Symmetry: Through the Eyes of Old Masters 9783110417142, 9783110417050

Table of contents :
Acknowledgments
Contents
Introduction
1 Fundamentalcategories
1.1 Symmetries of finite objects: Zero-dimensionalornaments
1.2 Friezes and guilloche
2 The convenient start: Plane groups of symmetry
3 Intertwined patterns: Layer groups of symmetry
4 Two-colored periodic ornamentation
5 Polychromatic patterns
6 Beyond 2D groups: Hypersymmetry, superstructures, two symmetries in one pattern, the “order-disorder” patterns, homothety and similarity, inversion and nonlinear patterns
6.1 Hypersymmetry
6.2 Superstructures and designs with several levels
6.3 Two or more symmetries in one pattern
6.4 Twinning
6.5 The “order-disorder” patterns
6.6 Homologous series
6.7 Inversion in a circle
6.8 Homothety and similarity
6.9 Chirality, enantiomorphs, hands
6.10 The nonlinear art of Hans Hinterreiter
7 Quasiperiodic patterns
8 Fractals and fractal character
9 Style and symmetry – symmetry and style
References
Index

Citation preview

Emil Makovicky Symmetry

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Emil Makovicky

Symmetry

| Through the Eyes of Old Masters

Author Prof. Dr. Emil Makovicky University of Copenhagen Institute for Geoscience and Natural Resources Østervoldgade 10 1350 Kobenhavn Dänemark [email protected]

ISBN 978-3-11-041705-0 e-ISBN (PDF) 978-3-11-041714-2 e-ISBN (EPUB) 978-3-11-041719-7 Set-ISBN 978-3-11-041715-9

Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Cover image: Emil Makovicky Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

| To Milota, Niki and Peter

Acknowledgments Many thanks to my wife Milota for her interest, support and an infinite amount of patience. Many thanks to Mrs. Berber-Nerlinger (DeGruyter) for the idea of the book and for gentle prodding along the long tortuous way of preparing the manuscript. Mrs. Britta Munch prepared many black-and-white drawings while she was employed at the institute. And my thanks to all those helpful people I met at four corners of the world while collecting and preparing these figures, including unnamed private collectors. And thanks to the Geoscience Institute of the University of Copenhagen for enabling me to prepare this contribution to the crystallography of culture.

Contents Acknowledgments | VII Introduction | XI 1 1.1 1.2

Fundamental categories | 1 Symmetries of finite objects: Zero-dimensional ornaments | 1 Friezes and guilloche | 20

2

The convenient start: Plane groups of symmetry | 35

3

Intertwined patterns: Layer groups of symmetry | 92

4

Two-colored periodic ornamentation | 111

5

Polychromatic patterns | 133

6

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Beyond 2D groups: Hypersymmetry, superstructures, two symmetries in one pattern, the “order-disorder” patterns, homothety and similarity, inversion and nonlinear patterns | 148 Hypersymmetry | 148 Superstructures and designs with several levels | 150 Two or more symmetries in one pattern | 151 Twinning | 157 The “order-disorder” patterns | 161 Homologous series | 166 Inversion in a circle | 166 Homothety and similarity | 169 Chirality, enantiomorphs, hands | 174 The nonlinear art of Hans Hinterreiter | 176

7

Quasiperiodic patterns | 179

8

Fractals and fractal character | 203

9

Style and symmetry – symmetry and style | 217

References | 235 Index | 239

Introduction Imagine yourself standing in the Patio de los Leones in the Alhambra, surrounded by the rich plaster ornamentation of the walls, ceramic panels composed of colorful tiles, ornate arches surmounting the corridors, ornamental wooden ceilings, and gracious columns (Fig. 1). Or, imagine yourself in front of the Gothic cathedral with an openwork spire, multiple turrets of different sizes, ornamental windows with stone tracery, and pointed or broken arches everywhere you look.

Fig. 1: A stucco illustration of paradise, Patio de los Leones, the Alhambra, Granada, Spain.

If it is too much for your taste, imagine yourself standing in front of a Renaissance palace with its regimented adornment of windows and ornamental entrance gates

XII | Introduction

(Fig. 2). All this beauty is composed of geometric elements; only some of them derive from, or resemble, (usually) floral forms. Finally, imagine yourself standing in front of glass cases with literally hundreds pieces of Pueblo pottery or of Iznik tiles in a large American museum (Fig. 3). Ornamental art has been with mankind ever since man started creating objects, whether for practical use or as adornment of his and her body or of their living space.

Fig. 2: A Renaissance door in the Vladislav Hall of the Castle of Prague. Point group of symmetry 2mm.

Throughout history, visual arts experienced a near permanent dichotomy between the depiction of visual reality on the one hand, and “pure” abstraction in more or less geometric forms on the other hand (Fig. 4). The boundaries between them may be uncertain, the object and subjects depicted may be, and always are, subjected to abstraction to very different degrees (for example “The Kiss” by Gustav Klimt), and the elements of ornamental art often come from the world of physical reality and/or biology in broadest sense, although then they were subjected to the rules of geometry. Attitudes to these two branches of visual art kept changing throughout history, sometimes one of them greatly prevailed, and especially the geometric ornamental art has been kept for something lower in recent European history, being “the applied art” of the artisans. In spite of this attitude, which often was a reaction to the previous, heavily ornamental period (Renaissance after Late Gothic, Classicism after Rococo, Loos’s

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XIII

Fig. 3: A superb Iznik tile with red tulips in a famous “Armenian bole” red pigment. Point group symmetry m. Rüstem Paşa Mosque, downtown Istanbul, Turkey.

constructivism after Art Nouveau (Fig. 4), etc.), geometric artists created a close to infinitely large, ever changing and evolving body of beautiful ornamental art. The history of ornamental art contained equally original genii as the well-known trailblazers of the “reproducing” art of visual reality (e.g., Leonardo or Bernini). On the other hand, both branches exhibit long periods of tiresome copying, reproducing the original canon of the founding fathers from the latest creativity revolution (e.g., the Chinese vases made and sold today or the pre-Impressionist academic art), in want of the next creative genius and style revolution. To close this paragraph, my answer to an anonymous influential Danish art history teacher is: you were wrong, dear sir,

XIV | Introduction

excellent designs created with help of compasses and a ruler are art; those created with free hand and brush must be equally excellent before they can be called art.

Fig. 4: Ornamental entrance to Castel Béranger (1894–1898; Hector Guimard), Paris. Art Nouveau with its love of complete asymmetry.

Geometric ornaments were an important part of human culture from at least Neolithic times, when the technologies of the day – bricklaying, as well as textile and basket weaving contributed directly to the notion and development of a symmetry concept. The majority of geometric ornaments are periodic in one or two dimensions. This implies that they are subjected to the laws of symmetry which allow “freedom of choice within rules”. These rules, often masterly utilized by the outstanding artists to their

Introduction

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XV

advantage, can be expressed by the exact language of crystallography instead of the usual, less precise artistic definitions.

Fig. 5: A vertical mirror plane (reflection symmetry) in a ceramic revetment of the mosque Masjid-eImam, Isfahan, Iran.

In this little book, we shall concentrate on two-dimensional ornaments which form the richest body of beautiful patterns. We shall deal also with the examples of finite (“zerodimensional”) ornaments, which are nonperiodic and are centered on a single point or line, and with one-dimensional “frieze” ornaments. And we shall use symmetry (or

XVI | Introduction

symmetries) as a common denominator, structuring and tying together the illustrated art objects. The fundamental symmetry operations – reflection on a “mirror” plane (Figs. 3 and 5), inversion on an inversion center, rotation axes of different order (Fig. 6), glide reflection on a plane (Fig. 7) and screw axes of different type – have been defined and described in numerous textbooks and contributions on mathematics and symmetry; here they are introduced in form of illustrations. For all types of periodic patterns, these symmetry operations can be combined in a precisely limited number of internally compatible combinations, the so-called groups of symmetry. In these groups, it does not matter in which sequence we apply the individual symmetry operations. After a complete set of operations has been exhausted, one obtains a complete and final set of positions and orientations of the initial motif, and a complete pattern. Symmetry groups of several categories will be introduced here by graphic means.

Fig. 6: Point group 6 (or 3 if details of lobe fill are considered). Islamic plaster ornament from the Archeological Museum of Iran, Teheran.

There are good accounts/listings for plane groups of symmetry (Schattschneider 2004, Washburn & Crowe 1998 and 2004, Kapraff 1991, Horne 2000, Abas & Salman 2007, among others), layer groups of symmetry (Kopsky & Litvin 1990), dichroic groups (Washburn & Crowe 1998, Crowe 1986, Shubnikov & Koptsik 1974), color symmetry (Loeb 1971, Coxeter 1987), and a detailed analysis of Islamic treatment of color symmetry (Makovicky & Fenoll 1999, Makovicky 2011b), as well as treatments of simi-

Introduction

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XVII

Fig. 7: Art Deco house entrance doors, Paris, 16th arrondisement. A vertical glide plane.

larity symmetry, frieze and rod groups (Shubnikov & Koptsik 1974, Kopsky & Litvin 1990, Jablan 1995, Castéra 1996, Grünbaum & Shephard 1987, Makovicky et al. 1998, among others). A complete collection of one- and two-dimensional monochroic symmetry groups is to be found in some of the volumes of International Tables for Crystallog-

XVIII | Introduction

raphy (especially the volume E) published by the International Union of Crystallography (IUCr); they are available on the net and they contain internationally approved symbols of symmetry groups (these symbols used to vary substantially in the past). Crystallography and art have been combined several times, for example by Washburn and Crowe 1998, Abas and Salman 2007, Shubnikov and Koptsik 1974, Makovicky and Fenoll 1999, Makovicky 2011b, Makovicky et al. 1998, Castéra 1996, Hann 2014, among others. In the present publication, the beauty and artistic value of the illustrated patterns will be weighted as equal to their symmetrological value and it is hoped that the reader will cherish them as much as the author did when he was facing them “in the field”. This publication does not intend to be yet another theoretical and systematic account of 1D and 2D symmetry as are some of the good books quoted above. It is not a book intended for specialists of symmetrology but written for other categories of a (somewhat) specialized audience. The aim of this little book is entertainment and inspiration, in whichever order the readers choose. I can only hope that you will not find it too difficult or, on the other side, not deep enough. Especially, I did not try to draw symmetry operations on every nice photograph – it is left to you to draw them mentally, as an exercise in symmetry. However, this book contains a number of problems which were not treated before, to the best of my knowledge, and perhaps are

Fig. 8: A Hopi bowl (Arizona, USA) from 1930s. Private collection. Point group 1 (acentric) with local symmetries and local color antisymmetries.

Introduction

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XIX

sketched here for the first time. The field of art symmetry is widely open for more research and publications. The illustrations come from the author’s camera, often obtained by the “boots and packsack” method which, besides the objects themselves, allows enjoyment of their position, general setting and social environment. These, however, are often difficult to reproduce in a book. The objects were photographed in their “natural” state, neither modified nor manicured. Some of them, however, underwent local renovation at a point in time, sometimes a careful professional one, another time a tourist-oriented or a romanticism-tainted variety. Those which were untouched by a later hand, with the traces of all what happened to them in the course of time, were preferred here. In this, this book differs from many books on art, which often illustrate the reconstructed buildings and their reconstructed details instead of the much less “representative” set of true remnants before reconstruction. One such example is the commonly used pictures of the facades of the Grand Mosque of Cordoba, which truly are a result of a rather romantic reconstruction by V. Bosco and M. Inurria in the years 1880–1928. So, you have to put up with chipped tiles and discolored glazes in this book. This is not an evenly balanced representation of all countries and cultures – it is a representation of beautiful or of very interesting symmetries as I managed to visit and record them. It is true that Islamic art is overrepresented here but it is the art with the most developed symmetries and often also with the most beautiful ornaments.

1 Fundamental categories 1.1 Symmetries of finite objects: Zero-dimensional ornaments Ornamental art abounds with finite elements/motifs, either placed in isolation or positioned on a periodic pattern. Those forming a part of a two-dimensional periodic pattern have their symmetries limited to those permitted by the periodic pattern – a selection of reflections and of rotations of the order (1), 2, 3, 4, and 6, and for two-sided layers also inversion (Fig. 1.1).

Fig. 1.1: A marble floor mosaic from the church San Miniato al Monte, Firenze, Italy. Point group 4mm in agreement with the plane group symmetry of the floor grid.

Free elements in a plane (Fig. 1.2) lack the restrictions imposed by periodicity and can, and very often do, display rotations of the order 5, 7, 8, 9, 10, 12, 16, 24, etc. to which spiritual meanings might or might not have been ascribed. The N-fold rotation axes perpendicular to the plane can be combined with mirror planes which intersect one another in the rotation axis. The multiplicity N of the rotation axis of free elements is unlimited, the number of intersecting mirror planes is determined by evenor odd-multiplicity of the rotation axis, with examples such as 2mm, 3m, 6mm, 7m and 8mm (Fig. 1.3). Total asymmetry is sometimes the means of expressing an artist’s intentions (Fig. 1.4). The combination of rotation with homothety will be mentioned later. The art objects with zero-dimensional extension can also be situated in space and developed with full three dimensions, e.g., pots, vases or architecture capitals, and not only in plane, e.g. drawings and reliefs. Symmetry groups found in space are correspondingly more complex than those found in plane (Figs. 1.5 and 1.6). Symmetry

2 | 1 Fundamental categories

Fig. 1.2: Fourfold and eightfold calligraphic ornaments on a madrasa wall, Isfahan, Iran.

groups in space may involve a possible symmetry plane perpendicular to the rotation axis and/or inversion in the center of the body, as well as rotoinversion axes which have composite symmetry operations which consist of rotation and inversion at every partial step. Furthermore, cubic and icosahedral systems involve cases with more than one unique axis of higher order. Any modern textbook of crystallography will serve for these, in art fairly rarely encountered categories of symmetries. A prominent artistic expression of point groups in plane, based on rotation symmetry, with or without reflection planes, is seen in the rose windows of the Gothic Age. These adorned the gables of Gothic cathedrals (Fig. 1.7) and together with the stained glass (Fig. 1.8) they created a mysterious celestial atmosphere for medieval

1.1 Symmetries of finite objects: Zero-dimensional ornaments

| 3

Fig. 1.3: Eightfold star with a fourfold fill, point groups 8mm and 4mm, respectively. Isfahan, Iran.

Fig. 1.4: Asymmetric ornament on a window shade. Castel Béranger, 16 Rue la Fontaine, Paris (Guimard 1894–1898).

4 | 1 Fundamental categories

Fig. 1.5: Capital and abacus of the column in the Alhambra Patio de los Leones. Fourfold rod symmetry defines the entire column and capital portions. Inscription: Izz li-mawla-na Abu Abd Allah (Glory to our master Abu Abd Allah).

worshippers. Abad-Zapatero (2014) quotes the statistics of rotation-axis multiplicity by Cowen (2005), obtained from 524 windows he studied: up to 27 % are windows with a 12-fold rotation axis, followed by about 21 % of windows with 8-fold rotation, about 17 % with 6-fold rotation axis, 9 % with 16-fold and 8 % with 4-fold rotation axis. Only about 4 % are 10-fold windows; the rest are 3-, 5-, or 24-fold windows while there are almost single cases with other rotation multiplicity. These statistics do not distin-

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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5

Fig. 1.6: A door knocker from Fez, Morocco. Symmetry 10.2.2; background door symmetry 16.2.2.

guish between the designs with mirror planes and the “spiral” windows (Cowen 2005) without reflection symmetry. A very interesting family of point-group designs is the infinite variety of famous Japanese heraldic kamon symbols (Fig. 1.9). Kamon were originally used by Japanese aristocratic families (Fig. 1.10), later also by commoners, guilds and temples. There can be one, three or five mon on a formal kimono. The medieval use of mon for friend-andfoe recognition in battles between clans is well known. The pictorial contents of kamon are rich: animals, plants, objects of everyday life and work, and pure geometric forms (Fig. 1.11). Collections of kamon, such as that by Hondasu Ichiro (1992), show 2D point groups (from 36.m.m to 1) and other symmetries which were skillfully derived from the original symmetry of the object depicted. Furthermore, they may contain combinations of 2–3 different symmetries in one mon, as well as (projected) 3D groups of symmetry, and further motif varieties situated outside a point-group definition. As we can see in the illustrations given here, kamon adorned or marked the property of nobles or temples, from castles (Fig. 1.12) to small objects (Fig. 1.13), and they still form part of the atmosphere of historic Japan. Statistics of representative kamon from the mentioned collection will be demonstrated on the variety of kiku and kiri kamons. Chrysanthemum (kiku) was an old Japanese Imperial kamon, protected by law. It is a frontal view of a schematized flower with sixteen petals, a point group 16.m.m, rarely also 8mm, 10mm, 12.m.m and 21.m, with the corresponding number of petals. A petal overlapping variety is 16.2.2. The symmetrical motif can be dissymmetrized to plane groups 1, m,2mm (lozenge-shaped chrysanthemum) and a rare case of 3. Chrysanthemum forms a number of compositions: pairs (point groups 2, m, 2mm), often triplets (m, 3m (presented as if enveloping a sphere). The trend of patterns “en-

6 | 1 Fundamental categories

Fig. 1.7: Notre Dame Cathedral, Paris (1163– about 1250). Two 12-fold rosette windows on a transept. Point group symmetry 12mm.

veloping a sphere” continues for groups of 4, 5, 8 and 9, composed of not always equally large chrysanthemums, with resulting symmetries 3m and 4mm. Chrysanthemum participates in combinations with leaves, plants, flowers and geometric motifs (Fig. 1.10) which form a bulk of point group occurrences of 3m, 2mm, m,2 and 1. The latter comprises especially entire “garden scenes” with growing chrysanthemums.

