closed sufficiently smooth space curve consists of planar triangles and torsal developable surfaces [Sed86], where each of these torsal developable surfaces interpolates parts of the curve. If a polyline is sampled sufficiently densely from a smooth space curve, its convex hull will closely approximate the convex hull of the curve. We observe that the torsal developable surfaces on the hull of the continuous curve correspond to regions, or charts, of con-

Subtracting each chart from the polyline by removing the portions of the polyline inside the chart and replacing them with the interior boundaries of the chart results in one or two smaller closed polyline subloops (Figure 7(c)). If the subloops lie on their convex hulls, their left and right envelopes will provide triangulations, which together with the removed chart will interpolate the original polyline (Figure 7(d)). If a subloop does not lie on its convex hull, we can identify charts on this convex hull and proceed recursively. By construction, charts on the subloop hulls will also correspond to torsal developable surfaces interpolating the original polyline. It is theoretically possible, though unlikely, for a hull to contain no valid charts. In this pathological case the algorithm treats each hull triangle as a separate chart. The recursion is guaranteed to terminate as the number of polyline vertices decreases at each iteration and a polyline with three vertices always lies on its hull. In any resultant triangulation, the only potentially non-convex edges will bound adjacent triangles computed at different levels of the recursion. All other edges are necessarily locally convex as they originated from within a convex hull, either that of the original polyline or of one of the subloops. As desired, the number of non-convex edges is very small and is related to the boundary complexity and not to the number of boundary vertices. However, as shown in Figure 7(d) and (e), the choice of different charts to proceed from leads to drastically different triangulations, raising the question of which choice the user would prefer. The subsequent sections analyse the desirable shape characteristics of discrete developable surfaces and describe an algorithm which guides the selection to efficiently obtain a good interpolating surface. 4. Desirable Triangulation Properties When considering triangulations which approximate a smooth developable surface, we require the majority of triangulation edges to be locally convex. An additional constraint, ignored in Section 3, is smoothness: requiring the dihedral angles between adjacent triangles to be low. Even c The Eurographics Association 2007.

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We observe that for polyline boundaries that lie on their convex hull, the two envelopes mentioned above are not necessarily the best solutions with respect to smoothness (see the pink and blue envelopes at iteration one in Figure 8). Therefore, our algorithm extracts not only these envelopes, but also the separate charts that are part of the convex hull. It then proceeds to explore possible interpolating triangulations that contain one or more of the identified charts. Fragmenting the envelopes into charts can slightly increase the number of non-convex edges in the final triangulation. However, their number remains a function of boundary complexity and does not depend on the number of boundary vertices, as desired. Figure 8: Algorithm stages on a simple example (interior triangles shown in black). The framed triangulation is the output. The cover pushed into the queue in iteration four will be discarded at iteration five in stage 1 (it is not better than the best triangulation). with these two restrictions, there may exist multiple boundary triangulations providing a valid solution (see Figure 7 (e),(f)), raising the question which of these the user expects to obtain when specifying a particular boundary. Clearly, when designing a modeling tool, predictability is a desirable property. Human perception studies indicate that “simplicity is a principle that guides our perception...” [Ben96]. This principle is well known in Gestalt theory and is commonly used in sketch interpretation [KH06]. In our setup, it implies that the surface the user expects is the simplest developable surface fitting a given boundary. Based on numerous examples, we hypothesize that a surface is considered simpler and hence more predictable if its normal map has fewer branches, or equivalently, if its directrix has fewer discontinuities. In addition to predictability, or instead of it, we can consider the fairness of the created surface. Frey [Fre02] and Wang and Tang [WT05] describe a large set of measures of surface fairness, including metrics of mean curvature and bending energy. We found that minimizing the integral l 2 mean-curvature described as the sum of squared dihedral angles across interior edges results in visually fair triangulations. The advantage of this metric is that it can be extended to provide a lower bound on the fairness of a boundary triangulation given only a subset of its triangles (Section 5.1.1). The next section presents a practical method for computing boundary triangulations that satisfy all of these requirements, and thus define which developable surfaces to output. 5. Branch-and-Bound Search Algorithm We now extend the basic methodology described in Section 3 to search specifically for smooth triangulations and describe a procedure to efficiently navigate the search space to obtain triangulations that are predictable and fair. c The Eurographics Association 2007.

To obtain smooth triangulations we require that any interior edge in a chart has a dihedral angle below a specified threshold. Charts with larger dihedral angles are not considered for future processing. For instance, in the first iteration of the algorithm in Figure 8, this invalidates the light and dark green charts. We also require the angles on edges between any chart and the adjacent interior triangles to lie below the threshold. We observe that since these edges are on the convex hull, the dihedral angle between the chart and any other triangle formed using these edges is bounded from below by the current angle. Charts which violate this property are also eliminated. In the first iteration in Figure 8, this invalidates the orange and dark yellow charts. To reduce the number of non-convex edges and to speed up processing, we only consider charts larger than a certain percentage of the convex hull area (we use 1%-3% in our examples). Both the angle and size thresholds can be adjusted depending on the input. If both are completely relaxed, our method will find a solution for practically any input. Given this definition of charts, our algorithm computes boundary triangulations that are unions of charts and envelopes. The algorithm uses a variation of the branch-andbound approach [CLR90], which helps drive the search towards a good solution while avoiding the exploration of nonpromising ones. The method uses a priority queue of sets of charts, or covers, that interpolate segments of the polyline (Figure 8). The queue is initialized with the empty cover. The priority function of the queue is based on a potential metric (Section 5.1.2) and orders covers such that the next popped cover is expected to lead to an acceptable boundary triangulation fastest. During processing, the method maintains the best boundary triangulation found to that point. The quality of a triangulation is measured with respect to the desired triangulation properties (Section 5.1.1). The same metric is used to measure the quality of a cover, as a lower bound on the quality of any possible triangulation containing this cover. At each iteration of the algorithm the following sequence of operations is performed as visualized in Figure 8 and the attached video. 1. Pop Cover: The algorithm pops a cover C from the pri-

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ority queue, based on the potential metric. If a boundary triangulation was already found, the method compares the quality of the best triangulation found to the quality of C. If the quality of C is worse, it is immediately discarded. Otherwise, the method obtains the set of polyline subloops S formed by subtracting (as defined in Section 3) the cover charts from the original boundary and computes their convex hulls. Explore Possible Triangulations: The method checks if the convex hulls of each of the subloops are separable into two envelopes. If the envelopes exist for all the subloops, then each permutation of them combined with the cover triangles defines a triangulation of the original boundary. Triangulations having interior dihedral angles above the smoothness threshold are discarded. In Figure 8, this discards all the boundary triangulations in iterations one through three. If there are multiple possible triangulations satisfying the smoothness constraint, the algorithm selects the highest quality one among them (Section 5.1.1). If this is the first triangulation found or if the new triangulation is better than the best triangulation found so far, then the best triangulation is appropriately updated. Form New Covers: The method then extracts valid charts from the convex hulls of all the subloops in S. If a chart shares a boundary with the cover C, it tests if the dihedral angle across the shared edge satisfies the smoothness threshold. Charts which fail the test are discarded. For each of the remaining charts the method forms a new cover N combining C and the new chart. Add to Queue: We observe that a subset of a new cover N may already be present in the priority queue. In this case, naively adding N to the queue can lead to repeated computations. To avoid this redundancy, the method checks if N contains a cover already in the queue. If this is not the case N is added to the queue. If a subset of N is in the queue and the quality of N is better than that of the subset one, the subset cover is removed from the queue and N is inserted. If it is worse, N is discarded. In Figure 8, iteration two, the blue-red cover is discarded since a better subset of it (the purple cover) was added to the queue at iteration one, and was not yet processed. Termination: The algorithm terminates if the queue is empty or if the best computed triangulation is deemed to be acceptable, using the measures described in Section 5.1.1. Otherwise, the algorithm goes back to Stage 1.

The pseudocode for the algorithm is summarized in Figure 9. As shown, the entire main loop consists of approximately twenty lines of code.

Input: Polyline orig best ← Null ; PriorityQueue pq ; pq.Insert(EmptyCover); while pq not empty and best not good enough do C ← pq.Pop(); if best is better quality than C then continue ; S ← orig.Subtract(C);

ComputeConvexHulls(S); if every subloop ∈ S has envelopes then foreach permutation P of envelopes do if P + C is smooth then if P + C is better quality than best then best ← P + C ; end end end end foreach subloop ∈ S do Charts ← ComputeCharts(hull of subloop); foreach chart ∈ Charts do N ← chart + C ; if N is not smooth then continue ; if N ⊇ some other cover R ∈ pq then if N is better quality than R then pq.Remove(R); pq.Insert(N); end else pq.Insert(N); end end end end return best

Figure 9: Pseudocode of main loop. ward and requires only minor data-structure modifications. When processing boundaries with multiple loops the method prioritizes processing of charts which connect separate loops before processing any other chart. If such charts are unavailable, the method connects the loops by the shortest tree of edges, treating those as interior edges for processing purposes. 5.1. Metrics 5.1.1. Triangulation and Cover Quality Triangulation Quality: When evaluating triangulation quality, we consider two of the criteria discussed in Section 4: predictability and fairness. We do not need to take smoothness into account as the algorithm automatically discards non-smooth triangulations. To evaluate predictability we compute the number of branching points on the surface normal map. In a discrete setup, these correspond to interior triangles in the triangulation and hence can be easily counted. Fairness is measured as the sum of squared dihedral angles across interior triangulation edges. Note that the optimum is zero for both metrics. In our setup, we consider predictability as more important than fairness. Thus, to compare two triangulations, we first compare predictability and only if the predictability is the same compare fairness.