1.1 Symmetries of finite objects: Zero-dimensional ornaments

| 7

Fig. 1.8: Ste Chapelle, Paris (1243–1248) is famous for its stained glass windows made in thirteenth century. A 16-fold rosette window, symmetry 16mm.

Fig. 1.9: Kamon: Katabami and kiri (at the bottom) mon from the roof of the Himeji Castle, Japan.

Chrysanthemum can also be centering a ring or wreath of other objects (leaves etc.) in which case it displays the above sketched gamut of symmetries, often different from the enveloping motif (e.g., 3m inside and 3 outside). By its character, kiri (Fig. 1.9) has predominantly point group symmetry m with small motif variations. Interestingly, next in abundance are asymmetric motifs (point group 1), and then the 3 and 3m cases, with three vegetal groups growing either from a center or from a limiting circle. The 2 and 2m cases are rare and 4mm, 5, 5m, 6, and 6mm are unique but impressive. A pure limiting circle is rather frequent. The spectrum of point groups 1 – m – 3m – 3 – 2 - 2mm seems to repeat for other examined kamon

8 | 1 Fundamental categories

Fig. 1.10: Imperial chrysanthemum (kiku) mon. Nikko Complex, Japan, built in the Edo period (from the seventeenth century on) under the “Japanese Baroque” period. Richly carved, gilded and brightly painted buildings are supplemented with rich furnishings.

Fig. 1.11: Kamon: Katabami (point group 3m), kiri (m), chichó (1), tomoe (3) and wameki (point group 6mm); mon type designations from Hondasu Ichiro (1992). Himeji Castle, Japan.

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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9

Fig. 1.12: Himeji Castle (built by Tokugawa Ieyasu 1601–1609).

Fig. 1.13: Inwards oriented aoi (probably Asarum plant) mon. Nikko Complex, Japan.

types (e.g., based on gingko or pine), except for the frequencies of 1 versus m cases which can alternate. Chinese ornamental art can be appreciated primarily in numerous temples and ceremonial halls in all larger towns. These consist of wooden pillars, mostly red lacquered and unadorned, except for the top portions where a complicated system of

10 | 1 Fundamental categories

rafters combined with brackets of great ornamental complexity (allegedly an antiearthquake arrangement) carries a heavy tiled roof and a cassette ceiling. This determines the distribution of pattern types. Interesting one-dimensional patterns occur (pmm, pgm, pm(along), pg, even p1, for symbols see Section 1.2) but the bulk of patterns on rafters are composed of zerodimensional ornaments at the rafter end (hezi), followed by a relatively short luxuriant pattern still at the apical portions of the rafters (head, cao), which itself is followed inwards by a cartouche (fang) with a fairly simple shape and variable contents: dragons

Fig. 1.14: Decoration of beams and brackets in a Chinese temple in Beijing.

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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and other animals, scenes, ornamental scrolls and the cut-outs of 2D patterns. The latter are mostly p4mm, sometimes p6mm, swastika patterns p4gm, and rare interlaced patterns (consult Chapter 2). Ma (1996) ascribes this division to the Ming dynasty. Among the luxuriant patterns mentioned, prominence is given to aggregates of Chinese “rotating flowers” (Ma 1996) which are rather open small spirals similar to the spiral clouds found on stone reliefs. They form one-sided friezes, clusters or rotating wheels in several concentric layers (Fig. 1.14), a style ascribed to the Qing dynasty by Ma (1996). The circular aggregate can have N-fold rotational symmetry or, often, just a mirror symmetry. The resulting aggregate in Fig. 1.14 (called “the style of 1 and 2 ½ × revolving flowers”) envelops the beam with symmetry pgm but, with blue and green coloring taken into account, actually pg′m with dichroic twofold axes. A prominent artistic practice was to have these motifs alternately colored green-and-blue, and blueand-green on adjacent rafters. The Grand Mosque (Ulu Cami) in Bursa was constructed by the Ottomans in the years 1396–1399 under Sultan Beyazid I in a Seljuk style, but it was finished only around 1420. The pronounced Quran citations on the walls and pillars of the mosque converted it into a shrine of Islamic calligraphy (Fig. 1.15). The present inscriptions in thuluth script, in black and occasionally also red on white background were made in the nineteenth century (Massoudy & Massoudy 2003). They are a superb set of examples for the point group symmetry.

Fig. 1.15: The Grand mosque (Ulu Cami) in Bursa adorned by Islamic calligraphy in the nineteenth century.

12 | 1 Fundamental categories

Fig. 1.16: A portion of Arabic inscription from the walls of the Alhambra, Granada, Spain. Pseudoperiodicities created with the use of tall “lam” and prominent “wa”. The complex foliage is sculpted by a repetitious detailed pattern for which we assume an endless repetition of the word “Allah”.

Arabic writing, with its inherent cursive character, lacks an array of local mirror planes and twofold axes which characterize the classical Roman texts hewed in stone. This lack of aesthetic symmetry features was often alleviated by Islamic artists by introduction of pseudoperiodicities of different kind (Fig. 1.16) or by placing several texts in different writing style and letter size on one panel. In Bursa, another feature was widely used – placing the text and its mirror image on the same panel, with full or partial overlap, a feast of reflection symmetry, point group m at a cursory look. Cases of exact reflection, appropriately resolved where the two parts overlap (Fig. 1.17), however, alternate with cases in which at least some contacts are modeled by inserting the blades of one orientation through the blades of the other orientation, i.e., a complete but delicate asymmetry (Fig. 1.18). Very rarely, a vertically oriented twofold rotation axis was employed instead. In several cases of citations from the Quran, the repetitious “wa” (and) was magnified and became the principal aesthetic element of the construction, interspersed by the quotations themselves in smaller Kufic script. Cyclic point groups 6 and 8, without mirror planes, and based on large repeating wa, are outstanding (Figs. 1.19 and 1.20) as also is an eightfold cyclic arrangement based on repetitions of “s¯ın” with a sweeping terminal curve (Fig. 1.21). The other striking example is illustrated in the chapter on homothety (Chapter 6). The cursive character of Arabic letters resulted in some cases in admirable asymmetric constructions, from the single “wa” (Fig. 1.22), to the com-

1.1 Symmetries of finite objects: Zero-dimensional ornaments

Fig. 1.17: Inscription with a perfectly reflected mirror image. Point group m. Ulu Cami in Bursa, Turkey: calligraphy from the nineteenth century.

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13

Fig. 1.18: Inscription with a partly violated mirror reflection. Point group 1. Ulu Cami in Bursa, Turkey: nineteenth century.

plicated thugra (official signature of a sultan) on the wall of Ulu Cami (Fig. 1.23) which expresses the dynamism of Arabic script by the repetition of vertical lines and curves. The early Italian Cosmatesque art covers excellent point-group design in 2D or in virtual 3D space, friezes and plane-group mosaics. All these floor and wall mosaics can follow either uncolored groups of symmetry or they were conceived as dichroic point, frieze and plane groups. Therefore, it is not possible to describe this art in its entirety in one chapter and we shall meet it again and again. Cosmatesque art production flourished from late eleventh century under papal clientele until 1305 when the papal residence moved to Avignon (France) (PajaresAyuela 2002). The Cosmatesque guild consisted of ornamentalists, mosaicists, sculptors and architects schooled on the ample Roman art material available everywhere in contemporary Rome. Although referred to as Cosmati, from the family name of the first known artistic clan, there were several clans (families) in the same business, with somewhat different periods of importance (blossom). Central Italy was the most important area of production although additional centers were Tuscany, Campania, Sicily and even London.

14 | 1 Fundamental categories

Fig. 1.19: A Quranic text in a sixfold wheel form based on large “wa” letters. Ulu Cami in Bursa, Turkey: calligraphy from the nineteenth century. Overall point group 6 without mirror reflections.

The Cosmatesque guild’s work is highly abstract and geometric. Their floor designs have bands, areas and discs of geometric ornament composed of small tesserae of Red Imperial Porphyry, green Greek Porphyry, as well as white and light-colored marbles alternating with bands of white marble. Circular and spiral configurations combined into complex aggregates abound among the floor mosaics whereas stripes of complex mosaics, mostly composed of glass tesserae were applied as inlay to columns and friezes. Red Imperial Porphyry and Greek green Porphyry create together the harmonious dichroic effect of the majority of Cosmatesque ground ornaments (Chapter 4). Red Imperial Porphyry was imported exclusively for imperial use in Roman and Byzantine times (beginning of first century AD to the middle of fourth century or even to the end of fifth century) from the quarries at Mons Porphyrites, Gebel Abu Dokhan, the Red Sea Mountains, Egypt (Lazzarini 2004, Makovicky et al. 2015). Green Greek Porphyry comes from Stefania, at Krokea, Peloponnesos, Southern Greece. At least the first one, a very expensive material, came from spolia, i.e., from professional pillaging of Roman ruins in Italy and elsewhere in the Romanesque and Renaissance times.

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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Fig. 1.20: A Quranic text in an eightfold wheel form based on large “wa” letters. Ulu Cami in Bursa, Turkey: nineteenth century. Overall point group 8 without mirror reflections.

Cosmatesque floor designs (Figs. 1.24–1.28) invariably contain bands of linear patterns. Characteristic features of these ornaments are curved and/or straight slabs of white marble between which a band of tesserae is sandwiched. The latter band suggests a grand “over and under” scheme of intertwining (Fig. 1.24 and following), because these bands do not cross and one of them is interrupted before contacting the other. In order to include this feature in terms of symmetry (which otherwise would fail to give a complete description of the ornament), we have to consider the slabs “transparent”, with the same composition on the invisible bottom side as on the visible top. Then they represent a truly interlaced pattern, with the symmetry of a 1D/2D object in a virtual 3D space. The tetragonal quincunx will have point group symmetry 422, whereas the orthorhombic one, or a short truncated guilloche will have symmetry

16 | 1 Fundamental categories

Fig. 1.21: A Quranic text in an eightfold wheel form based on large “s¯ın” letters. Ulu Cami in Bursa, Turkey: calligraphy from the nineteenth century.

222. An infinite guilloche will have a frieze symmetry p222, still in virtual 3D space. The grand design shows creative variability in the realm of style uniformity. Twisted designs form the bulk of outstanding examples; rectangular fields play secondary role. The twisted designs are large-scale guilloche designs of varying length (PajaroAyuela 2002 indicates a length of up to twelve eyelets with porphyry rotae), sometimes branched. The other type is a centered design – a quincunx with the central circle surrounded by four (usually smaller) circles in a square or rectangular fashion (Fig. 1.24). Nevertheless, it is a point-group design. The central circle may be replaced by a square, and the lateral circles may be used for interconnection with further smaller circles and

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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17

Fig. 1.22: Arabic letter “wa” (and) in ornamented form. Aesthetics based on complete asymmetry. Ulu Cami in Bursa, Turkey: nineteenth century.

Fig. 1.23: Sultan’s thugra (official signature) in Ulu Cami in Bursa, Turkey: nineteenth century. Intended asymmetry, point group 1 with false periodicities.

eventually even with another quincunx. The system is flexible: it allows adjustment to the given situation and allows interesting variations, some of which are in Figs. 1.25– 1.28. Further details of geometry are given in Pajaro-Ayuela (2002). The Cosmatic floor designs are “a creative freedom within a very limited spectrum of symmetry groups”.

18 | 1 Fundamental categories

Fig. 1.24: A complex Cosmatesque floor mosaic of two concentric circles enclosed in a large square frame; all interconnected by eyelets. Church of San Crisogono, Trastevere, Rome, Italy.

Fig. 1.25: A lozenge-shaped 222 Cosmatesque floor panel. Santa Maria Maggiore, Rome, Italy. This basilica was built already in 432–440 but remodeled several times, all the way to early seventeenth century. The pavement by the Cosmati, from twelfth century, featured in Figs. 1.25–1.28, was restored in eighteenth century.

1.1 Symmetries of finite objects: Zero-dimensional ornaments

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Fig. 1.26: A multilayered 8-fold screw-stapled star pile. A floor mosaic in Santa Maria Maggiore, Rome, Italy.

Fig. 1.27: A four-square group with a point group symmetry 422, a floor mosaic from the church of Santa Maria Maggiore, Rome, Italy.

19

20 | 1 Fundamental categories

Fig. 1.28: A spectacular 222 group of four interlocked rings with different inner markings (interpreted as identical on both their surfaces). Cosmatesque floor of the church of Santa Maria Maggiore, Rome, Italy.

1.2 Friezes and guilloche There are seven frieze (i.e., one-dimensional) groups of symmetry in which translation is eventually combined with a mirror plane or glide-reflection plane, and/or an array of twofold rotation axes positioned on the infinite axis of the frieze (Fig. 1.29). A periodic array of reflection planes perpendicular to this axis can complete this picture. The highest symmetry combines the mirror planes with twofold axes, p2mm, the lowest symmetry is pure translation, p1. Frieze groups p211, p1m1, p11m, p11g, and p2mg are symmetry-wise situated between them. In these frieze group symbols, the planar symmetry operation situated in the infinitely long frieze b axis is written at the last place in the symbol and those which are perpendicular to the 2D plane at the first place of the symbol (International Tables for Crystallography, vol. E). Friezes, frames, socles, pilasters, edges of two-dimensional panels, etc., are the objects of frequent adornment by unidimensional patterns. Dichroic examples of them are rather frequent. In their vast majority, one-dimensional patterns which can be interpreted as rods in space are actually guilloche, i.e., friezes of intertwined cords confined to a two-sided layer in space, i.e., they are two-sided rods/friezes. They abound in Greek, Roman, Celtic and Islamic art; some attractive examples will be dealt with here. What are the relations between the unidimensional and bidimensional patterns? As mentioned above, friezes are ornaments developed along a single translation direc-

1.2 Friezes and guilloche

| 21

a p1 a 2

2

p211

p11m

a/2

p11g

m p2mg m p1m1

p2mm

Fig. 1.29: The seven uncolored frieze groups in plane (redrawn from Shubnikov and Koptsik (1974) with changes). Symbols follow the International Tables for Crystallography vol. E. The a axis is perpendicular to the image plane; in the plane, the b axis is parallel to the length of the frieze and c axis perpendicular to it. Symmetry planes quoted are those perpendicular to these axes!

tion while the extent is limited perpendicular to it, resulting in a (usually) narrow area needed to develop the elements of the ornament. Sometimes the pattern preserves local symmetries of the plane group of a 2D-pattern from which the frieze was cut. As a virtual three-dimensional case, a rod pattern, with the rod axis occupied by a twofold rotation axis as the maximum multiplicity, is derived from the layer group (Chapter 3) of the structure from which a cut-out was obtained. The types of two-dimensional symmetry groups will be discussed in the following chapters. For other friezes, no “parent group” exists because they were originally designed as friezes. Therefore, we suggest that there are three “genetic” types of onedimensional frieze ornament in the arts: 1. A frieze which is a clear cut-out of a two-dimensional pattern. The two-dimensional pattern can be unambiguously reconstructed by appropriate juxtaposition of individual strips of the frieze. 2. A frieze with a probable derivation as a cut-out of a two-dimensional pattern. Its edges have been restructured so that the reconstruction of a two-dimensional parent pattern may be less definite or remains ambiguous. More rarely, the juxta-

22 | 1 Fundamental categories

3.

position of adjacent strips can be done in more than one way, leading to several potential parent patterns. A frieze originally conceived as a unidimensional pattern without two-dimensional affinities or extensions.

A little considered set of examples for artistic friezes are the church vaults and ceilings of Gothic churches or festive halls (s. Figs. 1.30–1.34). Disappointingly, however,

Fig. 1.30: A fan vault in the entrance hall/staircase leading to the dining hall of Christchurch College, Oxford. Built under the rule of Dean Samuel Fell (1638–1648) by “Smith, artificer from London” perhaps to Cardinal Wolsey’s original design (from 1525 on). A brief account by Mark Girouard can be found in Christ Church: A portrait of the House, edited by Christopher Butler (2006). Plane group pmm.