Darts† and Multiple Boundaries: Our method is the first to our knowledge to seamlessly handle darts as well as multiple boundary loops. The processing of darts is straightfor-

When determining if a triangulation is acceptable (Stage 5), the two criteria can be compared against lower bounds set by the user. Using such lower bounds can speed up the processing, as the algorithm will terminate once an acceptable triangulation is found.

† Darts, or duplicate edges, are frequently used in design setups such as garment making, to introduce points or lines of non-zero curvature onto the surface (see Figure 11).

Cover Quality: We consider the same two criteria when evaluating a cover, wherein a cover evaluation aims to provide a lower bound on the quality of any triangulation that c The Eurographics Association 2007.

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contains it. The lower bound on predictability measures the minimal number of interior triangles in any triangulation containing the cover. To compute it, we consider the set of subloops S formed by subtracting the cover charts from the original boundary. We observe that if a subloop shares edges with more than two cover charts, any triangulation of it will contain at least one interior triangle‡ . A subloop which is adjacent to one or two cover charts can potentially be triangulated without any interior triangles. Thus the predictability metric of a cover is the number of subloops adjacent to more than two cover charts.

Figure 10: Three of the alternative triangulations for the gazebo boundary in Figure 13 found by our method; the input boundary is shown on the left.

To measure the fairness of a cover we first compute the sum of squared dihedral angles within the cover charts and then add to it a lower bound on the sum of angles for the subloops in S computed as follows. If a subloop has two adjacent cover charts, we first fit a least-squares plane to the subloop and compute the dihedral angles α1 and α2 between the plane and the chart triangles adjacent to the subloop. The sum of the two angles gives us a lower bound on the sum of angles on any interpolating triangulation of the subloop and between this triangulation and the adjacent charts. To bound the sum of squared angles, we assume equal distribution on all the n − 1 edges involved, where n is the number of vertices on the subloop§ . Thus for each such subloop we add to the fairness metric (α1 + α2 )2 /(n − 1). If a subloop has more than two adjacent cover charts, we pick a random pair and do the same computation. If a subloop has only one adjacent chart, we return zero as an estimated lower bound for that subloop.

We use a fairly standard 3D curve sketching interface to create the polyline boundaries. The user can create the 3D boundary curves by first sketching them in one plane and then deforming them from a different viewpoint. Additionally, similar to [DJW∗ 06] our system infers the depth information from a single sketch when the polyline is drawn over an existing model (Figure 1 (a),(b)). The polyline is then set at a frontal distance to the model that interpolates the two distances at the extremities. This feature is especially useful for our garment examples, where we drew the desired boundaries on top of a 3D mannequin automatically keeping the boundaries at the desired distance from the body. The sketching system identifies darts as polyline sections that start from a closed boundary loop. When a dart is detected, this section is duplicated and added twice to the parent polyline while its orientation is switched, forming a single closed boundary. Lastly, when the tip of a dart reaches the same boundary again, the latter is split into two loops, enabling easy generation of a boundary network.

A cover and a triangulation or two covers are compared in the same way as two triangulations, by first considering predictability and then fairness. Since the cover quality is a lower bound, it can be safely used when deciding to discard a cover if it cannot lead to a triangulation better than the current one (Stage 1).

5.1.2. Cover Potential The purpose of this metric is to prioritize covers based on their potential to be part of the expected final triangulation. The final triangulation is expected to have a very small number of interior triangles. Thus a cover is more likely to lead to an acceptable triangulation if it contains a small number of charts, where at least one of the charts is quite large. We first order the covers in ascending order based on the number of charts, and then in descending order based on the largest consecutive chart area.

‡ The triangulation has n − 2 triangles and less than n − 3 edges on the original boundary, where n is the polyline size. Hence at least one triangle has no boundary edges. § We arrive at n − 1 as the number of interior edges in the triangulation n − 3 plus the two edges adjacent to the charts. c The Eurographics Association 2007.

6. Interface 6.1. Sketching

To further influence the result the user can also sketch a few of the rulings they expect to see on the final surface. These rulings are treated as triangulation edges which are constrained to be part of the final surface. For the purse model in Figure 12 we used this option to specify a ruling to the right of the handle, causing the purse to bulge outwards instead of curving inside. The specified edges segment the boundary into several separate subloops and the algorithm is run separately on each subloop, considering only the original polyline edges as boundary edges for chart extraction. 6.2. Overriding Optimal Selection The algorithm, as described, returns the best boundary triangulation computed, based on user indicated preferences in terms of quality metrics. Clearly, there might be cases when a user has additional constraints in mind. For instance, for the gazebo in Figure 13 we had a particular orientation in mind. In addition to user drawn rulings we provide another mechanism for obtaining alternative triangulations. Each time the algorithm computes a triangulation, it is immediately visualised and stored while the rest of the processing continues. The user thus has the option to interrupt the algorithm when they see a triangulation that they like, and they may also browse all the computed triangulations at any point during or after processing. The gazebo was se-

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(c)

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Figure 12: Metal and leather: helmet (six panels), purse (three panels), and glove (ten panels). (a)

(b)

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Figure 11: Modeling garments: poncho (modeled from two developable panels, front and back), dress (modeled from seven panels), tanktop (four panels) and skirt (two panels). (a) Input boundary networks. (b) Modeled surfaces (coloring shows the surface structure with torsal developable surfaces shown in blue and interior triangles, corresponding to planar transition regions, in white). (c) Textured garments worn by a mannequin (basic simulation [3DM] was applied to account for collisions and gravity). Last row (d) the planar patterns for tanktop and skirt. lected this way. Alternatives found by the method are shown in Figure 10. 7. Results and Implementation Throughout the paper we demonstrate the application of our method on a variety of inputs coming from different application areas where developables are used. Figure 11 shows several garments generated from simple sketches using our system. The modeling of each of the garments took only a few minutes compared to hours using traditional garment modeling tools such as [May] where the user is required to manually specify the 2D patterns for the garment. Real garments at rest are always piecewise developable since they are assembled from flat fabric pieces. Once worn by a character or a mannequin they stretch slightly due to gravity and collisions. The main challenge when modeling garments is obtaining the rest shape and the corresponding 2D patterns. Once these exist, standard simulation or procedural techniques can be applied to account for collisions and gravity [DJW∗ 06, May, 3DM]. In the examples in this paper, we fo-

cused on obtaining the rest shapes. We then used a standard simulation tool [3DM] to visualise the garment behaviour subject to the physical forces involved. As expected the results after simulation appear less stiff but remain very similar to the developable rest shapes. We note that in all the examples the garments are generated using a network of seams. Each individual panel surrounded by seams is a developable surface, but the surfaces are not developable across seams. As shown in Figure 11(b) in most cases the created surfaces are general developable surfaces, each containing several torsal developable surfaces connected by planar regions. Several examples, including the dress and tanktop, contain darts as part of the input polylines, robustly handled by our method. Since the created surfaces are analytically developable, the patterns (e.g. Figure 11(d)) can be used as-is to create reliable real-life replicas of the garments and the garment texture exhibits no distortion. Figure 12 shows a variety of objects designed from flat sheet materials: a helmet, a leather bag and a glove. Despite the complexity of the modeled surfaces (12 (b)) no modeling expertise is required when sketching them using free-form drawing. The examples also show the control mechanism available for the user, such as the use of rulings to guide the construction of the purse as explained in Section 6 and the impact of smoothness threshold in the helmet example, where we relaxed the threshold to create the appearance of metal ridges. Figure 13 shows examples of architectural structures generated using our method. The gazebo is an example of a complex general surface which cannot be projected to a plane without intersection and hence could not be generated by c The Eurographics Association 2007.

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tions, the method typically performs only a small number of iterations (less than a thousand for all the models shown in this paper). At each iteration the dominant component of the runtime is the convex hull computation, which takes O(n log n) time in the number of vertices on the input boundaries. Thus, in practice, the overall runtime is fairly small, varying from a few seconds for simple models such as the Opera House (Figure 13), to a few minutes for a complex model like the shoe (Figure 1). We observe that the runtime strongly depends on the number of charts formed at each iteration, which is directly linked to the complexity of the input boundary rather than to the number of vertices on it.

Figure 13: Architectural examples: gazebo and opera house.

Figure 14: Tulip paper lamp with developable paper petals and gold-foil leaves. any previous method for modeling developables. The Opera House model (Figure 13) was inspired by the Sydney Opera House and created by duplicating a single developable surface six times at different scales. The tulip lamp modeled from developable petals and leaves (Figure 14) is mimicking Art-Nouveau paper lamps. The flower is created by duplicating and scaling a developable petal surface. While the gold-foil leaves are general developable surfaces the paper petals are torsal ruled surfaces and thus could be modeled by previous techniques, e.g. [WT05]. However, in contrast to these approaches in our setup the user is not required to specify the ruling directions or even know what ruling directions are, allowing non-experts to use the system. Furthermore, the method, not the user, is able to determine that a single torsal developable surface interpolating the boundary exists, a non-trivial observation, making the method more attractive for non-experts. These examples demonstrate the robustness of our modeling method as well as the vast array of application of developable surfaces and the varied shapes they can have. Runtime: The search space for the algorithm is exponential in the number of charts found. However, using the priority queue combined with the potential and quality estimac The Eurographics Association 2007.

Robustness: We observe that the topology of a convex hull is easily affected by noise in the input polyline. This can drastically affect the algorithm runtime as it leads to chart fragmentation and can sometime also influence the resulting surface. To reduce the noise and simplify the obtained hull structure, the algorithm first re-samples and smooths the polylines by fitting a piecewise B-spline curve. The algorithm then computes the center of mass C of the polyline boundary and offsets each vertex radially from it. This offsetting effectively makes the curve more "convex". This pre-processing drastically reduces the number of interior triangles on the hull and improves stability. The offsetting also bends nearly all planar portions of the boundary, which would otherwise allow for ambiguous triangulations. Additional offsetting from a slightly shifted center is performed in the rare cases where C is in the same plane as part of the boundary. 8. Summary and Limitations To the best of our knowledge, we have presented the first algorithm for modeling developable surfaces that interpolate arbitrary polygonal boundaries. As demonstrated by the numerous examples in the paper, the method robustly computes smooth, predictable and fair developable surfaces interpolating the input boundaries and can be used to model a vast array of developable shapes. Since the method explicitly explores the space of possible solutions, users can easily select alternative solutions and explicitly specify which of the surface properties they wish to optimize. The presented approach is fairly generic and can be easily extended to handle other shape metrics. Limitations: The theoretical setup of our algorithm assumes that the polyline boundaries are sampled from a sufficiently smooth curve. As shown by the examples, the algorithm remains robust even when this is not the case. As noted earlier, though it is possible that a convex hull may not contain any valid charts, such situations are extremely rare. If such a situation occurs the runtime is significantly increased, but the method is still guaranteed to find a solution. We also observe that there may exist developable surfaces that are entirely contained by the convex hull of their boundary. Thus, our method would not locate them. Adding

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one or two extra rulings would typically break such surface into parts that partially lie on the respective boundary hulls and are thus computable by the method. Acknowledgements

Teddy: a sketching interface for 3d freeform design. In Proc. SIGGRAPH ’99 (1999), pp. 409–416. [JKS05] J ULIUS D., K RAEVOY V., S HEFFER A.: Dcharts: Quasi-developable mesh segmentation. Computer Graphics Forum (Proc. Eurographics) 24, 3 (2005), 581– 590.

We thank Christine Depraz, Laurence Boissieux, and Ciaran Llachlan Leavitt for their invaluable help with generating the example figures and proof-reading. This research was partially supported by NSERC, MITACS NCE, and IBM.

[KH06] K ARPENKO O. A., H UGHES J.: Smoothsketch:3D free-form shapes from complex sketches. ACM Transactions on Graphics 25, 3 (2006), 589–598.

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mathematics Article

Turning Hild’s Sculptures into Single-Sided Surfaces Carlo H. Séquin EECS Computer Science, University of California, Berkeley 94720, CA, USA; [email protected] Received: 15 December 2018; Accepted: 17 January 2019; Published: 25 January 2019







Abstract: Eva Hild uses an intuitive, incremental approach to create fascinating ceramic sculptures representing 2-manifolds with interesting topologies. Typically, these free-form shapes are two-sided and thus orientable. Here I am exploring ways in which similar-looking shapes may be created that are single-sided. Some differences in our two approaches are highlighted and then used to create some complex 2-manifolds that are clearly different from Hild’s repertoire. Keywords: Eva Hild; 2-manifolds; computer-aided design; 3D printing

1. Eva Hild’s Sculptures Eva Hild is a Swedish artist [1], who creates large ceramic sculptures (Figure 1) in an intuitive, incremental process [2,3]. I have been fascinated by her work for several years [4], before I was finally able to meet her in her studio in Sparsör, Sweden in July 2018 (Figure 1b).

Figure 1. Eva Hild: (a) A portrait; (b) working in her studio; and (c) with the sculpture Hollow [1].

Her creations are typically thin surfaces, which may take on the structure of bulbous outgrowths in plant-like assemblies (Figure 2a), or configurations of intricately nested funnels (Figure 2b). These surfaces may be bordered by simple circular openings or by long, sensuously undulating curved rims, which are connected by a system of crisscrossing tunnels (Figure 2c). These sculptures invite mental explorations, raising questions such as: how many tunnels are there? How many separate rims are there? Is this a 1-sided or 2-sided surface? These sculptures also have inspired me to create similar shapes [4]. I do not possess the skills to create large ceramic pieces myself, so I have tried to create computer-aided design (CAD) models of such surfaces, and then have realized the more promising ones as 3D printed models. My students and I found it rather difficult to capture Hild’s sculptures in CAD tools such as Blender [5] or Maya [6]. We had difficulties in defining the proper topologies—often just starting from a few photographs found on the World Wide Web. We also found it difficult to fine-tune these topologically correct models, so that they would reproduce the organic

Mathematics 2019, 7, 125; doi:10.3390/math7020125

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flow and natural elegance of Hild’s creations. We thus ended up developing some additional modeling tools to make the design of such shapes less cumbersome and frustrating.

Figure 2. Eva Hild’s ceramic creations [1]: (a) a bulbous surface with oculus-like openings; (b) various nested funnels; and (c) hyperboloid tunnels and undulating rims.

To place the results of this paper into proper perspective, I start with a brief review of the developments that led me to this current investigation and some of the many other contributions in this domain, made by artists as well as mathematicians. 2. Background and Previous Work For much of my life I have been fascinated by abstract, geometrical sculpture. As a student, some of my heroes were Alexander Calder, Naum Gabo, and Max Bill. In the 1980’s, when I started to teach courses in computer graphics and computer-aided design [7], I employed the help of such techniques to analyze and synthesize topologically interesting sculptural shapes. The key stimulation for my strong involvement in this field came from my interaction with Brent Collins [8]. This connection was prompted by an analytical write-up by George Francis [9] in a special issue on visual mathematics in Leonardo in 1992, which clearly showed the connection between the intuitive sculptural work of Collins and the geometry of minimal surfaces, the topology of interconnected tunnels and handles, and basic knot theory. I started to write a specialized, parametrized computer program, called Sculpture Generator I (Figure 3a) [10] that allowed me not only to capture some inspiring shapes created by Collins, but also enabled me to make several derivative designs that seemed to belong to the “same family”. Collins was fascinated by this prospect and was eager to sculpt additional, more complex shapes by relying heavily on my detailed, computer-generated printouts. Our first two collaborative pieces were Hyperbolic Hexagon II (Figure 3b) [11] and Heptoroid, (Figure 3c) [12], both of which were a generalization of a 6- or 7-story Scherk-Tower [13] bent into a closed, possibly twisted loop. Heptoroid was particularly exciting to us, because it introduced a non-orientable, single-sided surface with a heavily knotted border. Sculpture Generator I was a very specific program that could only generate sculptures based on Scherk’s Second Minimal Surface [13]. To capture any other one of Collins’ inspirational sculptures, such as Pax Mundi (Figure 4a) [14], I needed a new program with different geometrical primitives: in particular, versatile progressive sweeps that would allow me to move an arbitrary cross section along any 3D sweep path. This gradually led to the development of the Berkeley SLIDE program (Figure 4b) [15]. This modeling tool has a variety of parameterized geometric modules that can be combined in many different ways. By using its sweep construct, I was able to capture Pax Mundi and design a large collection of Viae Globi sculptures [16], in which a ribbon travels along the surface of a sphere. With this program we scaled up Pax Mundi, and with the help of Steve Reinmuth [17] realized it as a large Bronze sculpture [18]. The SLIDE program then allowed us to create other Viae Globi sculptures, some of which had knotted sweep paths, such as Music of the Spheres (Figure 4c) [19].

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Figure 3. Sculpture Generator I: (a) The user interface; (b) Hyperbolic Hexagon II; and (c) Heptoroid.

Figure 4. Roads on a Sphere: (a) Pax Mundi; (b) SLIDE user interface; and (c) Music of the Spheres.

Collins created his inspirational sculptures in an intuitive way, driven by the aesthetics of the emerging form, not by thinking about mathematics or topological issues. Other artists start out quite consciously with mathematics on their mind. Helaman Ferguson [20] has created stone and bronze sculptures based on mathematical equations, given knot structures, or topologically intriguing surfaces [21]. George Hart [22] is often driven by some high-order symmetry group and then finds ways to interlink many identical pieces in an intricate manner, while maintaining the envisioned symmetry [23]. He relies on computer-aided tools to define these shapes [24], and even to fabricate them with a computer-driven paper cutter [25]. A similar way of using CAD tools also enables the work of many other artists, such as Bathsheba Grossman [26], Vladimir Bulatov [27], Rinus Roelofs [28], and Hans Schepker [29]. Charles O. Perry [30,31], even though he often starts by defining crucial elements of his sculpture by shaping physical materials such as steel cables, is also quite conscious of the mathematical underpinnings of some of his sculptures [32,33]. For other authors, the key motivation is to generate a visualization of some mathematical concept, rather than creating a piece of art. Jarke van Wijk uses computer graphics to generate the Seifert surfaces supported by mathematical knots [34]. His demonstration tool is readily available on the web [35], and by choosing the right smoothing option, one can produce visualization models that can also stand on their own as pleasing sculptural shapes. Other mathematical concepts, where visualization models can readily turn into art, concern non-orientable surfaces—such as Boy’s surface (Figure 5a) [36,37] or Klein bottles (Figure 5b) [38,39]—and the depiction of regular maps [40–42], regular meshes [43,44], woven surfaces [45,46], knot theory [47–49], or high-genus objects (Figure 5d) [50,51].