1.2 Friezes and guilloche

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23

Fig. 1.31: Vladislav Hall , Prague Castle, Czech Republic. A late Gothic curvilinear vault of a series of concave spaces separated by broad supporting elements. Ribs of the vault form sixfold rosettes in the concave parts and continue smoothly down the supporting consoles. The rib plan is based on circle segments (in projection on the floor plane) which cross and seamlessly join one another while adhering to the curved surfaces of the ceiling. The hexagonal local configurations are inserted in the overall p2mm scheme, rosette-to-rosette, but without similar configurations sideways. Constructed by Benedict Riet (about 1500).

manuals on the art of cathedral and church ceilings (e.g., Mencl 1974, Binding 2000, Kavaler 2012) show that the vast majority of church vaults were constructed in the frieze group p2mm, notwithstanding the complexity of the design and their early or late Gothic character. Exceptions are rare: hall churches with broad spaces on tall pil-

24 | 1 Fundamental categories

Fig. 1.32: A square, curvilinear vault with circular “broken” ribs in the equestrian Rider Staircase of the Royal Castle, Prague, Czech Republic. In floor projection the circular segments of the ribs define symmetry p4. When the curvature of the vault and of the four corner consoles is ignored, the space ¯ arrangement (“above and below”) of truncated ribs defines a layer group p4.

lars (Fig. 1.30), curvilinear ceilings, generally with plane group symmetry p4mm or lower (Fig. 1.31), or with net vaulting which has the same symmetry, or is in p6mm and p2mm (Mencl 1974, Kavaler 2012). The rib pattern of these unidimensional sequences of shallow domes was obviously most often designed with a single dome in mind (Fig. 1.32) and subsequently adjusted to fit a periodic repetition along the church axis (Figs. 1.33, 1.34). Therefore they often contain a number of local symmetries not active in the frieze itself. More exciting are rare cases of a dynamic approach, such as the vault in the south aisle of the Augsburg church of SS Ulrich und Afra (Burkhard Engelberg, architect, circa 1478–1493) with a frieze group p211 and pseudohexagonal configurations, and with a distinct rhythm of alternation when looking down the aisle (Kavaler 2012). Exquisite examples of Cosmatesque friezes are in the cloisters at the church of San Paolo fuori le Mura (Rome suburbs), which were constructed in the first third of the thirteenth century (Pajares-Ayuela 2002). The bands of tesserae on the paired, winding and/or plain columns of the cloister galleries (Figs. 1.35, 1.36), together with the bands and panels of tesserae on the entablature above them (Figs. 1.37 and 1.38), give a special charm to the cloisters. At least in part, they are supposed to be the work of Vassalletto, a member of one of the six known Cosmatesque families (PajaresAyuela 2002). The adjacent basilica, however, was substantially rebuilt after the catastrophic fire of 1823. The brightly colored tesserae form a number of linear and (on the panels) twodimensional designs (Figs. 1.35–1.38). Tesserae essentially were shaped as small squares and small triangles. The variety of colored designs contrasts, however, with

1.2 Friezes and guilloche

Fig. 1.33: Divinity School, Oxford. William Orchard’s pendant vault (1480–1483). Perhaps the first pendant vault of English late Gothic. Carries initials of university dignitaries on bosses. A “flattened” pattern of the ceiling is p2mm.

| 25

Fig. 1.34: Christ Church Cathedral, Oxford (church rebuilt in twelth century). Church ceiling was perhaps made in 1528–1529, exact story unknown. Frieze group p2mm composed of mm2 fields.

the paucity of symmetry groups which were used. The majority of bands are in p2mm or in pb′ 2mm, if b is the infinite direction. The first symbol denotes a frieze group in which none of the symmetry operations change the color of the object; the second m in this symbol is parallel to the infinite axis, whereas the first m represents a periodical set of mirror planes which are perpendicular to this axis. The second symbol is for a dichroic frieze group in which translation (and, secondarily, some mirror planes which are perpendicular to it) changes the color of the object (tessera) under operation and creates a sort of 1-2-1-2-1… sequence of colors. To avoid disrupting further the Cosmatesque theme, we deal with them here although the reader may have to jump to Chapter 4 for details. Some designs, composed of dichroic configurations composed of square tesserae, which alternate with dichroic configurations of inserted tessera triangles, have the dichroic coloring of the latter set shifted by ¼b against the dichroic coloring of the former set. The only way to describe them without loss of symmetry information is that there are two colored lattices, each with corresponding pb′ 2mm structure, which

26 | 1 Fundamental categories

Fig. 1.35: A spiral scheme of different ornamental strips on a marble pillar from the cloisters of San Paolo fuori le Mura, Rome, Italy. Three dichroic and one monochromatic frieze (pb′ mm, pmm′, pmm) Fig. 1.36: Ornamental marble columns from the cloisters of the church of San Paolo fuori le Mura, Rome, Italy. Spiral and ring schemes of friezes, dichroic friezes and 2D pattern fragments (frieze groups p2mm, p2mg and pb′ mm; trichroic plane group pb (3) mm).

are shifted by ¼b against one another. This is a rare though not quite unknown principle (e.g., Makovicky 1986). The least frequent category are the pmm′ bands with a color change between the upper and lower side of the band, and a rare case of pb′ mm′ with alternation of gold-on-red and black-on-white squares containing smaller poised squares. Orientation of planes in the symbol of frieze groups has been explained above.

1.2 Friezes and guilloche

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27

Fig. 1.37: A twofold circular configuration with overall 222 point group symmetry but with a partly dichroic (frieze groups p2mm, pmm′ and p1) infill of the Cosmatesque strip. Tetragonal (pC′ 4mm on coloring) two-dimensional patterns as fill. Entablature of the cloisters of the church of San Paolo fuori le Mura, Rome, Italy.

Fig. 1.38: A 2 point group arrangement of red and green porphyry roundels interlaced by dichroic Cosmatesque strips of tesserae (pb′ mm). Entablature of the cloisters of the church of San Paolo fuori le Mura, Rome, Italy.

28 | 1 Fundamental categories

The two-dimensional panels of the entablature (Figs. 1.37 and 1.38) are pC′ 4mm (sorry, see Chapter 4), p4mm and p6mm mosaics, the easiest plane groups to construct from small square and triangular tesserae.

Fig. 1.39: A classical Roman corniche; alternating p1m1 and p2mm friezes. Piazza Bocca d. Verità, Rome.

Fig. 1.40: A Roman floor mosaic from Volubilis, Morocco. The portrait of Medusa is surrounded by a double-strand guilloche and the field is framed by a four-strand guilloche. Both have a frieze group p222 in a virtual 3-dimensional space. Coloring of double-strand guilloche colors intervals of the same string in three colors (a principle common in Irish Celtic manuscripts), that of the four-strand guilloche is consistent and results in color frieze symmetry (p2(2) 2(2) 2(2) )(4) for details of which you have to go to Chapter 5 (superscripts are number of permuted colors).

1.2 Friezes and guilloche

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29

The friezes and corniches of classical Rome (Fig. 1.39) are well known (and largely neglected) parts of artistic heritage. Several outstanding examples are illustrated here. Roman corniches are a combination of alternating p1m1 bands of acanthus leaves, p1m1 egg bands and kyma bands, with interspersed p2mm astragalus (bead) bands. The latter modify the “one-sided” overall symmetry of the p1m1 bulk. The Roman meander-swastika floor mosaic bands will be mentioned later. Here we illustrate a two- and four-strand guilloche from a Roman mosaic in Volubilis, Morocco (Fig. 1.40). Unfortunately, for the beautiful guilloche of Celtic manuscripts the reader has to go to the appropriate literature. The corniche, wall and floor friezes of the Romanesque and Renaissance architecture in Italy are represented here by a quadruple of parallel floor friezes with quite

Fig. 1.41: A multiple frieze band on the floor of Siena Cathedral, Italy. From the left, p1m1, a guilloche p222, p2mm, and pmg′.

Fig. 1.42: A p121 rod composed of two asymmetric sinusoidal curves braided through rings. The polarity of the rod is indicated by white eyelets. Marble floor of the Baptistery of San Giovanni, Firenze, Italy.

30 | 1 Fundamental categories

different symmetries (a cherished practice of artisans) from the cathedral of Siena, Italy. Starting from the left, Fig. 1.41 displays frieze groups p1m1, p222 (a guilloche), p2mm, and dichroic (Chapter 4) pmg′. Two unusual friezes from the Romanesque Baptistery of San Giovanni in Florence are in Figs. 1.42 and 1.43 (Makovicky 2015a). The former with strangely intertwined sinusoids and rings has symmetry p121 whereas the latter with quatrefoils interlocked with rings has a 3D rod group p2/b2/m21 /m (or pbmm). Finally, I add a vault frieze from Bukhara (Fig. 1.44) that has frieze symmetry p2 which is caused by the disjoint swastikas.

Fig. 1.43: A “flat rod” with rod symmetry pbmm composed of isolated two-sided quatrefoils with rings threaded through them. The rod axis is denoted as the c axis. If we negate the two-sided character, it is a p2mm frieze cut from a p4gm pattern. Marble floor of the Baptistery of San Giovanni, Firenze, Italy.

Fig. 1.44: A frieze of reversed swastikas combined with local mirror symmetries but still resulting in an overall p211 frieze symmetry. Kukeldash Madrasa, Bukhara, Uzbekistan.

1.2 Friezes and guilloche

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31

Fig. 1.45: Ceramic Pueblo vase from Cibola, Arizona, in a Tularosa style rectilinear (AD 1100–1200). Field Museum Chicago, no. 745.73776. External black-on-white pattern band with frieze symmetry p2′11, relating the black and hatched bands. Excellent analysis of Puebloan pottery is in Washburn et al. (2010).

Ancient Pueblo ceramics of the Southwestern USA are renowned for their mostly black-on-white decoration of homemade pottery. There is a question how to interpret these designs – as bands around the external or internal walls of the pot/plate (Figs. 1.45 and 1.46) or as cylindrical 2D designs on these walls, or even as point-group designs with a central rotation axis ± intersecting mirrors? In terms of the first interpretation, Fig. 1.45 is a dichroic p2′ band, with the black and parallel-hatched motifs related via twofold rotation. Figure 1.46 is a more complicated problem: it needs a modicum of abstraction to overlook the distortions of the crammed hand-painted design. Then, it emerges as a pg band with an excess width; it was extended by a half of the transverse translation period of the parent two-dimensional pgg pattern. The artist managed to (approximately) pack six periods of the pg pattern in a circular manner. Pueblo pottery is the best example of idealization that we apply when we undertake an evaluation of the symmetry of freehand designs. Chinese marble plates adorning the staircases in the Forbidden City and elsewhere are a prominent art object typical of official Chinese art (Fig. 1.47). A remarkable feature is their one- or two-dimensional patterns combined into “landscapes” (with or without additional, narrative motifs) in which individual strips/areas obey translation and/or symmetry only approximately (Fig. 1.48), although the overall symmetry transpires clearly from the pattern viewed as a whole. Isolated one-dimensional friezes, however, repeat the motif in an exact way. For other interesting aspects of Chinese art see Chapter 8.

32 | 1 Fundamental categories

Fig. 1.46: Ceramic Pueblo bowl from Kayenta, Arizona, in a Flagstaff style (AD 1100–1200). Field Museum Chicago, no. 707.81976. Internal pattern band with frieze symmetry p11g after correction for distortion (a 1 ½-period broad cut-out from pgg). The bowl was “killed” before deposition.

p2mg

▸ Fig.1.47 (top): A typical example of the approximate frieze symmetries in a Chinese stone-carving ornament. From the bottom: clouds in p1m1, an approximate pg pattern of waves, followed upwards by a one-dimensional ornament consisting of a row of approximately symmetrical peaks (p1m1), and a loose approximation to pmg in form of a cloudy sky with a flying dragon (cloud swirls appear chaotic). This panel is very characteristic for the historical Chinese “loose” approach to (especially 2D) symmetry. Forbidden City Beijing. ▸ Fig.1.48 (bottom): A typical Chinese pattern of swirling clouds in strips, a style which colonized Central Asian art and that of Islamic Turkey. Note the Chinese approximate approach to panel symmetry! Each strip of clouds and each inter-cloud space obeys approximately a horizontal glidereflection plane, including the swirl orientations. A frieze group p11g holds for the fully developed central strip, giving an approximate plane group pg. This ignores a “superstructure” of alternating larger and smaller “zigzags” in the central cloud strip. Swirls cancel a vertical plane of symmetry. At the Temple of Five Pagodas, Beijing.

1.2 Friezes and guilloche

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33

34 | 1 Fundamental categories

Two examples of more complex situations (Figs. 1.49 and 1.50):

Fig. 1.49: A complex net, studded with stars of increasing order that are not bound by symmetry of the underlying pattern. Darb-e-Imam Mausoleum, Isfahan, Iran.

Fig. 1.50: Muqarnas at the Masjid-e-Imam, Isfahan with mirror-symmetric individual motifs and with a two-niche horizontal p1m1 morphological periodicity.

2 The convenient start: Plane groups of symmetry Plane groups of symmetry describe symmetry of two-periodic patterns drawn or constructed in one plane. No element of the pattern lies above or below another element, and there is no upper and lower side of the pattern. There is only one plane in which the motif is situated and the third, perpendicular coordinate does not exist. When we look at the pattern, no bottom side exists and no symmetry elements converting the visible top into an unseen bottom are present. Thus, the mirror planes (planes of reflection), glide-reflection planes and rotation axes are all perpendicular to the plane of the pattern. These symmetry elements can also be understood as reflection and glide-reflection lines in the plane of the pattern (some authors interpret them as such). Periodicity of the pattern allows only 2-, 3-, 4-, and 6-fold rotation axes, other multiplicities are forbidden, including the potentially attractive general 5-, 8-, 10-, and 12-fold rotation symmetries which cannot be valid for the entire pattern. Only seventeen contradiction-free combinations of symmetry elements exist, forming the list of the plane groups of symmetry (Fig. 2.1). They can all be described in mathematical terms, which is not a topic of this contribution. A given pattern can contain one type of tile or several sets of different tiles. A tile set can be placed on an element of symmetry (these are so-called special positions and the tile shape is constrained by the symmetry element on which the tile is situated) or outside any symmetry elements, in a so-called general position (then the tile shape is a result of construction geometry, not of symmetry) (Fig. 2.2). Each element type has its own immutable color which is not involved in the symmetry operations (or all elements can be uncolored). In pure geometry there is a somewhat different approach to this problem: Grünbaum and Shepherd (1987) classify a complete collection of possible special and general positions from all plane groups as 51 henometric types of dispersed geometric patterns. Every two-dimensional periodic pattern contains a small area which reconstructs the entire pattern by simple translation along two translation vectors, and along their combinations. This area, outlined by straight lines, is called a unit cell of the pattern (Figs. 2.2, 2.3). Its shape may be a general lozenge, a rectangle, a square or an equilateral lozenge, the latter often with sides comprising a 120° angle, all according to the rotation axes and/or symmetry planes present in the pattern. In the arts we often find two-dimensional low symmetry patterns with a rectangular, hexagonal or even square cell, which is not supported by symmetry. The reason is that the ancient and present day designers prefer the metrically easy 90° and 120° coordinate systems with the same divisions on all axes (sides of the unit cell) even when they are not required by symmetry. A rectangular cell can sometimes have an additional node in the center (this node is translationally identical to those in the corners), and we describe the pattern as having a centered rectangular cell. A cell without the centration point is called “primitive”; the relevant symbols are c and p, respectively (Fig. 2.3).

36 | 2 The convenient start: Plane groups of symmetry

p1

p2

pm

pg

cm

p2mm

p2mg

p2gg

p3

p3m1

p6

p4

c2mm

p31m

p6mm

p4mm

p4gm

2 The convenient start: Plane groups of symmetry |

37

◂ Fig.2.1: Symmetry operations of the 17 plane groups of symmetry. Each drawing represents one unit cell (they should be multiplied by simple translations along the two axes (defined by cell edges)). Thick full lines: traces of reflection planes, dashed lines: traces of glide-reflection planes, twofold rotation axes: ellipses, remaining multiplicities: triangles, squares and hexagons. All symmetry planes and axes are perpendicular to the plane. Where applicable: a axis points downward, b axis points to the right.

o a2 a1

a

c d b

Fig. 2.2: The plane group p6 with six-, three- and twofold rotation axes on hexagons, triskelions and “eyes”, respectively. Special positions (on these axes) are a, b and c. General positions are d or those positions on any of the lines which do not overlap with the symmetry elements.

c

d

c

p

Fig. 2.3: The centered and the primitive unit cell with symmetries cmm and pmm.