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Figure 5. Mathematical visualizations as art: (a) a Boy’s surface [36]; (b) a Klein bottle [38]; (c) a regular map [40]; and (d) a high-genus object [50].

For most of the above mathematical concepts and mentioned artists, Berkeley SLIDE [15] or readily available programs such as Rhino3D [52] or TopMod [53] are adequate tools to capture the corresponding models, because these shapes typically can be depicted with a few clearly defined geometrical primitives. This is true even for the works of artists like Brent Collins [8], Keizo Ushio [54], or Bob Longhurst [55], who intuitively sense some underlying mathematical principles, but do not explicitly use a mathematical formulation or a computer program to define their shapes. Also, the “D-Forms” introduced by Tony Wills [56] or the developable surfaces by Ilhan Koman [57,58] have a clearly defined geometrical structure that can be captured in a procedural specification. Eva Hild’s work [1], on the other hand, presents a new modeling challenge. Her ceramic surfaces are truly free-form [2,3], and if the underlying topology would allow for some regularity or symmetry, Hild would deliberately remove it [4]. Thus, the predefined modules in SLIDE are no longer sufficient, and modeling her work requires some additional capabilities, which will be described in Section 4. Moreover, as discussed in Sections 7 and 8, I faced another novel challenge: how to make a significant topological change to some intuitively conceived geometries, while keeping the visual look-and-feel the same. 3. Classification of 2-Manifolds From a mathematician’s point of view, Eva Hild’s sculptures ( Figures 1 and 2) are smooth 2-manifolds with one or more borders [59]. As a brief reminder: every interior point in a 2-manifold has a neighborhood that is topologically equivalent to a small disk; and every border point has a neighborhood equivalent to a half-disk (Figure 6a,b). Branching of surface sheets, as in the spine of a book, is not allowed. Thus, if the 2-manifold surface is finely tessellated, then every facet edge is used by either one or by two adjacent facets.

Figure 6. Various 2-manifolds: (a) a 2-sided cylinder; (b) a 1-sided Möbius band; (c) a genus-2 handle body; (d) a disk: Euler characteristic χ = 1, genus g = 0; and (e) a disk with 5 holes and 5 cuts.

All possible 2-manifolds can be classified by just three numbers: b, the number of borders; s, its sidedness; and its connectivity described by its genus, g. For a given 2-manifold, the number of borders is easily determined: We start with some border point; we follow along this rim line until we arrive back at the starting point; we mark this rim as ‘counted’; and we repeat this until there are no further unmarked borders. To determine the sidedness, s, we start with an interior point and paint its

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neighborhood with a flood-filling algorithm; we continue this process as far as we can without ever stepping across a border line. If, in the end, all surface points have been painted, then the surface is single-sided, and also non-orientable. If only half of all surface areas have been covered, then the surface is two-sided, and orientable. The connectivity of a 2-manifold is more difficult to determine. Its genus, g, is defined as the maximal number of closed curves that can be drawn on the 2-manifold without dividing its surface into two separate “countries.” But how do we know that we have indeed found the maximal number of such curves? One more practical approach is to close off all punctures in the surface by gluing in disk-like surface patches into any open border loops, as long as this can be done without creating any self-intersections. If this results in a well-behaved handle-body (Figure 6c), we count the number of its handles or tunnels to find its genus. As an alternative, we may make cuts through the 2-manifold from some border point to another border point until the resulting 2-manifold has acquired the topology of a single disk without any holes (Figure 6d). If there are still interior holes in the surface, we need to make additional cuts from the current disk borders (Figure 6e). However, we must not split the surface into separate pieces; everything must remain connected. We know that the Euler characteristic, χ, of a disk is 1, and that every cut adds 1 to χ. Thus, we derive that the original manifold must have had an Euler characteristic of 1 minus the number cuts that we have made. The Euler characteristic and genus are then related according to the following formula: Genus = (2 − χ − b)/s,

(1)

An intriguing discovery is that the 2-manifolds created by Eva Hild [60] show a wide variety in their number of borders and in their genus—but they are all double-sided! Later, I will speculate how this may have come about. Furthermore, I will also try to transform some of these sculptures into single-sided 2-manifolds, while making only small style changes. 4. My Modeling Approach After some experimentation, I found that a good way for capturing the basic geometry of Hild’s sculptures is to locate some defining key features, in particular, “rims”, “funnels”, and “tunnels”. These features have been marked on a few sculptures in Figure 7. These features typically form the starting point in my attempts to model these geometries. They are placed in appropriate locations in 3D space, and the surface is then constructed, facet by facet, between points on these border curves in an interactive manner through a graphical user interface. Our home-brewed modeling system, Berkeley SLIDE [15], developed two decades ago, is well suited to define and place such key features in a parameterized manner. It provides a slider for every parameter, so that complex configurations of such features can be fine-tuned in an interactive manner. Unfortunately, it has no interactive capabilities to form the connecting surface between these features. Moreover, SLIDE cannot handle single-sided surfaces in operations such as surface smoothing and offsetting.

Figure 7. Key features marked on some Hild sculptures: red indicates rims; blue indicates funnels; and green indicates tunnels.

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To overcome these shortcomings, my research students have helped me to create a new CAD environment, called NOME (Non-Orientable Manifold Editor) [61,62]. Just like SLIDE, this program allows the procedural placement of the parameterized key features though a simple text interface. It then provides crucial editing capabilities to form a surface between these features. Moreover, there are pre-programmed generators for funnels and tunnels with all their defining variables. Once these features have been placed reasonably well, the user constructs the connecting mesh in a point-and-click manner. The user may select some number, n, of vertices and then create a n-sided facet (typically irregular and non-planar) spanned by those vertices. The key characteristic is that this facet behaves like a rubber sheet, staying attached to the different tunnels, funnels, or rims, as these elements are repositioned and re-shaped. The parametrization of the key features remains active through the entire design process. Even after subdivision smoothing has been applied and offset surfaces have been generated, the parameters can still be changed, and the user will immediately see the effect that this has on the final geometry to be sent to the 3D printer. This is a crucial capability of NOME that we have found difficult to implement in other CAD tools. I will now describe my current modeling approach with the example of a reconstruction of two relatively simple sculptures by Eva Hild. The first one is Interruption [60]; Figure 7a shows a bottom view of this sculpture, and Figure 8a is a side view. In Figure 8b, I placed a (blue) funnel at the bottom, fitted a (red) free-form B-spline curve along the top rim, and inserted three (greenish) tunnels to define the internal passages. Identifying, defining, and placing these key features is a crucial first step in creating a replica of an existing sculpture. So far, we have not been able to think of an automated process that would extract that information from a few images. Being able to adjust the parameters that define these features through the very end of the design process is thus an important capability for obtaining good-looking results.

Figure 8. Interruption: (a) Hild’s sculpture with key features marked; (b) key features modeled; (c) first quad faces added; (d) the connecting mesh; (e) the subdivision surface; and (f) the resulting 3D print.

The next step is to specify the polyhedral mesh components that properly connect these feature geometries to one another. First, a few connecting quads are introduced, shown in red in Figure 8c; then the various mesh patches are filled in, shown in yellow in Figure 8d. This composite mesh is subjected to three or four steps of Catmull-Clark subdivision [63] to produce a smooth, tangent-continuous surface (Figure 8e). This smoothing process has been robustly implemented in SLIDE as well as in NOME, and it typically produces soap film-like surfaces for polyhedral models with a variety of facets. The best results are obtained when the starting mesh is composed mostly of quadrilaterals that all have about the same edge length. Once we have a pleasing looking mesh, which is tessellated finely enough to display the desired level of smoothness, NOME generates two offset surfaces that lie at a distance t/2 on either side of the subdivision surface. Additionally NOME generates a set of rectangular facets along all border curves to connect the two offset surfaces into a water-tight boundary, creating a physical object of thickness, t, which then can be fabricated on a 3D printer (Figure 8f). Even at this late stage in the design process, the various parameters are all still fully in effect. For instance, by raising the position of the blue funnel, the tubular base of the sculpture can be shortened, or by changing the diameter or length of any of the tunnels, the corresponding hole-geometries can be fine-tuned. This final geometry is then captured as an STL-file [64]. In this

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procedure, the facets of the boundary representation are split into triangles, which are then listed with their three vertices and their face normal. This is a rather wasteful description, and the resulting files are typically tens of Megabytes in size; it is, however, a format that is understood by all 3D printers. Creating good 3D models represents additional challenges. 3D printers come with their own software for handling the tricky task of constructing adequate scaffolding structures for supporting hollow or overhanging geometries as needed. However, print orientation, print speed, and fill-in density must still be decided by the user, and they can strongly affect printing time and the quality of the print. 5. Modularity: Deriving New Topologies The CAD tools described above, allow me to now focus on designing new geometries inspired by Hild’s creations. In a first demonstration, I started from Hild’s Whole (Figure 9a,b). To model this structure, I created a new auxiliary geometrical shape, a cross-tunnel (Figure 10a). Two instances of this new feature were then surrounded with an undulating 3-period Gabo curve [65] to form the complete feature model (Figure 9c) defining my version of this sculpture. Figure 9d shows the resulting 3D print. In contrast to Hild’s sculpture, which has an organically flowing, slightly irregular shape, my model exhibits strict “C2h symmetry” (Schönflies notation), also known as “2*” symmetry (Conway notation), featuring a vertical mirror plane (most easily visible in Figure 9c) and a horizontal C2 rotation axes (Figure 9d).