Outlines of unit cells and the symmetry operators distributed in them can be seen in Fig. 2.1. The oblique system (the unit cell is a general lozenge) has two plane groups, the rectangular system is the richest one, with seven different plane groups, then the trigonal and hexagonal systems with three and two groups, respectively, and finally the tetragonal system with three plane groups. Figure 2.1 illustrates also how the combinations of symmetry elements dictate the shape of the unit cell, from two free translation vectors – comprising a general angle with one another – to two equal translation forming unit cell edges, and a strictly determined angle between them (90° or 120°, respectively).

38 | 2 The convenient start: Plane groups of symmetry

Two ornamentally important cases of the rectangular system are the plane groups cmm and pgg. The cmm group (Fig. 2.4) has reflection planes perpendicular to both crystallographic axes (i.e., unit cell edges), and interleaved by glide-reflection planes. Twofold rotation axes are at the intersections of reflection planes and at the intersection of glide-reflection planes, respectively. Special positions in the former host tiles shaped and oriented according to the point symmetry 2mm whereas the latter will have tiles oriented diagonally and then disposed according to the glide and reflection planes in the cell. Similar to this is the disposition of pattern elements which are not on operators of symmetry; those on mirror planes are aligned accordingly. The pgg pattern has glide planes at ¼ and ¾ of the edge lengths, and twofold axes spaced between them (Fig. 2.5). There are two mutually independent sets of twofold rotation axes which can host quite different motifs. Elements on glide plane intersections, and those in general positions, express demonstratively the glide plane conditions. The multiplicity of a general element (number of repetitions in one unit cell) in

c

b

a

c Fig. 2.4: Plane group cmm with mirror planes and twofold rotation axes indicated. Special positions (left column) are on 2mm intersections and on m planes alone, general positions (right column) lie outside the symmetry elements (three configurations).

Fig. 2.5: Plane group pgg with twofold rotation axes and intersecting glide-reflection planes. Special positions are only those on the two sets of rotation axes (positions a and b), the rest are general positions.

2 The convenient start: Plane groups of symmetry |

39

this group is four, whereas in cmm it is eight. The aesthetic values of these two groups are at the opposite poles of perception: cmm delivers a feeling of static solidity whereas pgg delivers dynamic interchange. Two outstanding plane groups of symmetry in the tetragonal system (unit cell with the axes a1 = a2 , interaxial angle γ = 90° are the holohedral (i.e., with the highest symmetry) p4mm and the hemihedral p4gm. The former group (Fig. 2.6) has fourfold rotation axes at the origin and at (1/2,1/2) site of the unit cell. Twofold axes halve the distances between fourfold rotation axes along the unit cell axes. One set of reflection planes are perpendicular to the unit cell axes, the other set perpendicular to the diagonals; glide-reflection planes are interleaved between the latter. Elements of the ornamental pattern situated on the fourfold axes of the plane group p4mm represent two sets independent of one another. The set at the origin is illustrated in Fig. 2.6, which also shows that the shape of this element has to obey the fourfold symmetry. The same principle holds for the elements on the twofold rotation axes. Furthermore, Fig. 2.6 shows a number of combinations where elements are straddling (and obeying) reflection planes and are at the same time close to a set of rotation axes. The highest multiplicity (number of repetitions in each unit cell), however, is achieved by a form situated in the space between symmetry elements (a so-called Wyckoff position g in Fig. 2.6), with eight incidences in a cell. It is shown clustered around the two sets of fourfold rotation axes in Fig. 2.6. Plane group p4gm (Fig. 2.7) has a similar disposition of four- and twofold rotation axes but a different disposition of reflection and glide planes leads to different results. The axial glide planes are disposed at ¼ and ¾ of the unit cell edges, whereas the only reflection planes are diagonal, through the twofold axes. In the p4mm group, the elements on fourfold axes have to obey a site symmetry 4mm whereas in p4gm their site symmetry is only 4 and these elements at (0,0) and (1/2,1/2) are interconnected by a glide-reflection operation, and the reflection on the diagonal mirror plane. Reflection planes direct the diagonal orientation of elements on them, whereas the elements in the general position outside symmetry elements are a half-subset of those in p4mm, in a pleasing glide-plane relationship (Fig. 2.7). The latter orientation is very different from the orientations of elements in the p4 plane group where all the orientations are in the same sense. The richness of ornamental elements in a complex p4mm pattern, together with a feeling of a static completeness of symmetry operations is the opposite of p4gm (and of p4) with their impression of movement and dynamism in the design. In the hexagonal system (a1 = a2 , γ = 120°) only the highest and the lowest group exist, the holohedral p6mm and a group without reflections, p6. In the former (Fig. 2.8), both the reflection planes perpendicular to the unit cell edges (crystal axes) and those perpendicular to the diagonals exist. Six mirror planes intersect in each sixfold axis of rotation symmetry so that a general position (outside any symmetry element) is repeated twelve times, a very useful element for many designs of intermediate or high complexity. Elements straddling either system of reflection

40 | 2 The convenient start: Plane groups of symmetry

g

d

d

f

c

b

g

a

c

a

Fig. 2.7: Plane group p4gm with symmetry elements, and motif elements in general positions (d in top row) and special positions on symmetry elements (fourfold and twofold rotation axes, mirror planes and their combinations). e

c

Fig. 2.6: Plane group p4mm with symmetry elements, and motif elements in general positions (around, not on symmetry elements; g in top row) and special positions on symmetry elements (fourfold and twofold rotation axes, mirror planes and their combinations). Here, and in analogous figures, individual types of positions are denoted by letters given to them by Wyckoff (listed in International Tables, vol. E).

planes repeat six times and are independent from those straddling the other system of mirrors. Important points of the p6mm patterns are elements in special positions on the sixfold, threefold and often also twofold axes (Fig. 2.8). Visually the reflections and rotations override the dense system of glide-reflection planes present (Fig. 2.1).

2 The convenient start: Plane groups of symmetry |

Fig. 2.8: Plane group p6mm with general positions (top) and various categories of special positions.

41

Fig. 2.9: A pattern from the mihrab of the tomb tower of Kharraqan (Iran) with brick ribs intersecting in the positions of six- and threefold axes and with twofold axes which center the S-shaped recesses. Plane group p6.

The rich static impression of a p6mm design is contradicted by the half-emptied and dynamic impression of a p6 pattern (Fig. 2.9). There are only six general elements around each sixfold axis of rotation, in a propeller-like arrangement and the twofold axis positions can be used for a dynamical element with suggested rotation as well. No reflection planes constrain the shapes of the pattern elements. Triangular elements on threefold axes can be used to create “stepped” patterns with small intriguing deviations from mirror symmetry. The two trigonal plane groups, p3m1 and p31m, are a frequent source of confusion. In the symbol, the first place represents the threefold rotation axis, the second place is for the mirror planes perpendicular to the a axes (it has value of “1” when there are no such planes) and the third place is for the mirror planes perpendicular to the diagonals halving the 60° angle between adjacent axes. This can be seen in Fig. 2.1 from which the different orientation of m planes with respect to the unit cell axes transpires. The resulting effect is very different. Finally, the plane group p3 can be used with advantage for simple patterns with threefold and twofold elements in combination. Plane-group patterns are widespread in practically all cultures. However, prominent concentrations of them occurred in several cultures at certain times, brought

42 | 2 The convenient start: Plane groups of symmetry

about by a mixture of cultural and technical reasons. Such is the Seljuk style of unglazed or scantily glazed brick patterns, paralleled by the brick-and-marble patterns at Cordoba, reliefs and mosaics of classical Mediterranean area, floral Iznik tilings, and Art Nouveau wall papers and tiles …On the other hand, certain motifs are recurrent in many cultures and at different times. One of them is a “key”, “meander” and “swastika” topic with the very interesting symmetry implications, with which we shall deal in some detail.

The Kharraqan tomb towers The Great Seljuks were a Turkish tribe who started as auxiliary troops to NE Iranian dynasties, conquered Iran and Azerbaijan after 1040 and built an empire stretching to the Mediterranean which lasted until 1194 when Mongols arrived. We present several examples of their fabulous ornamentation using unglazed bricks, which integrates with the technology of the day. We shall later mention another branch, the Anatolian Seljuks (1081–1307), in another connection. The two spectacular Seljuq tomb towers at Kharraqan in western Iran (Fig. 2.10), about 120 km to the southwest of Qazvin, were built by the architect Muhammad ibn Makki al-Zanjani for Turkoman chieftains in the years 1067 and 1086, respectively. Local stories, however, describe these towers as mausolea for an assassinated holy man and his sister. The tower tombs are octagonal prisms, about 13 m tall, each terminated

Fig. 2.10: The Seljuk tomb towers at Kharraqan in western Iran photographed from the east.

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with a brick dome, now restored. Both towers have powerful semicircular buttresses attached to all corners. The ornamental fields of the two towers consist of large arched blind windows and panels, horizontal friezes, rows of three small ornamental panels, as well as pillars and buttresses. All is covered by ornamental brickwork. Lower portions of the walls and buttresses have lost their ornamental coating due to seasonal humidity (Fig. 2.10). Most of the interesting patterns are in the raised-and-recessed brick technique. They are rather evenly distributed between plane-group patterns and impressive imitations of interlaced patterns. In the interior of the W tower there is a patterned mihrab wall. Both towers were expertly restored without forcing new additions upon the original brickwork. Three weeks after the author’s visit, this area was subjected to a powerful earthquake in which a part of the ornamental brickwork was damaged. It turns out that the most expressive or intriguing tetragonal patterns are all in plane groups p4 and p4gm (Figs. 2.11–2.13, 2.15) whereas nearly all hexagonal designs (e.g., Fig. 2.14) are imitations of interlaced patterns with the pattern plane in the “virtual 3D space” and are shown in the next chapter. The most intriguing pattern is in Fig. 2.15, with pentagons dominating the p4gm panel.

Fig. 2.11: Kharraqan towers; a raised-brick p4 pattern of squares and swastikas.

Fig. 2.12: Kharraqan towers; a raised-brick p4gm pattern.

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Fig. 2.13: Kharraqan towers; a raised-brick p4gm pattern; small patterns are p6 and p4/nbm.

Seljuk minarets Towns on the old Silk Road along the southern limits of the Elburz Mountains, and those encircling the vast Dasht-e-Kavir desert in Iran, were dotted with mosques that boasted exquisite minarets adorned with complex brick ornamentation. In most cases they are the last vestiges of the old building, the mosque itself having succumbed to numerous alterations. As a rule, top portions of minarets exhibit a variety of tetragonal patterns in raised and recessed brickwork (Figs. 2.16 and 2.17) whereas the bottom portions have tetragonal, or similar, patterns in solid brickwork, based on brick-stacking variations, producing an optical “damask effect”. I visited the minarets of this type at the Jami’ Mosque at Saveh, north-western Iran (built in 1110–1111; Hutt and Harrow 1977), as well as those of the Jami’ Mosque in Semnan (eleventh century), Jami’ Mosque in Damghan (about 1058), and the Tarik Khana minaret of the Great Mosque in Damghan (probably pre-Seljuq, 1027), all three situated in northern Iran. Books by Hutt and Harrow (1977), Michell (1978), Stierlin (2002), and Clévenot and Degeorge (2000) offer additional, although rather sparse material.

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Fig. 2.14: A frieze-like cut-out of a large p6mm pattern (Archimedean tiling) for which a less obvious p622 interpretation is also possible. Kharraqan towers; a raised-brick pattern.

Fig. 2.15: A complicated p4gm pattern with prominent pentagons. Divided squares are positioned on 4-fold rotation axes. Kharraqan towers; a raised-brick pattern. A p6 pattern of shaped bricks is situated above.

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Fig. 2.16: A minaret of the Jami’ Mosque in Damghan with four tiers of different p4mm patterns in raised brick design.

Outside this region of Iran lies the famous Kalyan Minaret in Bukhara, Uzbekistan. It was built in 1127 together with the Kalyan Mosque which, however, was completely rebuilt in the fifteenth and sixteenth centuries. Among ten broad ornamental bands circling the body of the minaret, the three lower bands are based on differently orientated solidly packed bricks whereas the upper bands consist of raised and recessed bricks. Kalyan Minaret is crowned by an ornamental lantern. Two cmm patterns, a p4gm pattern, two more p4gm patterns affinely altered into cmm as a result of brick and mortar dimensions and a p1 “lace-like” pattern are those of raised-brick type (Fig. 2.18 and Fig. 2.19).

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Fig. 2.17: A raised p4mm pattern on the wall of the minaret of the Saveh Jami’ Mosque, north Iran (built in 1110–1111).

Fig. 2.18: Kalyan minaret, Bukhara, two ornamental belts: top p4gm, bottom p4gm altered to cmm.

Fig. 2.19: Kalyan minaret, Bukhara, two more ornamental belts: top p4gm, bottom p1 “a lace ornament”.

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Fig. 2.20: A cmm mosaic worked out in Maragha/ /Kond style tiles. The Friday Mosque in Isfahan, Iran. Turquoise pentagons, black-tipped composite lozenges and ten-fold stars.

Iran and Uzbekistan As witnessed by the monuments of Maragha, Isfahan, Konya, Sivas etc., since the times of the Seljuks the Iranian, Turkish and Central Asian tilings have appeared in several styles defined by shapes of fundamental tiles. The tiles of the Kond type (Mofid & Raeeszadeh 1995 and references within) are the large tiles in Fig. 2.20 with a regular pentagon as the basic shape, a composite lozenge (decomposed into body and tips in the original Kond understanding), a “butterfly” (see Figs. 2.20 and 2.21 and figures in the quasiperiodic chapter) and 10-fold stars. These are the Maragha-style tiles of Makovicky (1992) and (as pure shapes) they allow a number of periodic arrangements (cmm in Fig. 2.20, another cmm pattern in Fig. 2.21). Interesting facts about them are in the discussion (Chapter 9). The Tond style has a “Soviet-style” pentagonal star with acute points as the basic shape, combined with an array of darts, stars and half stars and other derived shapes. Some of them can be seen as the large elements in Fig. 2.22. In this figure, they are filled by small Kond tiles in a half-orderly fashion. The remaining tile type, Shol tiles, include an obtuse pentagonal star, concave and convex irregular hexagons, acutepointed half-stars and overlapping darts in the registry. A typical Shol cmm pattern is in Fig. 2.23. The stars in this figure and in Fig. 2.20 are filled by minute Tond–Shol tile aggregates in proper orientation. For decagonal rosettes, the cmm arrangements are the natural result of all these three tilings.

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Fig. 2.21: A periodic cmm mosaic composed of Kond style elements which include black butterflies. Friday Mosque in Isfahan, Iran.

Fig. 2.22: A large-scale mosaic in Tond style with large “tiles” filled by semi-regular arrangement of multicolored Kond-style tesserae. Bazar-street Madrasa, Isfahan, Iran.

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Fig. 2.23: A cmm mosaic composed of Shol style tiles – blunt-tipped fivefold stars and twinned stars, irregular hexagons and large 10-fold stars. The same style applies to the minute tiling painted in the interior of the latter. Friday Mosque in Isfahan, Iran.

The principal occurrence of plane groups of symmetry in the Alhambra (Granada, Spain) are patterns incised in the plaster of the walls of corridors and the complicated ornamental plasterwork adorning the upper wall portions, up to the roof level. In this locality, the latter mostly shows high complexity, especially several symmetries mixed over one another, and will therefore be dealt with in Chapter 6. Three simpler wall patterns are in Figs. 2.24, 2.25 and 2.26. The first two patterns are in p4, the last one in p6. The pattern in Fig. 2.24 has many interesting derivatives (Makovicky and Fenoll 1994), and the pattern from Fig. 2.25 occurs also as a dichroic pattern at various localities. They can be joined by a lovely small-tessera p4mm mosaic from Fez, Morocco (Fig. 2.27). A rich set of sculpted plaster ornaments from the Alhambra is illustrated in Chapter 6 because they combine two or more plane groups of symmetry in one ornament.

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Fig. 2.24: A tetragonal pattern traced in plaster, from the Puerta del Vino and Patio de los Leones in the Alhambra, Granada, Spain. Plane group p4. A potential order-disorder pattern (Makovicky and Fenoll 1994).

Fig. 2.25: A p4 pattern of eightfold Solomon stars and angular elements. A stucco pattern from the Patio de los Leones in the Alhambra, Granada, Spain.

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Fig. 2.26: A p6 pattern of sixfold stars and linear elements. Sgraffito from the corridors in the Alhambra.

Fig. 2.27: A street mosaic from Fez, Morocco, composed of small tesserae, plane group p4mm.