Figure 9. Modeling Whole: (a,b)two views of Hild’s sculpture; (c) skeleton computer-aided design (CAD) model; (d) 3D printed model.

Figure 10. Wrapped Cross-Tunnels: cross-tunnel sculpture.

(a,b), a one cross-tunnel sculpture; and (c,d) a three

These same basic features can now be readily used in different configurations to make new derivative geometries. Figure 10a,b show what happens when I use just a single cross-tunnel and augment it with a simple 2-period Gabo curve (as found in the seam of a baseball). In yet another variation, I have placed three such cross-tunnels side-by-side and wrapped a 4-period Gabo curve around them (Figure 10c). NOME [62] was used to fill the voids between the Gabo ribbon and the inner core formed by the three cross-tunnels. The resulting 3D printed object is shown in Figure 10d.

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Figure 11 shows a sculpture that results, when three cross-tunnels are placed in a circular arrangement, so that the lower tunnels point in radial directions. This inner core is surrounded by a 3-period Gabo curve, and an additional large funnel is placed at the bottom of this arrangement to produce a stand for the sculpture. Figure 11a shows the key features placed in 3D space. To this I then add one third of the connecting control surface (Figure 11b). Two copies, rotated by ±120◦ , result in the complete topological model. After three levels of Catmull-Clark-subdivision [63] and offset surface generation (Figure 11c), a corresponding STL file is saved and sent to a 3D printer (Figure 11d).

Figure 11. The Lighthouse: (a) the key features; (b) 1/3 of the control surface is added; (c) model after smoothing and offsetting; and (d) the resulting 3D printed object.

This model has overall 6-fold C3v symmetry. Eva Hild would not create such geometries with perfect symmetry. Even if the topology offers some inherent symmetry, she would deliberately deform the geometry to break that symmetry and obtain a more organic, natural look. A drastic example of this can be found in Wholly, a metal sculpture located in the town of Borås, Sweden. 6. Wholly—A More Challenging Modeling Task Free-form surfaces, such as Wholly (Figure 12a), offer a bigger modeling challenge. The difficulties in capturing the shape of this piece demonstrate the need for a well-selected procedural description of the basic geometry. The first step is to figure out the topology and connectivity of the given surface. Here it is captured in the relatively simple model shown in Figure 12b. This is an orientable surface of genus 4 with a single border. Geometrically, it can be seen as a chain of eight side-by-side tunnels separated by seven saddle surfaces. The bottom half shows a simple polyhedral model that yields the proper topology for Wholly. However, it would be overwhelming to ask the user to move all 72 vertices of this model into the appropriate locations to recreate Wholly. Thus, we need a higher level of control!

Figure 12. Hild’s Wholly: (a) the sculpture in Borås; (b) models capturing its topology.

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For this model, I defined nine cross-sectional planes, seven of which go through the walls between pairs of adjacent tunnels, and two more that are located at the two ends of the chain. Each of these sections can now be non-uniformly stretched, rotated, and shifted. A result of such coordinated edits is shown in Figure 13a. All of the nine cross sections remains planar and symmetrical, but their sizes and positions define the shapes and orientations of the tunnels between them. The shape in Figure 13b starts to show the look and feel of Hild’s Wholly sculpture.

Figure 13. Modeling Wholly: (a) a deformed control mesh; (b) the resulting shape after smoothing.

The described approach was not yet satisfactory for modeling this particular sculpture. A key difficulty was keeping all eight tunnels close to circular. So, I looked for a higher-level procedural model that comprises circular tunnels as one of its basic primitives, accompanied by a convenient user interface to appropriately scale each tunnel and to place them snuggly next to one another at the desired rotation and tilt angles. Figure 14 shows such a procedural model, positioning eight partial toroids into a flexible chain. I did not use complete rings, because the walls between adjacent tunnels must be a single shared surface, rather than two almost coinciding surfaces contributed by two adjacent toroids. The missing connectivity between the open gap in one toroid and the ends of the adjacent one were then established in an interactive editing session in NOME.

Figure 14. A procedural model of Hild’s Wholly: (a) eight toroidal elements in appropriate positions; (b) smoothed model after linking the building blocks.

But even this remaining gluing-operation was relatively tedious and did not immediately lead to the smooth saddles found in Hild’s sculptures. Thus, in a further iteration, I focused on these saddle elements. I defined a procedural model for a saddle geometry in which I could vary the two principal curvatures individually (Figure 15a). Now the gluing operations can be done between the four open ends of two adjacent saddles. This operation now takes place in a more open space, and it was thus less difficult to produce nice, smooth transitions (Figure 15b).

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Figure 15. A procedural saddle model: (a) the basic element, (b) a chain of three elements, and (c) the end module.

For added expediency, I also introduced a special end module, consisting of a complete, well-formed toroid with a single saddle connector on one side (Figure 15c). This then allowed me to obtain a reasonably good match with the original sculpture. 7. Introducing a Möbius Rim With a reasonable modeling technique in hand, I will now focus on topological issues in Hild’s sculptures. Most of her sculptures seem to be two-sided. I have not yet found a picture of one that is clearly single-sided; but it is difficult to analyze the more complex ceramic pieces from just one or two images. This prompted me to try to construct some Hild-like 2-manifolds that are single-sided and non-orientable—like a Möbius band or a Klein bottle. A first approach is derived from the Sue-Dan-ese Möbius band (Figure 16a) [66]. It has a single circular border connected to a roughly spherical bag. But it does not look very Hild-like; its rim does not resemble a funnel, and part of it is obscured by the geometry of its bag-shaped body. Thus, I extend the circular rim into a loopy figure-8-shape, letting the border follow the tangential pull of the attached surface (Figure 16b). This leads to an undulating 3D border curve that is more akin to what is found in Hild’s sculptures. In Figure 16c, this new Möbius rim (red) is combined with a bottom funnel (blue) on which the sculpture may rest stably; in addition, part of the connecting mesh is shown in green. Figure 16d shows the resulting smooth, single-sided surface.

Figure 16. (a) The Sudanese Möbius band [66]; (b) relaxing the rim into a 3D figure-8 shape; (c) the key features: Möbius rim plus bottom funnel; (d) the resulting single-sided surface.

In a related experiment, I tried to make a single-sided version of Hild’s Interruption. With this goal in mind, I replaced the bent oval border loop of Interruption with a stretched version of the Möbius rim (Figure 16c), and I used the same internal combination of three tunnels as in Figure 8b. Figure 17a shows the skeleton formed by these key features and a few initial connecting faces (shown in red). NOME made it easy to add all the other facets to form the complete connecting mesh (Figure 17b). Two steps of subdivision already form a nice, smooth surface (Figure 17c). Figure 17d shows the resulting 3D printed model.

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Figure 17. Modeling Single-Sided Interruption: (a) a skeleton with a Möbius ribbon on top; (b) the connecting mesh; (c) the subdivision surface; (d) a resulting 3D printed model.

8. Rings of Dyck’s Disks Another approach to generate a non-orientable surface starts with a single circular disk from which two tubular stubs emerge in opposite directions (Figure 18a); this is known as Dyck’s surface [67]. If these two stubs are joined together with a toroidal loop, one obtains a single-sided 2-manifold (Figure 18b). However, such a prominent toroidal handle is not a typical occurrence in Hild’s artwork. Thus, I replaced the toroidal handle by inserting additional Dyck disks into the loop (Figure 18c,d). Any such loop formed by an odd number of disks will result in a single-sided 2-manifold.

Figure 18. (a) Dyck’s surface; (b) with a toroidal loop added; (c) five disks in a circuit; (e) seven Dyck disks in a circuit.

Eva Hild typically avoids rigid symmetry and makes the tunnels and lobes in a sculpture of somewhat different sizes (Figure 9a,b). Thus, in Figure 19, I scaled subsequent instances of Dyck’s disks by 10% and let this logarithmic spiral sweep through only 300◦ . The remaining 60◦ are then filled with a bulbous element (as found in some of Hild’s sculptures), which in this case, is a convenient way to connect the two tubular stubs of rather different diameters. This sculpture was exhibited at the Bridges 2017 Mathematical Art Gallery [68].

Figure 19. Five Dyck disks of different sizes connected into a loop: (a) the placement of the defining key features and part of the connecting mesh; (b) the resulting 3D printed model [68].

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9. Dyck Clusters of Higher Genus The Dyck disk has only two tubular connectors, so it can form only open chains or simple loops. Therefore, I created a modification with two stubs emerging from either side of the disk. Now we have a valence-4 element that can realize more intricately connected graphs. A single Dyck disk with the stubs emerging from the same side connected to one another, yields a two-sided surface (Figure 20a). If, instead, loops are formed between stubs emerging from opposite sides, a non-orientable surface results (Figure 20b). In Figure 20c, four disks have been connected in a symmetrical manner to form an orientable, two-sided surface. In the following constructions, I have moved in the opposite direction of where Eva Hild would be going; I have aimed to maximize symmetry as well as the connectivity of my 2-manifolds. To obtain a high degree of symmetry, I start with the symmetries of the Platonic solids. In Figure 21a, I have placed six 4-stub Dyck disks at the edge-midpoints of a tetrahedron. I then interconnected groups of six stubs that point towards the same tetrahedron vertex so that they form a 3-branch ring, reminiscent of a truncated corner. In Figure 21b,c, I apply the same process to an octahedron. In these latter figures, we are looking down onto a truncated valence-4 corner. The genus of these surfaces is always one higher than the number of disks, and every rim of a disk represents a topological “puncture” in these 2-manifolds.

Figure 20. Dyck disks with four stubs: (a) local connections resulting in a two-sided surface; (b) local connections forming a one-sided surface; (c) four disks forming a two-sided surface.

Figure 21. Clusters of 4-stub Dyck disks: (a) 6 disks in tetrahedral symmetry; (b) 12 disks in octahedral symmetry: CAD model; (c) octahedral 3D printed model.

In Figure 22a, twelve 4-stub Dyck disks have been placed at the mid-edge points of a cube; here we are looking at one of the original cube faces. Finally, in Figure 22b, 24 disks have been placed perpendicularly to the edges of a rhombic dodecahedron. The result is a single-sided surface of genus 25 with 24 punctures. It has been exhibited at the JMM 2018 Mathematical Art Gallery [69].

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Figure 22. Platonic Dyck clusters: (a) based on cubic symmetry; (b) based on rhombic dodecahedron symmetry [69].

10. Discussion and Conclusions In my modeling efforts assisted by CAD tools, I start with the placement of border curves and other defining features, and then I fill in surface patches between them (Figure 17 and Figure 19). Alternatively, I may place modular surface elements, so that some of their borders coincide and then merge them into a single cohesive 2-manifold (Figures 21 and 22 ). Eva Hild, on the other hand, conveyed to me, in my visit to her studio, that she mostly thinks about spaces. Her ceramic surfaces serve to define those spaces and to separate adjacent spaces. She may start with some bowl-shaped surface or some cylindrical wall and then gradually grow that surface in an organic manner. In this process, the surface may form funnels or bulbous outgrowths. Tunnels may split into two or more branches and then recombine in different ways, and border curves may warp into sensuous undulations. Twisted ribbons are not natural elements for defining spaces, so they are not typically found in Hild’s sculptures. Tubular loops, starting and ending on different sides of the same surface patch, as in Dyck’s disk (Figure 18b), would be another way to produce single-sided surfaces. Such loops rarely show up in her 2-manifold sculptures. Thus, it took some effort to introduce non-orientable surface constructs into my models, while hiding them, so that the result does not immediately stand out as something that one would not find in Hild’s studio. Our emerging NOME modeling environment [62] makes the design process for 2-manifold sculptures in the style of Eva Hild manageable, even though it is still very much in the development stage. It is particularly convenient and powerful for geometries with a high degree of symmetry. On the other hand, it is unlikely that with our current approach I will ever create something approximating the fluid, natural beauty of Hild’s sculptures. Eva explained to me that her pieces grow slowly and organically. She has an initial idea and a starting point, but this will then change and develop gradually during a process, which may take months or even years. She is often surprised when she looks at the resulting final sculpture and wonders: “Where did that shape come from?” In contrast, my own computer-based approach is much more “top-down.” I start with a well-defined plan, and specify the overall symmetry that I want to maintain. The use of symmetry significantly reduces the amount of detailed design work that I have to do. Moreover, the use of computer-aided procedures allows me to create structures of a complexity that would be difficult to achieve in a gradual, bottom-up approach. While my own creations may have a quite different look-and-feel to them, I still would like to thank Eva Hild—and many other “intuitive” artists—for the inspiration they provide.

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Funding: This research received no external funding. Acknowledgments: I would like to thank the staff of the Jacobs Institute for Design Innovation at UC Berkeley for their help in fabricating many of the sculptural models presented. Conflicts of Interest: The author declares no conflict of interest.

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D-Forms and Developable Surfaces John Sharp 20 The Glebe Watford, Herts England, WD25 0LR E-mail: [email protected] Abstract Every so often you learn of a new concept that is so simple you wonder why it was not thought of before. This paper is about such a case: the exploration of D-Forms, where new three dimensional forms are created by joining the edges of two flat surfaces that have the same length of perimeter. This then leads onto other developable surfaces, all of which offer a host of ideas for sculpture.

Introduction The concept of D-Forms was invented by the London designer Tony Wills. New surfaces are being discovered as the ideas are explored and this paper just shows a few possibilities. As a product designer Tony has developed such products as the D-Form street furniture range which uses D-Forms as moulds into which is cast artificial stone to create elegant architectural elements and they have been investigated for aircraft propeller shapes [1]. They have much in common with the sculptural forms of artists such as Barbara Hepworth, Constantin Brancusi and Naum Gabo. D-Forms are three dimensional forms created by joining the edges of two flat surfaces that have the same length of perimeter. The flat surfaces should be made of material that does not stretch or shear. This excludes woven material, this does not mean that the concept cannot be extended in that direction except that the surface will deform. Depending on where you have chosen to start the join the two surfaces, each face ‘informs’ the other what three dimensional form to finally produce. The emerging D-Form continually changes shape as the edge joining progresses. The final D-Form that results only appears when the process is complete. Two circles can only join to a flat disk. If one of the surfaces is a circle then only one D-Form results from joining it to another surface. When the perimeter curves of the two surfaces are different, a whole host of D-Forms can be created, depending on which points on the two perimeters are joined. Further possibilities arise when creases are added to the surfaces. Since D-Forms are a new mathematical discovery, their geometry has not been fully explored. The construction process is remarkable simple, but the mathematical solution of what is happening is remarkably complex. Once they have been constructed as physical objects, it is possible to gain some insight into their properties, but it is not easy to predict the surface properties from the original planar shapes, and so far no one has been able to create them in a computer system other than to approximate them using Evolver [2]. D-Forms are a special type of surface called a “developable surface”, which means the surface can be cut open and flattened into a plane. Every developable surface is also a ruled surface, which means that the surface can be defined as the path of a line that moves across the surface. This allows some experimental study of reverse modelling by creating surfaces which can then be developed.

Simple D-Forms The simplest D-Forms are created from a pair of ellipses. These can be the same ellipse or different ones, although drawing a pair of different ellipses with the same perimeter requires computer software which

can match the perimeters. There is no formula for calculating the perimeter of an ellipse. It is not easy to appreciate the way a D-Form grows before your eyes as you construct it unless you carry out the process. There is as much beauty in making a D-Form as there is in the sculptural sensuality of the finished form. To comprehend this, I suggest you try making some with the same ellipse for both surfaces being joined. Even if you do not have computer graphics software, it is easy to create an ellipse by drawing a circle in the graphics feature of a word processor and then squashing it in one direction to make the ellipse. Photocopier paper is ideal for making them. The fastest way to make D-Forms is to join the edges together by using small pieces of sticky tape. You need to keep the width small so that the edges of the two shapes join easily and are not distorted. The use of a heavy but small tape dispenser (12mm ‘invisible’ drafting tape is ideal) is highly recommended. I have also made models where the edges are joined with overlapping “fingers” but this forces the joining in a discrete number of ways. Tony Wills has also worked with thin metal and used a similar technique, using laser cutting. The surfaces can be quite large. Figure 1 shows Tony Wills with some steel D-Forms and you can see the joining technique along the edges.

Figure 1: Tony Wills with steel D-Forms One of the most interesting aspects of D-Forms is the range of surfaces obtainable from a single pair of shapes. A good illustration of this is shown in figures 2 and 3. The D-Forms are all created from the same pair of ellipses. The left D-Form of figure 2 has the end of the minor axis of one ellipse joined to the end of the major axis of the other, so forming the D shape. At the right, the two ellipses are rotated slightly before joining. This gives a slight twist which increases (figure 3) until the two ellipses are planar.

Figure 2: D shape plus slight twist

Figure 3: further rotations with more pronounced twists There are obviously many more possibilities using other curved shapes as well as multiple ones, but there is no space to show any more, since I need to consider other variations.

D-Forms with creases Some shapes seem ideal to attempt to make a D-Form from, but they don’t always behave as you might expect. For example, if you take a circle and a square with the same perimeter, the circle does not have a preferred state and crumples. However, if you crease the circle along two perpendicular diameters, the resulting D-Form takes on a definite shape. When Tony Wills discovered it, he called it the “Squaricle”. Figure 4 shows the square and circle in the correct proportions together with a brass version made for Tony Wills.

Figure 4: Squaricle There are many more possibilities using this creasing concept. For example, rectangles and other polygons could be used. The Squaricle is one D-Form surface that can be analysed [3]; and it is found to be a set of four sections of a cone from the circle and a planar square (with vertices which are the midpoints of the square) together with four parts of cylinders from the corners of the square.

Anti-D-Forms As originally conceived, D-Forms are created by joining the outside edge of two shapes with the same perimeter. In writing a book on D-Forms [3], I also considered what I called anti-D-Forms where holes are cut in surfaces and these inner edges are joined in the same way. Figure 5 shows various views of the same anti-D-Form.

Figure 5: Anti-D-Forms from elliptical holes There is also an intermediate case where the outside edge of a shape is joined to the inner edge of a hole and further refinements such as adding creases can also increase the possibilities. Figure 6 shows some variations by Tony Wills using the squaricle concept.

Figure 6: Anti-D-Form squaricles We have also explored the possibilities of using the outside edges as part a further D-Form construction.