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Iznik Tiles About the 1470s, the Iznik potters begun making high-quality vessels from a pure white stonepaste with underglaze painting (Porter 1999).The first tiles from (presumably) Iznik are known from about 1507–1513, reaching perfection at ≈1520–1540, with the famous gamut of flower compositions and ornaments (Yerasimos 2000). The traditional cobalt blue-and-white combination (“cobalt blue” is ideally CoAl2 O4 ) was enriched by turquoise (obtained from the combination of alkaline glaze, tin and copper) after about 1530, and by olive green after 1540, as well as by mauve after 1550 (all used in combination). The Chinese motifs disappeared in favor of naturalistic motifs – nature-true flowers, pomegranates, artichokes, and leaves in the serrate Saz style (Figs. 2.28 and 2.29). Introduction of “Armenian bole” (kil-i ermeni) (βoλoς = lump of earth), iron-rich clay that fires into intense “coral red”, dates to the second half of the sixteenth century (e.g. Rüstem Paşa Camii, completed in 1561). Around 1560, under the court architect Sinan, Iznik works concentrated upon tiles for wall revetment. The popularity of the bright red color lasted for only ≈25 years. After 1580 it was replaced by orange or dark brown and by ≈1600 it disappeared completely, perhaps because in 1603 Turkey lost Armenia to Persia (or a decline in firing techniques?). At this time a decline of Iznik works started; the “late blue and white” style was typical for this period (Henderson 2000). In the seventeenth century, Iznik was replaced by workshops

Fig. 2.28: A detail of Iznik tiling with a traditional Turkish plant ornament. Rüstem Pasha Mosque, Istanbul.

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Fig. 2.29: Iznik tile with a stylized flower on a diagonal. Point group symmetry m (mirror plane on a diagonal). Rüstem Pasha Mosque, Istanbul.

at Kütahya (Yerasimos 2000). In spite of the promising title, the paper by Kiziltepe (2005) never got to a real analysis of tile symmetry. The three Istanbul mosques richest in Iznik tiles are: The Sokollu Mehmet Paşa Camii which was built by the Turkish top architect, Sinan, in years 1570–1572 under the rule of Selim II. The Blue Mosque (Sultan Ahmet I Camii) was built in 1609–1617 by Sedefkar Mehmet Ağa under the rule of young sultan Ahmet I. The light character and luminous beauty of the mosque come from the unusual profusion of colorful tiles (> 20,000). The Rüstem Paşa Camii (Fig. 2.30), completed in 1562, is a small mosque built by Sinan between his two masterpieces, Süleymaniya in Istanbul (1557) and Selimiya in Edirne (1575). The rich revetment of tiles reaches up to the level of red-andwhite arches, and in the triangular fields between the arches. Although there are small good-quality books illustrating the tiles of Rüstem Paşa (e.g., Cimok 1998), lately a similar book appeared in which the computer artist (Dalgalidere 2012) trimmed tiles at will, and composed them in combinations entirely foreign to the original symmetry. Turkish Iznik tilings are very different from other ornamental genres. Their symmetries are determined by a combination of predominantly floristic and vegetal subjects, ornamentation of tile centers and of tile boundaries (which are combined from two or four adjacent tiles), and by the square shape of tiles. The statistics of symmetries of

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Fig. 2.30: Interior of the Rütem Pasha Mosque in Istanbul with a rich tile revetment from the flowering period of Iznik workshops.

the Iznik tilings on mosque walls (i.e., not symmetries of isolated tiles) is summarized in the discussion. Figure 2.31 shows a representative of the richly decorated Iznik tiling in p4mm. Tile symmetry is a reflection plane on a diagonal; all the other symmetry operations result from tile composition in the tiling. Figure 2.32 shows a disordered tiling with tiles which have internal symmetry mm2 (different elements on the two diagonals) but the symmetry of design along their four edges is 4mm. This allows a free choice between two orientations of each tile, 90° apart, with the orientation of distinct tile di-

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Fig. 2.31: A richly decorated Iznik tiling (stylized vegetal motifs) typifying a family of tilings which obeys the p4mm symmetry.

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Fig. 2.32: A disordered Iznik tiling of tiles with local symmetry mm2 but with boundary configurations following 4mm. Details in the text.

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Fig. 2.33: Iznik tiling with plane group cmm. A prominent pair of serrated leaves surrounding the twofold rotation axes.

agonals selected without regard to that in adjacent tiles, i.e., a completely disordered pattern. Figure 2.33 is a cmm pattern with interesting pairs of serrated leaves surrounding twofold axes. A racemic composition of enantiomorphic tiles will be discussed as Fig. 6.36. In many cases, orientation of floral, vegetal and ornamental elements favors plane groups pm (Fig. 2.35), pg (Fig. 2.36) and cm (Fig. 2.37). Geometric ornaments have symmetries p4 (Fig. 2.34), and p2 (Figs. 2.38 and 2.39). Especially the latter diaper ornament creates a pleasing impression by its abstract “rotors”. (Have you noticed the faint 2 × 2 superstructure created by variations in little eyelets?). A number of peculiar mistakes are hidden in several of the tile panels: a disorder similar to Fig. 2.32, two-tile sets from which one set was forgotten in some panels, enantiomorphous sets of two tiles obviously combined in a way not envisaged by the Iznik designer …. Still, nothing can detract from the charm of tiled walls and pillars, especially in the two smaller mosques. The statistics of symmetries of the Iznik tilings on mosque walls (i.e., not symmetries of isolated tiles) are summarized in the discussion. How do the Chinese cloud patterns, which spread in different forms over Asia from Japan (Nakamura 2009) to the Islamic art of Turkey look? Figure 2.40 shows

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Fig. 2.34: A geometric p4 pattern of Iznik tiles.

slightly more than two tiers of a pg pattern of isolated clouds, for which the outer portions are approximately m symmetric but a central frill negates this symmetry and defines the pg symmetry of the pattern. The end of the nineteenth century brought about a cultural revolution, discarding principles which governed academic art in several European countries, and abandoning the traditional language of ornament which was oriented mostly to historical examples. Somewhat different schools in different countries (in spite of frequent interactions) developed into Art Nouveau in France and Belgium (Fig. 4), Jugendstil (Sezession) in Austria and Germany, and into their own art forms in Italy, Spain, Scotland and so on. Wonderful examples of new ornaments were produced in all these countries, some of which will be shown here. What is interesting is that lower symmetries were preferred (Fig. 2.41 and Fig. 2.42, pg and cm, respectively) and asymmetry was often selected as an ideal form (Fig. 2.43 and Fig. 2.44, both p1). This choice is necessarily modified in the tile industry (van Roojen & Navarro 2008), popular in the Art Nouveau times, but more than often a tile has only a single reflection plane, axial or diagonal, the rest being taken care of by tile composition. Symmetry breaking using the final and minute detail of design, e.g., when two leaf blades on the centerline overlap one another, the rest of leaves being separated and mirror-symmetrical, was a favorite practice of Art Nouveau artists. There is no mention of classical art without treating swastikas and meanders which gave indelible character to so many Greek and Roman antiquities. They are not

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Fig. 2.35: One of the most attractive Iznik tilings from the Rüstem Pasha Camii in Istanbul. Plane group of symmetry pm. Stylized tulips are colored in Armenian bole, bulk of the figure is in cobalt blue and turquoise.

limited to these two periods and European Classicism, however; they played an important role in the Indian and other Buddhist art as well as in early Islamic art and architecture. Swastikas can form linear (although easily branching) arrays or they can occur in two-dimensional arrays of different types. They can be “simple”, with the straight arms created by a simple right-angle turn of the central bar (Fig. 2.45), or “recessed”, when the arms turn back again into the direction parallel to the central bar before merging (in both cases) into a connecting line of the pattern (Fig. 2.46). Orientation

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Fig. 2.36: A rich tile panel with a tulip-clad sinusoid, plane group pg. The broad frame has frieze symmetry p2 down to its small details.

Fig. 2.37: A broadly designed vegetal tiling with plane group of symmetry cm.

Fig. 2.38: A broad band cut out of a p2 pattern with hypersymmetrical flower heads. It is surrounded by a frieze pattern in which two overlapping p2 friezes (tulips and hyacinths, respectively) are mutually shifted by a quarter of the translation period.

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Fig. 2.39: A diaper pattern of “rotors” and hypersymmetrical flower heads with symmetry p2. Blemishes in the glaze result from the firing processes.

Fig. 2.40: Chinese ornamental clouds forming a pg marble panel. At the Temple of Five Pagodas, Beijing.

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Fig. 2.41: A flowery wallpaper with plane group pg in the Art Nouveau Museum, Darmstadt, Germany. Glide planes run through flower stacks.

Fig. 2.42: A wallpaper with plane group cm. Darmstadt, Marienhöhe, tower.

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Fig. 2.43: Art Nouveau vase with a surface decoration in p1. Darmstadt Art Nouveau Museum.

of the arms gives a sense of rotation to the swastika. The sense of rotation can be the same for all swastikas in the aggregate or it can alternate. Swastikas and their connecting lines can form the entire pattern or they can combine with other geometric elements, e.g., squares (Figs. 2.47 and 2.48; the two orientation varieties were used side by side as floor mosaics in Pompeii – Fig. 2.49 is p4 whereas Fig. 2.50 is p4gm), triangles or lozenges. When adjacent swastikas are sharing one interconnecting line, they are “single-returned”; when sharing two adjacent lines, they are “double-returned”, etc. (Balmelle et al. 2002a).

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Fig. 2.44: Wall paper, National Gallery of Canada, Ottawa. Plane group p1.

Fig. 2.45: A vortex series of rotation N = 1 to 3 homologues of I-beam type patterns with simple swastikas.

Egyptian and Minoan art appear to favor patterns with spiral vortices. Vases of the Geometric Period in Greek art often display friezes of lines broken at 90° in complicated sequences, which might have developed into the later more regular meanders and swastikas. However, a perfect 2D pattern of simple swastikas framed in squares, with swastika arms interconnected across the frame (Fig. 2.51), with plane group symmetry p4gm, served as a standard which was probably carried by priests in the kingdom of Hatti, central Turkey, which became later a Hittite kingdom. The date is about 2100– 2000 BC.

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Fig. 2.46: Two members of a vortex series of I-beam type with recessed swastikas, N = 1 and 2.

Fig. 2.47: Combination of simple swastikas and commensurate squares.

Friezes formed by two single strings of swastikas can have symmetry p211 (most frequent) or p11g (Figs. 2.52 and 2.53), with a clear dynamic swirling character, although the angular nature of individual swastikas and of their interconnections imparts a certain solidity to the pattern. From the Late Roman Empire come rather dense mosaics of the villa Torre Palma (Roman Lusitania, now Portugal) with p211 strings of doublereturned simple swastikas of the N = 3 (see Fig. 2.45, right) category (Blazquez 1975). A frequent form of interlaced double strings of swastikas is a “latchkey ornament” which has symmetry p11m if the consecutive pairs of swastikas have the same orientation, or pmm when the consecutive pairs of meanders in the double string have opposite orientations (Fig. 2.54). This pattern lacks the dynamic character of the single strings.

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Fig. 2.48: Combination of recessed swastikas and commensurate squares.

Fig. 2.49: A floor pattern of swastikas and squares from Pompeii, Italy. Plane group p4.

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Fig. 2.50: A floor pattern of swastikas and squares from Pompeii, Italy. Plane group p4gm.

The versatile swastika meanders were utilized for borders or for dividing a mosaic panel into compartments (Fig. 2.55). Branching of meander strips could be achieved by insertion of square-shaped spacers or by a combination of single- and double-returns to the surrounding swastikas. Our illustrations come from the ruins of Volubilis, an important African city of the Roman Empire near Meknes in Morocco. Mauritanian Volubilis became an important outpost of the Roman Empire and between AD 45 and 285 it was capital of the Roman province of Mauritania Tingitana. Following a period of dark ages after the Roman withdrawal, Volubilis became briefly the capital of the Islamic Idrisid dynasty (from 788 on). By the eleventh century it was abandoned and subsequently quarried for outstanding architectural material, now found in Meknes buildings. Among 2D patterns, there is a prominent family of “I-beam patterns”, always with four swastikas in a local mm2 configuration, and two and two of these swastikas interconnected. These are tetragonal p4gm patterns (Figs. 2.45 and 2.46) but their symmetry is lowered to pgg in affinely distorted patterns of the I-beam category (Fig. 2.56). What are the differences between different members of this family? Firstly, there is a subfamily with simple swastikas and a subfamily with the “recessed” (visually “broken”) swastikas (Figs. 2.45 and 2.46). Then there are differences in the number of partial rotations by π/2 that each swastika in the pattern must undergo before the interconnection

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Fig. 2.51: An openwork p4gm grid of simple swastikas (I-beam pattern, homologue N = 1, on a background of a square grid). Hatti Empire; Museum of Anatolian Civilizations, Ankara.

lines between swastikas are drawn. In the N = 1 member of the subfamily with simple swastikas, each swastika arm transforms directly into a straight connecting line; in the N = 2 member it becomes a connecting line after one line break (this can be modeled as a rotation of the swastika itself by π/2 against its orientation in N = 1), in the N = 3 member after two breaks (i.e., rotation by π), etc. Inspection of the series in Fig. 2.45 shows the resulting apparent “rotation” character of the series: it is a vortex series of homologues; and we can see that the ornamental value of this pattern type increases from N = 1 to 3. This concept of rotation while preserving the interconnections is very close to the concept of Liu and Toussaint (2010) who were producing swastikas by

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Fig. 2.52: A p211 frieze of simple swastikas.

Fig. 2.53: A p11g frieze of simple swastikas.

Fig. 2.54: A latchkey pmm frieze from Volubilis, Morocco. A cut-out from an I-beam p4gm pattern with simple swastikas.

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Fig. 2.55: Compartmentalization of a mosaic floor by means of single swastika p211 strips. Branching by means of intercalated squares.

Fig. 2.56: The N = 1 I-beam pattern of green-glazed tiles with symmetry pgg from the Forbidden City, Beijing, China.

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“spinning” two intersecting diagonals positioned on prospective swastika sites, interpreting it as a possible working procedure of artists themselves. The “I-beam” series of “recessed” or “broken” swastikas has the connecting line attached only after the arm returned to the orientation of the central bar (Fig. 2.46). Already its N = 1 member is of a fair ornamental value. It is a favorite pattern for affine distortion into the pgg pattern of Japanese and Chinese ornaments (Fig. 2.56). One of the classical 2D patterns I find interesting is the pattern of poised swastikas (simple, single-return with four neighbors) enclosing lightly colored lozenge fields which, in spite of first impressions, has only symmetry p4, with two independent sets of swastikas (look at the configuration of interconnecting lines). This remarkable pattern in Fig. 2.57 comes from Ravenna, the church of San Vitale (AD 530–550). For the other pattern (Fig. 2.58) I have no locality. This dizzying pattern has the same disposition of square-shaped swastikas but the lozenge fields are occupied by the oppositely rotating lozenge swastikas. Interconnections of (square) swastikas from the San Vitale pattern are replaced by interconnections of square swastikas with lozenge-like ones. Islamic art contains a number of outstanding examples of swastika and swastikalike patterns. One of the prominent localities in which swastikas were used to create two-dimensional patterns is the Mezquita Aljama de Córdoba (The Great Mosque of Cordoba), Spain, which survived until the present day in a functional although much battered form (Fig. 2.59). It contains a unique collection of brick-and-marble patterns created especially during approximately 65 years of Cordoban caliphal art. The Golden Age of this art started with accession of the Ummayad emir Abd al-Rahman I (AD 750). It blossomed under caliph Abd al-Rahman III (after AD 929) and ended in 1008, marking the death of Abd al-Malik and start of civil wars. The Great Mosque served the capital of al-Andalus. It was built in 784 and subsequent years; enlarged for the first time in 833–848,remodeled in 855–856, enlarged again in 962–966 (S part of W portion) and finally enlarged substantially (the entire E half) starting in the year 987. Its vast and impressive interior is a forest of columns and white-and-red arches (Fig. 2.60), with Byzantine mosaics in the area of mihrab. Adornment of the exterior consists of red-and-white brick-and-marble patterns surrounded or accompanied by carved vegetal ornamentation. After conquest, two Christian cathedrals were placed in the mosque interior. The battered exterior has mostly been neglected, only to be extensively restored by V. Bosco and M. Inurria (Fig. 2.61). Bosco’s somewhat Romantic restoration of some portions was based on a careful study of ornamental panels and their remains so that the pattern reconstructions are based on the historic examples. Patterns seen in the interior (Fig. 2.62) were originally the exterior patterns before enlargements. The most outstanding are numerous brick-and-marble panels with geometric ornaments, together with analogous ornaments from the nearby royal city of Madinat al-Zahra. Many of them are based on (mostly recurved) swastikas combined with filled squares; some had to be reconstructed from pitiful remnants (Fig. 2.63). A simple, p2, pattern of this type is Fig. 2.64. A homologous series of such combinations is in

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Fig. 2.57: A p4 pattern of poised swastikas from San Vitale, Ravenna, Italy, with inscribed lozenges.

Fig. 2.58: A complex swastika pattern composed of alternating and interconnected lozenge-shaped and square swastikas. Redrawn; locality unknown. Related to Fig. 2.57 and also as a “broken” version of the I-beam pattern N = 3 in Fig. 2.45. Plane group p4 with a nice demonstration of all rotation symmetries and the nonequivalence of the special positions on fourfold rotation axes in the corners of the unit cell and in its center.