Developable surfaces and ruled surfaces As I said above, D-Forms are a special type of surface called a “developable surface”. Strictly speaking developing a surface means bending it so as to change its form, but the term “developable surface” is commonly restricted to such inextensible surfaces that can be developed into a plane, or, in common language, “ smoothed flat”. This process is in the direction of taking the surface (which may need to be cut open) and flattening it into a plane. D-Forms are perhaps better described as “applicable surfaces”

since the process is one of taking a pair of plane surfaces and bending them to a pair of curved surfaces with a common edge. Another property of developable surfaces is that they are also ruled surfaces. It should be stressed that whereas all developable surfaces are ruled surfaces, the converse is not true. For example, the hyperboloid and hyperbolic paraboloid which are probably the best known ruled surfaces are not developable.

Figure 7: hyperboloid and hyperbolic paraboloid (as a stereo pair) The condition for a ruled surface to be developable is that two adjacent rulings must intersect (at infinity in the case of a cylinder) so adjacent rulings must lie in the same plane. The hyperboloid and hyperbolic paraboloid are both double ruled surfaces. In the case of the hyperboloid, a single and double ruling is shown. In either case, only one set of rulings (generators) need to be considered for development. It is adjacent rulings of each set which must intersect, not rulings of different sets. If you imagine the development being one of rotation about the ruled line acting as a hinge then it is easy to see that the lines must intersect in order that such a hinging will result in a flat plane. The hyperboloid and hyperbolic paraboloid are both generated by rulings which are skew, and so they are not developable. This is more evident in the case of the hyperbolic paraboloid. The consequence of this is that most D-Forms consist of pairs of surfaces which are either parts of cylinders or cones. Cylinders are special cases of cones where the vertex of the cone is at infinity. (Note that the cylinders are not necessarily circular cylinders nor are the cones necessarily right circular cones.) The rulings in the surface of a cone intersect at the vertex (figure 8).

Figure 8: cone with ruling lines This shows why the cone is the most well known developable surface. Since the surfaces of most D-Forms are either parts of cylinders or cones, then if you take a ruler and move it over the surface, the ruler either moves in a direction perpendicular to the rulings or rotates about a point. Figure 9 is a triple surface D-Form which is clearly seen to consist of two conic surfaces and a cylindrical surface.

Figure 9: a crescent D-Form Whereas it is not possible to compute D-Forms by starting with two planar shapes, because they are either intersections of cylinders or cones, it is possible to “reverse-engineer” some shapes by for example taking two cylinders and intersecting them and then developing the result. An example of such a D-Form is shown in figure 13.

The Möbius strip paradox Taking a strip of paper and twisting the ends and joining them creates what is probably the most famous developable surface, the Möbius strip or band. This is often described mathematically as a combination of two rotations of a segment of a line. Take a line segment which starts off parallel to an axis and rotate the segment about the axis while at the same time rotating the segment about its centre as it goes round the circle of revolution formed by the centre.

Figure 10: Möbius strip So we have a paradox that as everyone knows the Möbius strip to be created as a developable surface but the common textbook illustration as shown at the left of figure 10 is not developable even though the surface is obviously a ruled surface! Adjacent lines of the surface generated in this way do not intersect but are skew by the very nature of the rotation about their centre. However, if you triangulate the surface, then it is possible to develop it using software like Pepakura [4] as shown in the right of figure 10. This compounds the paradox, and there are many cases in books of sheet metal work which show theoretically non-developable surfaces which are developed for example Abbott, p 279.

Tangential developables If the generating lines of a developable ruled surface are not parallel they must meet since they are in a plane when the surface is plane. If they do not meet all in one point, they must meet in several points and in general, each one meets its predecessor and its successor in different points. Such a surface is called a tangential developable since the developable lines will in general be tangents to a curve (the locus of the points of intersection when the number is infinite). This curve is called the edge of regression of the surface. In the case of a cone, the edge of regression is a point which is at infinity for a cylinder. One of the ways to create a model of tangential developable is described by Thompson and Tait [5] as follows with their figure redrawn as figure 11.

Figure 11 “To construct a complete developable surface in two sheets from its edge of regression. lay one piece of perfectly flat, unwrinkled, smooth-cut paper on the top of another. Trace any curve on the upper, and let it have no point of inflection, but everywhere finite curvature. Cut the two papers along the curve and remove the convex portions. If the curve traced is closed, it must be cut open (see second diagram). Attach the two sheets together by very slight paper or muslin clamps gummed to them along the common curved edge. These must be SO slight as not to interfere sensibly with the flexure of the two sheets. Take hold of one corner of one sheet and lift the whole. The two will open out into the two sheets of a developable surface, of which the curve, bending into a curve of double curvature, is the edge of regression. The tangent to the curve drawn in one direction from the point of contact, will always lie in one of the sheets, and its continuation on the other side in the other sheet. Of course a double-sheeted developable polyhedron can be constructed by this process, by starting from a polygon instead of a curve.” If only they had not cut the curve open and rotated it slightly, they would have discovered anti-D-Forms, which would probably have taken them on the path to D-Forms some 120 or so years ago. Their method is also described in Koenderik [5] who says to take a series of circles on two sheets of paper. These form the tangential developable helicoids of a helix which is the edge of regression. He shows how the two halves of the tangents to the circles form the two surfaces which meet at the edge of regression (figure 12).

Figure 12 The helicoids are not as prominent as in the anti-D-Forms of figure 5. Tangential developables are easy to program as ruled surfaces. If you have a space curve, you can just extend the tangents. Figure 13 shows a D-Form which has been created by the intersection of two elliptical cylinders. The edge curve was then extracted from this and the two tangential surfaces created as shown as two views of figure 14.

Figure 13: simulated D-Form

Figure 14: tangential developable The two surfaces result from the two halves of the tangent of the space curve. Thompson and Tait and Koenderik’s example of figure 12 took two separate circles and produced the edge of regression by twisting to form the space curve. When you take a space curve as in the edge curve extracted from the example of figure 13, the two tangent halves create the surfaces. This is not an intuitive result, and like constructing D-Forms, needs to be performed to be understood. What is interesting in this case (and also counter-intuitive) is that the two tangential developable surfaces intersect one another.

Concluding thoughts The more you explore this simple concept the more you wonder why it was not thought of before. There has only been space to touch on the fundamentals of the subject in this paper. It has also not been possible to show the dynamic way a D-Form takes on a life of its own as is constructed. The only way you can understand this is to make one for yourself. The subject raises many questions and suggests many avenues to explore. References 1. Tony Wills has examples of his D-Form work at www.wills-watson.co.uk/proj_street_01.html Figure 1 is copyright Tony Wills 2004 as is the Squaricle shown in figure 4. 2. Paul Bourke http://astronomy.swin.edu.au/~pbourke/surfaces/dform/. He may reorganise his site, so this may change. 3. John Sharp, “D-Forms: surprising new 3D forms from flat curved shapes”, Tarquin 2005 4. Jun Mitani, Pepakura Designer, www.tamasoft.co.jp/pepakura-en 5. Sir William Thomson (Baron Kelvin) and Peter Guthrie Tait, “A Treatise on natural philosophy”, 1888 6. Jan J. Koenderik, Solid Shape. MIT 1990 7. W Abbott, “Practical Geometry and Engineering Graphics” Blackie London1943

A Technique for Constructing Developable Surfaces Meng Sun Eugene Fiume Department of Computer Science University of Toronto 10 King’s College Road Toronto, Canada, M5S 1A4

Abstract

sheets. In computer graphics, we are interested in modelling and animating objects seen in everyday life, and many objects can be approximated by piecewise continuous developable surfaces. Our aim is to work directly with developable surfaces as first-class modelling primitives for computer graphics. Our focus in this work is on the application of the theory of developable surfaces to the interactive creation of simple geometric models. Developable surfaces may be deformable, but they have strong isometric properties. Because they can be easily parameterised so as to preserve arc lengths, they are excellent candidates for texture mapping [2][12]. However, developable surfaces are a small subclass of the polynomial or algebraic surfaces. When manipulating surfaces defined using common piecewise polynomial surface formulations, it is easy to violate isometry properties. For instance, if we simulate the tearing of a piece of paper using an elastic model, the corresponding surface defined using a conventional formulation would appear to stretch and shear unnaturally during the tearing process. In general, it seems fruitful to devise modelling systems that are able to represent developable surfaces directly, and provide manipulation techniques that preserve their isometric properties. In Section 2, we define and introduce properties of developable surfaces. In Section 3, we describe our new developable surface modelling technique. In Section 4, we present a hanging scarf and a bow modelled using our new approach. In Section 5, we discuss possible extensions to the system and areas of further research.

Paper, sheet metal, and many other materials are approximately unstretchable. The surfaces obtained by bending these materials can be flattened onto a plane without stretching or tearing. More precisely, there exists a transformation that maps the surface onto the plane, after which the length of any curve drawn on the surface remains the same. Such surfaces, when sufficiently regular, are well known to mathematicians as developable surfaces. While developable surfaces have been widely used in engineering, design and manufacture, they have been less popular in computer graphics, despite the fact that their isometric properties make them ideal primitives for texture mapping, some kinds of surface modelling, and computer animation. Unfortunately, their constrained isometric behaviour cuts across common surface formulations. We formulate a new developable surface representation technique suitable for use in interactive computer graphics. The feasibility of our model is demonstrated by applying it to the modelling of a hanging scarf and ribbons and bows. Possible extensions and interesting areas of further research are discussed. Keywords: computer-aided geometric design, surface modelling, developable surface, surface flattening

1 Introduction The study and use of developable surfaces has a long history [4]. Real developable surfaces have natural applications in many areas of engineering and manufacturing. For instance, an aircraft designer uses them to design the airplane wings, and a tinsmith uses them to connect two tubes of different shapes with planar segments of metal

2 Developable Surface Modelling Any surface whose (Gaussian) curvature vanishes at every point can be constructed by bending a planar region. These are developable surfaces. By their definition and 1

their intrinsic properties, developable surfaces can be flattened onto a plane without stretching or tearing. In theory, the length of any curve drawn on such a surface remains the same, and the area of the developable surface also remains the same [6]. This is to say that curves on developable surfaces admit an easy arc-length parameterisation, and subregions of these surfaces have a direct surface-area parameterisation.