Fig. 2.65. The homologues differ in the number of linear spacers between swastikas and squares (a line-intercalation series of patterns). A real example of the N = 1 member is in Fig. 2.66. Another outstanding pattern from both localities contains poised swastikas (Fig. 2.67a) which have been partly exchanged by squares which, ordered in idealized form, are in Fig. 2.67b. An interesting variety of an expanded and filled swastika pattern from Cordoba is in Fig. 2.68. A detailed treatment of Cordoban brick-and-marble ornaments is in Makovicky and Fenoll (1997a and b). It might be mentioned that beside the early Eastern Islamic influences and ample swastika/meander examples left in the ruins of the Roman Empire, the just conquered Visigoths also preferred recessed swastika ornaments. A rich variety of swastika patterns was also created by substituting the swastikas for squares, lozenges and triangles (as triskelions) of the Archimedean patterns. Such is the rich swastika pattern of rings composed of interconnected swastikas on one of the Kharraqan towers (Iran) (Fig. 2.69) which has been schematically rendered in Fig. 2.70, and the majority of brick-in-mortar patterns from a lantern of

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Fig. 2.59: A brick-and-marble pattern of squares and swastikas, p4, with three lines of separation. The Great Mosque of Cordoba, Spain.

Fig. 2.60: A forest of columns and white-and-red arches in the interior of the Great Mosque of Cordoba.

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Fig. 2.61: The Great Mosque of Cordoba, Spain. A restored eastern façade of the mosque.

Fig. 2.62: A formerly exterior p2mm pattern, now in the Mosque interior.

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Fig. 2.63: A brick and marble pattern based on simple swastikas: present-day situation and a suggested reconstruction. A tympanum of The Great Mosque of Cordoba.

Fig. 2.64: A p2 pattern with continuous doubled separation lines in the tympanum; bottom: a randomized swastika pattern in the window trellis.

the Şifaiye Madrasas (Seljuk Sultan Izzettin Keykavus I Hospital) in Sivas, Turkey, from 1217. In the latter case, the fundamental element are modules of four, but sometimes only two, tightly joined swastikas, always in parallel orientation and double returned. Three examples shown here are based on recessed (broken) swastikas, not on simple swastikas; they can be called “modular structures” because they use blocks of four (rarely only two) parallel swastikas as ornamental elements. One of these modular structures resembles that from Kharraqan but the larger ring is modular (plane group p6) whereas the others are p4, either as octagonal rings with two-swastika modules (Fig. 2.71) or based on 2 × 2 blocks interconnected by poised swastikas (Fig. 2.72). Links to similar Turkish patterns and related localities are found in Schneider (1980). One of the most impressive 4.3.4.32 patterns comes from a broad framing band around the entrance to a nameless Seljuk tomb tower at Maragha, Iran (Fig. 2.73). Broken swastikas are positioned on fourfold axes; the central arms of twinned triskelions are on twofold axes whereas the strange-looking 7-pointed fields are in general positions. We hypothesize that the latter’s existence is given by their similarity to a

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Fig. 2.65: A homologous p4 series of lineintercalation type (N = 1 to 3) from the tympana of the Great Mosque of Cordoba. One-to-three lines intercalated between the swastika and square elements are partly interrupted for aesthetic reasons. Uninterrupted lines are in Fig. 2.64.

Kufic form of the name Allah or Ali. Two such elements can interlock in two nonequivalent orientations although their immediate key-and-lock scheme is identical for both orientations. Using this possibility of variation, we constructed a p6 pattern (Fig. 2.74) with the interlocking orientations which are absent in the original. Triskelions become part of sixfold rotors whereas the fourfold groups around swastikas are altered into twofold configurations. Figure 2.75 shows a column with a very pleasing I-beam pattern with simple poised swastikas from the eccentrically ornamented complex of Ulu Cami & Darüşşifa in Divriği (Turkey). The Ulug Beg Madrasah (built in 1417–1420), in Registan, Samarkand, hosts on its outer wall a panel of poised recessed swastikas combined with sixfold stars (Fig. 2.76). The p4gm pattern drawn in thin black lines on the uniform buff background is an em-

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Fig. 2.66: A restored façade with the N = 1 member of the homologous series from Fig. 2.65 and an I-beam type pattern of swastikas which are poised in relation to the I-beams.

Fig. 2.67: An I-beam type p4gm pattern with poised swastikas (a) and replacing squares (b). Cordoba.

bellished version of the I-beam pattern seen in Madinat al-Zahra and is outstanding in its gyral character set in a framework of reflection and glide-reflection planes. The most sumptuous group of swastikas combined in complicated patterns is represented here by the wall pattern closing the large iwan of the Karatay Madrasa (1251–

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Fig. 2.68: The most complicated swastikabased pattern in the Great Mosque of Cordoba; now in the interior of the mosque. Plane group p4; square configurations are hypersymmetrical (see Chapter 6).

1252) in Konya, Turkey. The blue glaze was a new achievement of the age and the local eightfold symmetries do not project into the plane group p4 of the entire design based on radiating modules, each composed of two different swastikas (Fig. 2.77). The Chinese and Japanese use of swastikas appears ruled by a much stricter kanon. The archetype I-beam pattern with symmetry p4gm is often affinely modified to rectangular pgg. Another modification that we have seen in China is replacement of swastikas in p4gm by winged squares and replacement of the recessed lozengeshaped swastika in pgg by only one of the two crossing lines which together make a simple swastika. This reduces the overall symmetry to p2. Manji, the Japanese Buddhist swastika symbol which can be oriented either way, is an element of a typical Japanese p4gm pattern. Besides this “parent” pattern, the affinely modified pgg variety and other derivatives are widely used. Japanese swastika patterns belong to the I-beam type. On gold-plated (or imitated) cartouches both the N = 1 tetragonal p4gm and affine pgg and p2 patterns, analyzed below, with recessed swastikas are a standard. The same patterns are on the bases of metal columns and wooden framework. On metal plaques, they are most often dichroic: the shiny and matt fields alternate, with thin boundaries and dichroic group p4′g′m. An example is in Fig. 2.78 from the Nijo¯ Castle, Kyoto (built in 1601–1603, additions in 1624–1624). An¯ other pattern (Fig. 2.79) observed on the buildings of the Tosho-gu complex of Nikko, 128 km north of Tokyo, is dichroic with darkened strips and brighter “latchkeys” of the same thickness enclosed in their maze. This particular and effective configuration was found to be a selection of lines from the N = 2 member of the I-beam subfamily with simple swastikas. Additional bars were inserted between the long parallel connecting lines and the zigzag composites of horizontal interconnecting lines. The horizontal central bars of adjacent swastikas and their vertical arms joining the interconnections were selected, omitting the same vertically oriented/based elements. In this way, one subset of original lines was broadened whereas the same subset oriented at 90°, to

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Fig. 2.70: Ideal drawing of the pattern in Fig. 2.69 rotated by 90°.

Fig. 2.69: A complicated p6 brick pattern with rings of swastikas from a Kharraqan tomb tower in NW Iran. Plaster portions of swastikas are partly dissolved by rains.

the first one was omitted together with the space it originally covered. The resulting cmm pattern (it is not dichroic although on cursory observation it may appear so) is reduced to p2 by affine deformation. Two-dimensional patterns in Chinese art are mostly confined to ornamental doors (Fig. 2.80), lattice windows (Fig. 2.81), and dados (Fig. 2.56). Figure 2.80 shows a meander pattern of broken swastikas, a version with two dividing lines. The plane group of this (Buddhist) Chinese openwork pattern is p4gm with fourfold rotation axes on the swastikas; the glide-reflection planes run between the swastikas. It is located in the Forbidden City, Beijing, China.

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Fig. 2.71: A p6 pattern of hexagonal stars and rings (left) and a p4 pattern of octagonal stars and rings, both composed of two-swastika modules. Openwork with a mortar-clad background. Shifaiye Madrasa, Sivas, Turkey.

Fig. 2.72: An openwork brick pattern with a mortar background. Based on four-swastika blocks, plane group p4. Shifaiye Madrasa, Sivas, Turkey.

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Fig. 2.73: A reconstructed p4 pattern of reFig. 2.74: A p6 pattern that I constructed on the cessed swastikas and triskelions from an basis of the p4 pattern in Fig. 2.73. unnamed tomb tower in Maragha, Iran. The original is an uninterrupted broad frame around the pishtak.

Fig. 2.75: I-beam pattern with poised swastikas in p4gm carved on a column in Ulu Cami mosque in Divrigi, Turkey.

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Fig. 2.76: A p4gm pattern of swastikas and sixfold stars on a wall of Ulug Beg Madrasa, Registan, Samarkand

Figure 2.81 shows deliberate variations of the openwork meander lattice. The left-hand panel shows a p2 pattern of alternative blocks formed by two or four parallel lines, respectively, for the two block orientations, and “two-armed” meanders. The righthand panel is the classical affine pgg pattern derived from p4gm. This ornamental door comes from the house-block “Jiu Tou Ma”, located at Qi Yang village, He Shang town, Changle city, Fujian province, China. The Silk Road to China was dotted by caravanserais, spaced at intervals equal to one day’s journey of a caravan. One of the prominent royal caravanserais was Rubati-Sharif in eastern Iran, built in 1128 on the road from Mashad to Herat (Fig. 2.82). The two surviving tympana (Figs. 2.83 and 2.84) display plane groups p4mm and p4gm, respectively, with patterns constructed of raised bricks in the best Seljuk tradition. Old textiles are an integral part of the story of symmetry, although they required special preservation conditions to avoid their demise. Later I shall show some special cases with interesting polychromatic symmetry. Here is a typical subrecent example of the plane group pg from a museum in Tashkent, Uzbekistan (Fig. 2.85). The same

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Fig. 2.77: Eightfold rosettes of twinned swastikas in turquoise tiles. A p4 pattern in the iwan of Karatay Madrasa, Konya, Turkey.

plane group envelops a Zulu basket from South Africa, with attractive white triangles and with the size adjusted to the bulging body of the basket; they define the pg plane group of symmetry (Fig. 2.86). I finish this chapter with four nice and instructive examples of the plane group p6 in colorful tiles. Figure 2.87 is from an old Ilkhanid mosque (1300–1310) in Natanz with a precious turquoise and blue decoration. The second example (Fig. 2.88) is a (reconstructed) ceramic p6 pattern from the Reales Alcazares in Sevilla, Spain, with tiles on six-, three- and twofold axes of rotation nicely separated by glaze colors. In this example, the twofold axes received their own tile in a special position. Figure 2.89 is a panel of four deltoid kites on the walls of the Friday Mosque in Isfahan with a spider web of white connections of sixfold axes with adjacent axes connected via arms running through the intervening twofold axes. Each kite-like field has its own pattern, interrupted by kite boundaries. Relative tile sizes in Fig. 2.90 differ from those in Fig. 2.88. Three unsual mosaics are illustrated in Figs. 2.91–2.93.

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Fig. 2.78: An ornamental metal panel at Nijo¯ Castle in Kyoto. The background I-beam pattern was worked out as a dichroic swastika pattern, dichroic group p4′g′m.

Fig. 2.79: Manji p4gm pattern worked out as p4′g′m; also p6mm is present. What looks like pg′g′ actually is p2 because the matt areas were worked out as continuous lines, also through the “ideal” swastika sites and the bright areas became isolated “double T” islands. Nikko¯ complex, Japan.

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Fig. 2.80: A meander pattern p4gm of recessed swastikas. A door panel, Forbidden City, Beijing, China.

Fig. 2.81: Two of the ornamental windows from Qi Yang village, Fujian province, China. Left: a p2 pattern of meanders; right: a pgg pattern of broken swastikas.

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Fig. 2.82: A caravanserai Rubat-i-Sharif on Silk Road in E Iran with two of the three ornamental gates preserved.

Fig. 2.83: A raised-brick pattern in a caravanserai tympanum; plane group p4mm.

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Fig. 2.84: A raised-brick pattern in a caravanserai tympanum; plane group p4gm.

Fig. 2.85: A painted textile strip with plane group pg. Art Museum in Tashkent, Uzbekistan.

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Fig. 2.86: Plane group pg fitted to the bulging surface of a recent Zulu basket (South Africa).

Fig. 2.87: A small-tile pattern in p6 from the portal of the Ilkhanid mosque complex in Natanz, central Iran.

Fig. 2.88: A p6 wall pattern in the Reales Alcazares, Sevilla, Spain.

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Fig. 2.89: A panel of four deltoid “kites” with a p6 pattern of continuous and broken white lines. The courtyard of the Friday Mosque in Isfahan.

Fig. 2.90: A modern example of p6. Morocco.

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Fig. 2.91: Plane group p6mm in an unusual type of small element mosaic. Friday Mosque in Yazd, Iran.

Fig. 2.92: A cut-out from the plane group p4gm. A pelta pattern from the excavations of Volubilis, Morocco.

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Fig. 2.93: A p4mm tile pattern in Qajar style of painted tiles with flowers as the chief element. Iran.

3 Intertwined patterns: Layer groups of symmetry From the symmetry point of view, the main difference between the symmetry groups of intertwined (interlaced) patterns and the plane groups of symmetry is the presence of symmetry elements transforming the upper (i.e., visible) portion of the pattern into the lower portion, which is turned away from the viewer (unless he can inspect the opposite face of the pattern in the case of free-standing trellis). These pattern-reversing elements can be represented either by a mirror plane or a glide-reflection plane situated in the plane of the pattern, or by an inversion center, or a twofold rotation – or twofold screw (i.e., rotate-plus- glide) axis lying in the plane of the pattern. Except for the rather rare cases of free-standing two-sided panels (trellis), these operations are only revealed by interlacing/intertwining in a “virtual 3D space”. As long as only the upper and lower faces of the layer exhibit a pattern, the actual thickness of the layer itself is immaterial. When we include the plane groups of symmetry as a special non-flipping case, there are 80 different contradiction-free combinations of symmetry elements of this type, the so-called symmetry groups of two-sided layers (Table 3.1). When we make the “up-and-down” alternation of tiles or “above-and-below” weaving of strands invisible (as suggested in the previous chapter), several distinct layer groups can be reduced into one plane group. The same pattern of tenfold rosettes in an interlaced c222 edition and in a plane-group cmm form is shown in Figs. 3.1 and 3.2. Tab. 3.1: Layer groups of symmetry arranged according to the “parent” plane groups of symmetry which describe their surface configurations. p1 → p11a, p11m, rectangular: p211, p21 11, c211 p112 → p112/a, p-1, p112/m, rectangular: p222, p21 22, p21 21 2, c222, square: p-4 pm11 → p2/m 11, p21 /m 11, pm2a, pm21 n pb11 → p2/b 11, p21 /b 11, pb21 m, pb2b, pb21 a, pb2n cm11 → c2/m 11, pm2m, pm21 b, cm2m, cm2e pmm2 → pmmm, pmma, pmmn, square: p-4m2 pma2 → pmaa, pmam, pman, pbma pba2 → pban, pbaa, pbam, square: p-4b2 cmm2 → cmmm, cmme, square: p-42m, p-421 m p4 → p4/m, p4/n, p422, p421 2 p4mm → p4/mmm, p4/nmm p4bm → p4/nbm, p4/mbm p3 → p-3, p312, p321, p-6 p3m1 → p-3m1, p-6m2 p31m → p-31m, p-62m p6 → p6/m, p622 p6mm → p6/mmm

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Fig. 3.1: A c222 interlace pattern of tenfold rosettes. A widely used Islamic and Mudéjar pattern.

Fig. 3.2: A cmm equivalent of the pattern in Fig. 3.1. A widely used Islamic and Mudéjar pattern as well.

The “parent” plane group is put at the start of each line, the “daughter” layer groups follow after the arrow. The “layer-generating” operation is either defined by the last letter of the symbol or it is placed after the forward slash which follows a fourfold or sixfold rotation-axis symbol (eventually it is included in the inversion operation of the axis present). m is a reflection plane, a, b and c are glide-reflection planes with a half-period translation parallel to the axis of the same name, n glide plane has this

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shift along a diagonal, 21 is a twofold screw axis with a half-period shift parallel to the axis, and −N is a rotoinversion axis (a combination of rotation immediately followed by inversion). When we proceed along any individual string/line of the pattern, good interlaced/ intertwined patterns have the “above” and “below” configurations in regular alternation. Two “aboves” or two “belows” in a row indicate a faulty pattern. At the edges, all strings of a two-dimensional pattern can be cut off or, more frequently, they are interconnected with other strings of the same pattern along the edge, in a way which is absent in the pattern interior. In this way, the pattern is converted from an infinite two-dimensional pattern into a finite, 1- or 2-dimensional “carpet” design. This introduces ambiguity: either the symmetry can be understood as that of the infinite two-dimensional pattern or it is the frieze symmetry of a one-dimensional interlaced ribbon, or even it is a point group symmetry of a finite object (a “carpet” of certain dimensions). Symmetry of a finite object has one definite center in which a rotation axis can reside and in which intersection of potential reflection planes of symmetry and of potential rotation axes takes place (think of a symmetry of a cube in three dimensions). A cut-out of a two-dimensional pattern with a “properly worked-out border” can be only one unit cell wide or with a width of several unit cells of the original twodimensional design. What can we do in such cases? We can indicate symmetry of the underlying, infinite two-dimensional design and the m × n size of the design area in terms of a number m and n of its unit cells along the two shortest translation vectors, and besides it, we can also indicate a point-group symmetry of the resulting “carpet pattern” (or a frieze symmetry of the resulting frieze, see below) with properly finished borders. Both are valid and combine in the way we perceive the pattern. Islamic interlaced patterns are characterized by simple or complicated, regularly developed grids/nets. As an example, the periodic decagonal c222 pattern from the Reales Alcazares in Sevilla (Fig. 3.3) exhibits a regular alternation of the vertically running unit and τ interspaces (τ = [1 + √5]/2). For the steeply oriented systems of white dividers, alternation of two τ intervals with one unit interval is observed, and for the shallowly dipping systems, these bars are broken up and staggered. The tenfold local symmetry 10.2.2 of the rosette is embedded in a much lower, 222 site symmetry. The hexagonal “box” design of the wall mosaic from The Alcazares in Sevilla (about 1360) (Fig. 3.4) had been employed in a rich form with a Cosmatesque flavor as the altar floor mosaic in the Baptistery of San Giovanni in Pisa (after 1152) (Fig. 3.5) and at several Turkish Seljuk localities constructed in the years 1200–1250 (see Makovicky 2015a). It is a member of a small family of patterns: a more complicated pattern in the Norman Capella Palatina in Palermo, Sicily (1132–1140) was obtained by shearing the rectangles (Fig. 3.6) and displacing them by their width (Makovicky 2015a) and the pattern on the mihrab wall of the Kharraqan tomb tower (Fig. 3.7) has the sheared rectangles displaced by only a fraction of their width. The latter is not a two sided interlaced pattern – it violates the under-over sequence of line crossing and has three lines

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Fig. 3.3: A periodic decagonal c222 pattern of white ribbons over dark blue background. Reales Alcazares, Sevilla, Spain.

meeting in one place. The plane group is p6, whereas the previous family members were in p622. A very interesting subject of Islamic interlaced patterns is the typology of interlacing. If we forget the problem of carpet and frieze development with a “false” finish of interlaced lines at the edges, an interesting question is the complexity of design versus the complexity of interlacing. Surprisingly, there is no direct correlation of these two complexities. A rather complicated pattern in Fig. 3.6 has been created from one type of line only, and the fairly complex interlacing of the c222 type with decagonal rosettes, mentioned above and common for all corners of Islamic world (Fig. 3.3) is

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Fig. 3.4: A hexagonal “box-like” pattern from the pillars of the Reales Alcazares, Sevilla, Spain. Layer group p622.

composed only of three zigzag lines: [010], [110] and [310]. The pattern in Fig. 3.9 consists of two different closed loops, the same for the pattern in Fig. 3.10, whereas the complicated interlacing in Fig. 3.11 was created by application of only one loop (in a form of an irregular octagon). Combinations of loops and infinite line segments occur as well (two loops and one open line for Fig. 3.12). How did they construct these interlaces? Purely a feeling is that they were working them out from noninterlaced

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Fig. 3.5: A rich version of the “box-like” pattern from Fig. 3.4, from the floor mosaics of the altar in the Baptistery of San Giovanni in Pisa. Islamic design combines with a Cosmatesque adornment. For more details see Makovicky (2015a).

Fig. 3.6: Interlacing scheme of the wall mosaic in the Norman Capella Palatina in Palermo, Sicily (1132–1140). A p622 design family with the pattern in Fig. 3.4.

Fig. 3.7: A relief pattern on the mihrab wall of a tomb tower at Kharraqan, NW Iran. Although it is related to the patterns in Figs. 3.4 and 3.6, the “box-rectangles” are sheared, and interlacing is disturbed, resulting in a plane group symmetry p6.

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Fig. 3.8: A tetragonal wall mosaic from the Reales Alcazares in Sevilla. Layer group p422. Figs. 3.8– 3.12 illustrate the relationship between symmetry and the underlying line scheme.

designs, varying the line connections until the proper up-down-up-down sequence was achieved. And what about constructing the interlace of quasiperiodic tilings from Islamic Spain, described in Chapter 7? There are other outstanding cultures with intertwined patterns – Viking and Celtic art, and that of late antiquity (fifth to sixth centuries, e.g. Ravenna, Italy), as well as the Tamil cultural circle. The Viking and Celtic intertwined patterns are animal designs in which highly stylized animals occur alone or in pairs. Whereas the individual creatures do not appear to have identical front and back sides, many intertwined pairs have a horizontal twofold rotation axis between the two intertwined individuals (point group 2 instead of a mirror), the others have a vertical twofold rotation axis instead. The famous intertwined, partly zoomorphic frieze of the Urnes church (late eleventh, to early twelfth century) is a nonperiodic intertwining of sinusoidal and 8-shaped cables with two thickness and size levels (Anker 1970, Meehan 1995a). Celtic knotwork on stones and stone crosses, as well as in the Book of Durrow is mostly modular – square-like complexly knotted blocks are interconnected in a looser way into pairs but mostly into quadruples (Bain 1973). Point groups of the interlacings composed of these fourfold blocks are 222 or 422, in agreement with their knotted structure. There is a rich literature which guesses the meaning of different knots and aggregates but we should recall that these were the times when every individual nail had to be hammered separately

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Fig. 3.9: A hexagonal interlaced p622 pattern from the Reales Alcazares, Sevilla.

by hand from not so cheap iron, and tying a series of specialized knots belonged to the important practical skills for solving many tasks. Still, Farley et al. (2015) suggest that the interlacing Celtic Insular style is a product of seventh century fusion of Celtic, Germanic and Mediterranean influences. A similar character is observed for the hand-drawn floor designs of the Tamil cultural region in India (Fig. 3.13). Here knots are not worked out but the under-over interlace is assumed by the design (Gerdes 1989). In the blossoming times of this magic design (to ward off harmful spirits) the entire design represented one twisting closed line but already by about 1937 “degraded” designs of several closed lines were common (Gerdes 1989).

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Fig. 3.10: A p422 pattern with concentric squares. Reales Alcazares, Sevilla.

Fig. 3.11: A p422 pattern with interlacing modified in comparison with Fig. 3.10. Reales Alcazares, Sevilla.

3 Intertwined patterns: Layer groups of symmetry |

Fig. 3.12: A p422 pattern with interlacing in form of “L-squares” (details in Chapter 7). Reales Alcazares, Sevilla.

Fig. 3.13: A virtually interlaced Tamil floor pattern. Hindu Temple in Chicago, USA.

101

102 | 3 Intertwined patterns: Layer groups of symmetry The Roman mosaics of late antiquity have a different character, seen for example in the excavations of the old pavements in Ravenna (Farioli 1975). They consist of circles, squares, lozenges, and their interspaces (as many other Roman mosaics do), but their points of contact are points of up-and-down reversal, being arrayed on horizontal twofold axes of rotation. The resulting layer symmetry is p422. The vegetal or pictorial contents filling the fields enclosed by the outlines are independent of this geometry. An illustration of this principle is in Fig. 3.14 for a circle-shaped mesh of a floor mosaic and in Fig. 3.15 for a square mesh on a limestone wall panel, which has a lovely

Fig. 3.14: An interlaced pattern of the late Antique type consisting of interconnected circles composed of black tesserae. Castle Museum, Rhodos, Greece. When extended, it has a two-sided layer group p422.

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Fig. 3.15: An interlace square p422 motif with mesh filled by oriented vegetal motifs. An early Christian marble panel at the entrance to Santa Maria in Trastevere, Rome, Italy.

naïve early Christian character. Both have p422 symmetry and both are composed by interlocking of four-lobed topological rings. Confirming a predilection of old masters for the tetragonal p422 group is a cluster of white marble windows of the old Grand Mosque of Cordoba. The modestly twosided interlacing of these windows (Figs. 3.16 to 3.18) mostly points to p422 which is further reduced to p222 or p211 by vegetal inserts in the little mesh of the windows (Figs. 3.16 and 3.17). Other mosque windows are directly p222 (Fig. 3.18) and p622 or a carpet design (Fig. 3.19). Preference of the p422 two-sided layer group is seen also in the linear framework of the plaster panel from the ancient synagogue in Cordoba (Fig. 3.20). This “quiet” pattern, however, is underlain by a polar, cm, leaf-studded pattern. The same predilection transpires from the complicated tetragonal p422 pattern in the patio of Reales Alcazares in Sevilla (Fig. 3.21) with fourfold axes in rosettes and in composite squares. This is an old tradition: the interlaced plaster ornament in p422 from the Silk Road caravanserai Ribat-i-Sharaf in eastern Iran (Fig. 3.22) was made in twelfth century AD. A similar looking pattern is in the false window in the tympanum of the Kalyan mosque, Bukhara, Uzbekistan (Fig. 3.23). However, the fourfold groups have only local symmetry, their vertical rows have a vertically oriented twofold rotation axis in the plane of the pattern, horizontally oriented twofold axes through and between the 8-fold groups, and mirror planes between the vertical rows. The layer group is p2mb, with a b-glide plane in the plane of the pattern. Its glide component is from a row to the adjacent row (a complete structure consists of columns alternatively oriented “from” and “into” the projection plane). It is a nonstandard orientation of the layer group pm2a from Table 3.1.

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Fig. 3.16: A marble window of the Grand mosque of Cordoba (restored). Layer-group symmetry p422 is violated by the oriented fleur-de-lis and reduced to p211.

Fig. 3.17: A marble window of the Grand mosque of Cordoba (restored) with a circular motif. Symmetry as in Fig. 3.16.

Fig. 3.18: A marble window of the Grand Mosque of Cordoba. Although it simulates higher symmetry, certain details make it only p222.

Fig. 3.19: A hexagonal interlaced window from the Grand Mosque of Cordoba; layer group p622.

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Fig. 3.20: A stucco pattern in the ancient synagogue of Cordoba, Spain. Two-sided layer group p422. Foliage in interspaces, however, follows a polar plane group cm.

Fig. 3.21: A tetragonal p422 pattern from Reales Alcazares in Sevilla in a style of quasiperiodic patterns.

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Fig. 3.22: An interlaced plaster ornament from the caravanserai Ribat-i-Sharaf, Khurasan, Iran, twelfth century AD, with layer group p422 for the interlace and only a plane group p112 (with mistakes) for the vegetal fill. In spite of the much lower symmetry, the latter symmetry created by mechanical application of the leaf motif does not disturb the impression gained from interlaced ribbons.

Hexagonal designs on the wall of the Kharraqan towers (Chapter 2) are all in p622 except for the simple diaper patterns composed of barely touching rhomb-shaped bricks on tympana. Large examples are Figs. 3.24 and 3.26 – a complicated and a simple edition of the layer group, respectively. Figures 3.25 and 3.27 are two nice “frieze-like” cases of p622 with less prominent interlaces. I admire the mastery of old bricklayers in creating hexagonal designs from ordinary bricks and the flawless execution of virtual overlaps in them. Art Deco lost taste for regular interlacing and chose other composition principles (Fig. 3.28).

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Fig. 3.23: A pattern in the false window of the tympanum of the Kalyan Madrasa, central Bukhara, Uzbekistan. The layer group is p2mb, a non-standard orientation of pm2a.

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Fig. 3.24: A “virtually interlaced” p622 pattern from a tympanum of the Kharraqan towers, NW Iran, composed of infinite wavy lines.

Fig. 3.25: A virtually interlaced p622 pattern from a Kharraqan tower, composed of infinite zigzagged lines.

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Fig. 3.26: A p622 pattern from a Kharraqan tower composed of large virtually interlaced dodecagons.

Fig. 3.27: A p622 pattern from a Kharraqan tower composed of large virtually interlaced hexagons.

110 | 3 Intertwined patterns: Layer groups of symmetry

Fig. 3.28: In spite of its construction on several depth levels with rudimentary interlacing, this American Art Deco pattern from central Chicago avoids a truly periodic interlacing. With the exception of the central panel, it is a carpet design with symmetry mm2.

4 Two-colored periodic ornamentation In several important cultures, much of the ornamental decoration is based on the use of contrasting colors. They occur either as two-colored patterns or as more complex, polychromatic ones. Their complete visual symmetry often exceeds that of simple plane groups of symmetry and must also include the nongeometric part, i.e. the color permutations, whenever they are present. Among two-colored patterns, those where each color is assigned to a tile set which has its own shape and position in relation to the symmetry elements, with both of these properties different from the other tile sets of the pattern, are just cases of simple plane group symmetry, because none of the symmetry operations involve a color change. However, the cases in which a set of, in all respects, identical tiles is divided into two or more differently colored subsets, are cases of colored symmetry. It means that some of the symmetry operations of the original plane group of symmetry also alter the tile color. We call dichroic the groups with symmetry operations which alter one color (let us call it “white”) of a tile set into another color (call it “black”) (Washburn & Crowe 1998).

Fig. 4.1: A fragment of wall mosaic from the Alhambra, Granada, Spain. “Maple-leaf tiles” are positioned on mirror planes of the dichroic group p4′g′m.

The dichroic plane groups can also be called groups of black-and-white symmetry or antisymmetry. Remember, these patterns are combinations of only two colors (Fig. 4.1). As briefly mentioned before, antisymmetry does not apply only to the two-

112 | 4 Two-colored periodic ornamentation

dimensional patterns but the point groups and frieze groups can become dichroic as well. Dichroic, color-changing operations can be reflection planes, glide-reflection planes, 2-, 4-, and 6-fold rotation axes (Fig. 4.2b,c) and also translations, including the centration vector of a unit cell. Threefold rotation axes obviously cannot be dichroic elements of symmetry. Not all symmetry operations in a dichroic symmetry group are color-changing operations. Therefore, from one plane group of symmetry several distinct dichroic groups can be derived, with different subsets of dichroic operations which give distinct coloring schemes. As already mentioned, the plane group p4gm has fourfold axes in the corners of the square unit cell and in its center; furthermore it has twofold axes which halve its edges, sets of glide planes at ¼ and ¾ of the cell width, and parallel to the respective edges and, finally, reflection planes parallel to the cell diagonals (Fig. 4.2a). These are neatly arranged in its plane-group symbol (twofold axes are not written, they follow automatically from a combination of other symmetry elements). The dichroic groups derived by black-and-white coloring of this plane group are (color-changing operations are primed, e. g. m′; color-preserving operations remain unprimed): either p4g′m′ (Fig. 4.2b) with all reflection planes dichroic but with fourfold axes which do not change the color of the motif on which they act, or p4′gm′ (Fig. 4.2c) and p4′g′m with the fourfold axes producing a black-white-blackwhite sequence on one full rotation, and with alternative types of symmetry planes acting as dichroic, and pC′ 4gm in which only centration and additional, interleaved, reflection planes are dichroic (note the subscript C′) . The p4g′m′ group will change a “black” cluster of four general positions into a symmetry related “white” cluster around (1/2,1/2) (Fig. 4.2b). The plane group p4′gm′ has a dichroic fourfold rotation axis, creating a “propeller” of four general positions with alternating colors; their disposition obeys a color-preserving glide reflection (Fig. 4.2c). What is important for the pattern makers is that all special positions on the axes and mirror planes are “gray”, combining white and black in the same object (Fig. 4.2b,c). A profusion of such elements in a simple small-scale pattern (Fig. 4.1) would disturb the arresting beauty of its black-and-white character. It has consistently been avoided by artists of all cultures. A different situation exists in large-scale complicated patterns with many tile types and shapes, where such a “gray” element, e.g., in the form of a prominent colored rosette, serves as an artistic tool. In the hexagonal system, the sixfold axis can be dichroic, b-w-b-w-b-w, but a threefold rotation axis cannot (it is trichroic by its nature). Standardization of dichroic groups is not yet at the same stage as layer groups; variable symbols and axis orientations can still be found in literature. Coxeter (1985) introduced an alternative notation set for dichroic (and by extension other colored) groups of symmetry. He realized that for each two-colored pattern, a set of three groups is characteristic: when G is a group of the uncolored pattern, G1 is a subgroup which preserves colors and the group G/G1 is the group (of the order two in this case) which permutes the colors. Only two groups have differentiation problems in this notation

4 Two-colored periodic ornamentation

d

113

d

c

b

c

b

c

a

c

a

Fig. 4.2a: Symmetry elements, general positions and special positions (on various symmetry elements) of the plane group p4gm.