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Figure 1: Generators of a developable surface.

2.1 Definitions


  

We take our definitions of ruled and developable surfaces from [4]. Let be a closed real domain. A one-parameter family of lines for a differentiable space curve and a vector field is a correspondence that assigns to each a point on 0. For each and a vector , , the line passing through that is parallel to is called the line of the family at . For a oneparameter family of lines , the surface

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is the ruled surface generated by that family. The lines are the rulings, and the curve is a directrix of the surface X. A ruled surface is developable if

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perspectives. The modelling problems can be formulated in different ways. Given two distinct space curves, the classic problem of constructing a continuous developable surface connecting the two space curves has been extensively studied [5, 15]. A way is described to reconstruct a smooth developable surface from a given set of data points, where the data points are the spherical images of corresponding points of a geodesic of the original developable surface [11]. Redont approximates the spherical image of the geodesic, and builds a family of circular cones, each with a geodesic segment that corresponds to one segment of the original geodesic. Then Redont forms the desired developable surface using patches of the circular cones. In [1], Aumann discusses a different developable surface modelling problem: given two distinct line segments and in and a number of constraints, how do we determine a developable surface whose four boundaries are , , a B´ezier curve with end points and , and another B´ezier curve with end points and , provided that the constraints are satisfied. Aumann derives a number of theorems concerning the properties of developable surfaces under such constraints. Simulating the bending of a developable surface is also an interesting problem. In [8], Kergosien, Gotoga and Kunii studied the bending of a developable surface under external and internal forces. In [3], Bodduluri and Ravani introduce a representation for developable surfaces in terms of plane geometry,

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are coplanar for all points on the surface. The notation signifies the inner product operator. The simplest examples of developable surfaces are cylinders and cones, and the simplest non-developable surface is a sphere. A generalized , cone is a ruled surface generated by a family , where is contained in a plane and the rulings all pass through a point . Every surface enveloped by a one-parameter family of planes is a developable surface. Each plane in this family is tangent to such a surface along a line that is obtained as the limiting position of the line in which two neighbouring planes intersect. Since the totality of these straight lines covers the entire surface, as shown in Figure 1, these straight lines are called the generators of the surface. In some cases, the generators envelop a space curve at which the developable surface has a sharp edge called the edge of regression or cuspidal edge, as shown in Figure 2.

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using the concept of duality between points and planes in 3D projective space. The idea is to design a developable surface using control planes with appropriate basis functions. In [3], this approach is demonstrated using B´ezier and B-spline bases. Geometric construction techniques, such as the de Casteljau and Farin-Boehm-type construction algorithms, are extended to the design of developable surfaces. The conversion from the dual form to a point representation is based on the line of regression. In [10], Pottmann and Farin present an approach to constructing B´ezier and B-spline surfaces, based also on the dual representation in the sense of projective geometry. They transferred projective algorithms for NURBS curves to constructions for developable NURBS surfaces in dual rational B-spline form, using control planes, frame planes and two reference planes. The two reference planes are chosen dependent on the application. In [10], a new method for converting a dual NURBS surface to the usual NURBS tensor product form is discussed. This method takes advantage of fact that the planar intersection curve of a developable NURBS surface is a NURBS curve. In this paper, we are primarily concerned with geometric considerations for modelling with developable surfaces, although we hope to generalise our work to physical models. Our new technique is inspired by the formulation of the classic problem and some ideas related to Redont’s work.

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3 A New Modelling Primitive Our new developable surface modelling technique reduces the geometric concept of a developable surface to a relatively simple visual specification. We use generalized cones to outline the shapes of segments of a developable surface. We divide the surface into several patches based on the geometry of the surface, as shown in Figure 3(a). continuous deIn this paper, we restrict ourselves to velopable surfaces without cuspidal edges. Currently our modelling tool is set up to deal with sequential developable surface patches only, and it handles sheets with polygonal boundaries. We can easily extend the implementation to handle flattened sheets of arbitrary shape, but the underlying principle is clearer in the polygonal case. We approximate each patch by a generalized cone as shown in Figure 3(b). To define a generalized cone, we specify a cross section and the position of the apex in relation to this cross section as shown in Figure 3(c). The specification of a piecewise developable surface is done at four levels of detail, as shown in Figure 4. From level 4 to level 1, the main task of constructing a developable surface is broken into more elementary subtasks. At level 4, a developable surface is desired. At

(c) Use piecewise continuous Bezier curves to define a cross section of a generalized cone.

Figure 3: Dividing a developable surface into patches and defining each surface patch by a generalized cone.

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level 3, the developable surface is divided into surface patches. At level 2, each patch is associated with a generalized cone. The shape of a patch can be determined by the shape of the corresponding cone and the cut the curve makes through the cone. Several patches can be associated with the same generalized cone, possibly with different initial conditions. At level 1, each cone is specified by a cross section of the cone and the relative position of the apex and the cross section.

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3.1 Implementation Using the data flow diagram shown in Figure 4, we can construct a developable surface in three steps. 3

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Figure 4: Flow of specification. The user specifies curve segments drawn from cross-sections of cones. The subsequent patches are inferred by the implementation.

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3.1.2 2D Mapping Between Cones and Surface Pieces In this step, our task is to flatten out the user defined cones and determine the relationship between a patch and the corresponding flattened cone in 2D. First, we want to flatten a user defined cone. As shown in Figure 6, when a cone is flattened out, for a given point on the surface patch, we can easily locate the generator passing through it. Clearly, when the cone is flattened out, the user-specified cross section corresponds to a plane curve. When dealing with a generalized cone, it is difficult to write down a closed form formula for this plane curve. However, we may use a system of differential equations and solve the problem numerically using a procedure described in [9]. Assume the curve is parametrized in terms of . We can determine the profile of this plane curve using the system of differential equations

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Figure 5: A 2D partitioning of a surface.

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dc , where [  is the arc length of the X plane curve,  is the angle subtended by the tangent of the plane curve on the H -axis as shown in Figure 7 [9]. Note that \/ is the curvature at point G  of the cross section of the cone in 3D, and the curvature at point G  of a curve on the developed surface is equal to the curvature of the projection of the original curve onto the tangent plane of the surface at G  [5].

3.1.1 Surface Subdivision and Cone Constructions First, the user observes the developable surface properties of the object to be modelled. Then the user has to decide on the subdivision of the surface. To specify the subdivision, the user needs to flatten the surface and inputs the coordinates of a sequence of points along the boundary of the surface in 2D. Then he/she can use these points to specify the partition of patches. An example is shown in Figure 5. When the surface is subdivided into patches, we would use the shapes of generalized cones to outline the shapes of patches. For each patch, the user can define a cone by specifying a cross section and the position of the apex in relation to an interactively specified cross section as shown in Figure 3(c). The initial conditions will determine the region that needs to be trimmed from the generalized cone to obtain the patch, as we shall see in step 2. Up to this point, each cone is defined in its own 3D local coordinate system.

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As shown in Figure 8, is parallel to in the plane embedding the flattened surface. In this case, the shape of the developable surface in 3D cannot be defined by a generalized cone, because after development we would still have in 3D. The shape of the developable surface in 3D can be defined by a generalized cylinder instead. In our system, this case is not currently implemented. To incorporate generalized cylinder into the model is easy, and it follows similar principles as those of generalized cones, i.e., flattening the cylinder out in 2D, finding the correspondence between points on

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sect at a point . In this case, the apex of the corresponding cone has to coincide with the point . This is because there is a one-to-one correspondence between generators of the patch and generators of the associated cone patch. Since the last generator and the first generator corresponds to two distinct generators of the cone, and two generators of a cone must intersect at the apex of the cone, clearly, the apex should coincide with the point . The user can specify a special generator of the cone which corresponds to either or as one of the initial conditions mentioned in step 1. Then the relation between the cone and the patch can be determined as shown in the figure below.

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,f , , CASE IV: The case where , I and " are distinct , , N I vertices, and coincides with is symmetric to 3.1.3 Constructing a 3D Developable Surface Using the 2D mapping shown in Figure 6, we can find, for each point on the surface patch, its counterpart on the cone. Since each cone is defined in its own 3D coordinate system, we can also represent the ribbon patch in this 3D local coordinate system. Next, we position the surface patch in the 3D world coordinate system. Since currently the system is set up to deal with sequential developable surface patches, let the developable surface patch be numbered , and let the last surface generator of the surface patch be the first surface generator of the surface patch. To properly connect the two adjacent patches, we want the shared surface generator to be correctly aligned and the surface normals of each patch at that boundary to be parallel. A unique linear transformation matrix can be determined using these constraints. Assume that surface patch is already in its proper position. Our algorithm connects the patch to the surface in the iteration for , i.e., by the end of the iteration, patches through are in their proper positions in the 3D world coordinate system. At the beginning of the iteration, we consider patches and . We need some notation. In the world coordinate system, let be the final generator of patch , the unit direction vector of is g , the unit surface normal of patch along is n, and the position of is .

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