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Fig. 4.2b: Symmetry elements, general positions and special positions (on various symmetry elements) of the dichroic plane group p4g′m′.

d

c

c

b

a

Fig. 4.2c: Symmetry elements, general positions and special positions (on various symmetry elements) of the dichroic plane group p4′gm′.

114 | 4 Two-colored periodic ornamentation

e

c

a

e

c

a

Fig. 4.3: (a) Plane group p2 with a general position and special positions. (b) Dichroic group p2′ with a general position and “gray” special positions. (c) Dichroic group pb ′2 in a centered edition (see text).

(they are indicated in Table 4.1 by m after the symbol for the group in which all reflections preserve color and by m′ for the group in which some reflections change color; Crowe 1986). The two notations respectively represent analysis of individual symmetry operations versus a synthetic group approach – a matter of personal choice and convenience. For the present audience we use the former analytical approach, as shown in the appendix of this chapter, where the dichroic plane groups and their “parent” achromatic plane groups are listed and illustrated by means of general positions in a unit cell and their symbols. One of the plane groups which can be used for dichroic coloring of simpler, smallscale patterns is p2 in the oblique system. There are four sets of twofold rotation axes of symmetry in the unit cell, at (0,0), (1/2,0), (0,1/2), and (1/2,1/2) (Fig. 4.3a), so that they can host four different ornamental elements, each of them subject to a twofold rotation symmetry. Elements situated outside of the rotation axes (in so-called general positions) can be asymmetric and are related by any of the 180° rotations present in the pattern. If all rotation axes become dichroic, p2′, the unit cell preserves its size; the elements on rotation axes will be black and white at the same time, i.e., they will be “gray” (and will be avoided in small-scale patterns). The dichroic character is seen only in the black and white coloring of the elements in the general positions (Fig. 4.3b). If only every second of these twofold rotation operations becomes dichroic, it leads

4 Two-colored periodic ornamentation

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115

to a doubling of the crystallographic axes and the rotation-related pairs of asymmetric elements become alternatively of opposite colors and of the same color (Fig. 4.3c). With the original vectors we have a centered large cell in this figure, a quadruple of the original one. In the oblique system, however, we can choose another down-pointing a vector, a diagonal of the old small cell, so that only the horizontal b vector (and the cell area) will be doubled; this determines the dichroic plane group as pb′ 2 (appendix). All dichroic twofold operations are generated by the combination of the dichroic translation, expressed by a subscript b′, with uncolored twofold rotations so that it is not necessary to express them in the symbol. If we want to penetrate to the underlying plane group of a colored pattern, all colors have to be “erased” and only blank tiles (tesserae) with their line-thin boundaries are to be studied. If we do not include the mathematically admissible cases in which all tiles are “gray” or one-colored, but only the groups with a color change, there are 46 dichroic groups (appendix) to be considered in the practice of periodic ornamental tiling. A large body of dichroic patterns (Figs. 4.4–4.10) forms a part of the Romanesque Cosmatesque mosaic art of central Italy (the story of Cosmatesque art was given in Chapter 1). In these cases, the dichroic “black-and-white” pair is a red Imperial Porphyry–green Greek Porphyry pair, supplemented by white (yellow) marble as a (usually) neutral medium. The underlying geometry consists of squares (if they come

Fig. 4.4: A Cosmatesque floor pattern with symmetry p4′g′m with a fractal character. Church Santa Maria in Cosmedin, Rome, Italy.

116 | 4 Two-colored periodic ornamentation

Fig. 4.5: This pattern combines two different dichroic plane groups: solid tiles have pb′ mm arrangement whereas the fractal aggregates have p4′g′m color scheme. Yellow tiles are a neutral background. Cosmatesque floor mosaic in the church Santa Maria in Cosmedin.

Fig. 4.6: A pattern with two portions, each with its own symmetry: pb′ mm for the aggregation of small tiles, and pa′ mm for the larger tiles. The two periodicities cross one another at 90°. Cosmatesque floor mosaic in the church Santa Maria in Cosmedin, Rome, Italy.

4 Two-colored periodic ornamentation

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117

Fig. 4.7: Original design in p4mm was colored in such a way that the circles have symmetry pC′ 4mm whereas the small squares are pb′ mm. Cosmatesque floor mosaic in the church Santa Maria in Cosmedin, Rome, Italy.

in two sizes, a diagonal of the smaller square is equal to the edge of the larger square), triangles (sometimes according to the Sierpinski fractal principle), and sometimes lozenges. A recoloring of a certain element set of the pattern as an array of red and green rows was the preferred means of creating a dichroic pb′ mm pattern. Nine out of 25 patterns from several historic churches in Rome (especially Trastevere and Cosmedin) are uncolored plane groups in which colors were applied to distinct elements; six are pb′ mm patterns produced from p4mm but in most cases as a second stage coloring, the first stage (not preserved) being pC′ 4mm as a “black”-and-white edition. Only the black tiles were recolored red and green in the final stage of coloring of these patterns; the white tile subset was taken out of consideration and remained white. Three patterns ended up as pC′ 4mm, some as p4′g′m (Figs. 4.4 and 4.5). Furthermore we found pb′ 1 obtained from cm (Fig. 8.2), pb′ m from pm (Fig. 8.1), and interesting cases of coloring which produced two partial patterns with different symmetries from the uncolored one: pa′ mm and pb′ mm valid for two different types of squares (Fig. 4.6), pC′ 4mm and pb′ mm from p4mm (Fig. 4.7), cmm produced from p6mm (Fig. 4.8), cmm (frames) and c′mm (stars, with unit cell at 30° from the previous) from p6mm, and two pC′ 4mm shifted against one another by a × √2 of the original uncolored p4mm

118 | 4 Two-colored periodic ornamentation

Fig. 4.8: Coloring-produced plane group cmm from the original p6mm pattern. Cosmatesque floor mosaic in the church Santa Maria in Cosmedin.

pattern. This indicates that dichroic translations were the principal means of coloring for the two-dimensional Cosmatesque floor patterns. Three outstanding dichroic patterns from the Alhambra show the characteristic ceramic colors: green, rusty yellow, black and blue. In all cases, the pattern is enlightened by white interspaces. The tetragonal pattern (Fig. 4.9) is rather misleading by its highly symmetric architecture (uncolored symmetry p4mm): all colored tiles are on special positions, even the green and orange ones which advertise the dichroic character of the pattern. A careful reanalysis indicates plane group pC′ 4gm with the clearly visible uncolored glide planes vertical and horizontal in the panel and the uncolored mirror planes, on which the green and orange rosettes dwell, running diagonally between the large black rosettes with fourfold rotation axes. Reflections on planes situated through the black rosettes are all dichroic. The complex pC′ mm pattern in the Patio de los Leones (Fig. 4.10) is based on an uncolored p6mm tracing. The block-like coloring leads to poised lozenges with alternating green and honey color. Six- and threefold symmetries are lost and the unit cell origin can be placed in the center of a honey-yellow lozenge; the green lozenge is the antisymmetrical centration. Black elements are boundaries (sharing both colors) whereas the horizontal blue strips avoid the dichroic coloring. From the tower complex of Comares comes the third panel (Fig. 4.11) with a light p4′g′m pattern with white elements on 4′ positions and the colored elements on m split by a white stripe.

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119

Fig. 4.10: A c′mm pattern obtained by coloring a p6mm tracing. Patio de los Leones complex, the Alhambra, Granada.

Fig. 4.9: A complex large-scale pC′ 4gm pattern with the green-orange rosettes that reveal the dichroic plane group situated on diagonal mirror planes. Sala de Dos Hermanas, the Alhambra, Granada, Spain.

An interlaced pattern from the corridor adjacent to Sala de la Barca (Fig. 4.12) shows how the artists used two incompatible principles in one and the same pattern: the interlacing, creating the layer group p422 and coloring which, with exception of the upper orange row (a frame of the panel) has a dichroic alternation of orange and black columns of stars (all the complicated interspaces are black, color-neutral). The dichroic group is pb′ mm, with a doubled horizontal translation, the uncolored mirror planes perpendicular to the b axis are situated on the orange and black arrays, respectively, and the dichroic ones, m′ which follow from the combination of translation and m in the group symbol, are situated on the blue arrays which themselves are the neutral “gray” elements.

120 | 4 Two-colored periodic ornamentation

Fig. 4.11: A light-colored wall mosaic p4′g′m from the complex of Comares, the Alhambra, Granada.

Coming from a quite different environment, an impressive dichroic pattern (Fig. 4.13) adorns the family house designed and built for himself and his family by Joseph Maria Olbrich in the Mathildenhöhe artists’ colony, Darmstadt (Germany); built in 1901. The artists’ colony was founded by the enlightened patron Grand Duke Ernst Ludwig von Hessen with mostly young artists as a counterweight to the ponderous pathos of Imperial German art of the day. J.M. Olbrich and Peter Behrens were the principal personalities of the movement. The ornamental broad band envelops the ground floor of a typical German suburban family house. Dark blue and white tiles with a scroll design follow the dichroic plane group pg′g′, derived from an uncolored pgg tracing. Can you find a mistake in this dynamic pattern?

4 Two-colored periodic ornamentation

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121

Fig. 4.12: A combination of an interlaced pattern (p422) with independent coloring of interlace “eyes”; except for the upper rim, the dichroic group is pb′ mm (blue areas are the gray positions).

Fig. 4.13: Dichroic plane group pg′g′ with two systems of dichroic glide planes and monochromatic twofold rotation axes. Single “Hokusai waves” are overlain by vertical sinusoids, both in two alternating colors. Designed by Joseph Maria Olbrich, detail of architect’s house in the Mathildenhöhe artists’ colony, Darmstadt (Germany), built in 1901.

122 | 4 Two-colored periodic ornamentation

Superficial similarity is shown by the sgraffito of M.C. Escher’s metamorphic drawing “Metamorphosis III” on a house in central Madrid (Fig. 6.40). It starts from the bottom, with a (slightly distorted) dichroic square pattern pC′ 4mm and through a gradual metamorphosis to arched lizards it converts into p4′; from all the symmetry elements of the former group, only subsidiary dichroic 4′ rotations are preserved. It is a perfect copy of Escher’s work. Dichroic effects in Cosmatesque friezes were mentioned in Chapter 1. They can be used with success in nonperiodic designs as well. Figure 4.14 shows an Arabic in-

Fig. 4.14: Black-and-red inscription with dichroic reflection symmetry (above) and only approximate dichroic reflection (below). Ulu Cami (Grand Mosque) of Bursa.

4 Two-colored periodic ornamentation

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123

scription on a pillar in the Ulu Cami mosque in Bursa, Turkey, with the top panel in red being an exact mirror image of the text in black; however, in the bottom panel the same kind of correspondence is only approximate. An amazing ornament built on this principle comes from the Friday Mosque in Isfahan in which only one important element is presented with a black versus white principle (Fig. 4.15).

Fig. 4.15: A locally dichroic point group m′ pattern from Friday Mosque, Isfahan.

The next figure, Fig. 4.16, has been drawn by a Canadian Northwest Coast Indian artist, in the traditional half-geometric, half-pictorial style. The upper and the lower wings of the eagle are colored in opposing manner, red in one wing being black in the other, and vice versa, in spite of only approximate geometric correspondence. The red upper portion of the beak contrasts with the black contents being swallowed. In the central portions, the black eye and black beak contents, and the red beak top and the red head base, introduce an approximate twofold monochromatic “reflection” as contradiction of dichroism. Without this internal tension in the coloring, the picture would lose much of its power. The internal tension between color and/or shape antisymmetry around (a) selected point(s) and symmetry about selected (approximate) points or “reflection lines” lies behind a permanent dynamic impression emanating from Fig. 4.16 and from a number of successful modern abstract paintings. A lack of this tension lies behind a very fleeting interest generated by other abstract paintings.

124 | 4 Two-colored periodic ornamentation

Fig. 4.16: A symbolic Canadian NW Coast raven figure, signed by D.D., 1989. An example of artistic combination of color antisymmetry and symmetry which results in an exciting inner tension in the figure.

The mosaic marble floor of the Battistero di San Giovanni in Firenze (Makovicky 2015a) contains very fine examples of dichroic symmetry. Among the simple ones, Fig. 4.17 represents a sequence of black and white sinusoids, dichroic plane group pa′ gm. However, Fig. 4.18 in which each strip has two different boundaries has a completely different symmetry: the asymmetric strips of two opposing colors “turn towards one another” with the same kind of sinusoid, and produce a pmg′2′ floor pattern, with vertical dichroic twofold rotation axes in tile boundaries. Figure 5.5 and similar have a dichroic basis but have been recolored in schemes which will be dealt with in the next chapter. The uncolored p4mm tracing of Fig. 4.19, with inlaid almond-shaped elements has been colored in a dichroic plane group p4′mm′. Conspicuous dichroic g′ planes through the tile centers are interleaved by m′ planes running through the “dissected” tiles. The last pattern of this group, Fig. 4.20, is full of visual ambiguity. Its eightsegmented dichroic circles in mutual contact, the dichroic “hyperbolic squares” they carve out, as well as the square and its complex dichroic infill, clash visually in this deceitful pC′ 4gm pattern tucked away from the main floor sectors. Wall mosaics in the city of Fez, Morocco, are distinguished by patterns of carefully shaped relatively small ceramic tesserae. Figure 4.21 has black and white propeller tiles on threefold rotation axes, the dichroic group is p6′. Figure 4.22 has the same dichroic group, in spite of a different, somewhat confusing appearance with special positions on threefold axes alternately black and yellow. Figure 4.23 consists of a centered pattern of “flames”. The white skeleton has a two-sided layer symmetry c211 but the coloring disregards it and the dichroic group is pb′ m, with the black flames on special positions, on subsidiary m′ planes of symmetry. We bring one modern mosaic

4 Two-colored periodic ornamentation

Fig. 4.17: A pa’ gm floor pattern in the Baptistery of San Giovanni in Firenze (a axis horizontal).

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125

Fig. 4.18: pmg’2’ floor pattern in the Baptistery of San Giovanni in Firenze (a axis horizontal).

Fig. 4.19: A p4’mm’ floor pattern in the Baptistery of San Giovanni in Firenze.

126 | 4 Two-colored periodic ornamentation

Fig. 4.20: A pC′ 4gm floor pattern in the Baptistery of San Giovanni in Firenze, Italy.

Fig. 4.21: A wall mosaic from Fez, Morocco, dichroic group p6’.

Fig. 4.22: Another wall mosaic from Fez, Morocco, dichroic group p6’.

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Fig. 4.23: A cm pattern altered into pb′ m, the city of Fez, Morocco.

from a workshop in Fez (Fig. 4.24) because of its very interesting character. The original uncolored group, p4gm, was colored as two independent sets: black and dark blue lozenges obey the tetragonal p4′g′m symmetry whereas the quadruples of square tiles in white and beige follow c′mm, in a very pleasing mosaic with a quiet character. Very interesting framed fragments of dichroic patterns on the walls of the Casa de Pilatos, Sevilla, Spain, consist of white and light-blue editions of the same halfcircular tile (Fig. 4.25a–c). The simplest combination is cm′m (a axis along the strip) with blue and white tiles oppositely oriented, the next one pb′ mm with tiles in two very different positions, forming colored strips “dotted” with pairs of oppositely colored tiles, and the third type is pb′ m′m in which tile pairs which “center” the space between the principal tiles are quartered into white and blue quadrants. It is a virtual “dichroic groups for sale” collection. To finish this section, let us discuss a magnificent mosaic from a small, hidden madrasa along the bazaar street in Isfahan (Fig. 4.26) with dichroic swastikas in turquoise and yellow. The dichroic group of this design is only p4′, is spite of conspicuous lozenges in a glide-plane arrangement. On the contrary, the mosaic in Fig. 4.27 is very plain.

128 | 4 Two-colored periodic ornamentation

Fig. 4.24: A mosaic and tile workshop in Fez, Morocco: a modern mosaic with clusters of four white and brown square tiles; the composite p4′g′m and c′mm coloring is analyzed in the text.

Tab. 4.1: Notations of dichroic plane groups of symmetry. Column 1: Belov–Tarkhova; column 2: Coxeter Table (source: Crowe 1986) pb′ 1 pb′ 1m pg′ pb′ gm pm′g pgg′ cm′ cmm′ p4′m′m pC′ 4gm p31m′ p6′m′m

p1/p1 pm/pm(m′) pg/p1 pmm/pmg pmg/pg pgg/pg cm/p1 cmm/cm p4m/cmm p4m/p4g p31m/p3 p6m/p31m

p2′ pb′ g pb′ 1g pb′ mm pm′g′ pg′g′ pC′ gg cm′m′ p4′mm′ p4′gm′ p3m′ p6m′m′

p2/p1 pm/pg pg/pg pmm/pmm pmg/p2 pgg/p2 cmm/pgg cmm/p2 p4m/pmm p4g/pgg p3m1/p3 p6m/p6

pb′ 2 pb′ m pmm′ c′mm pb′ gg pC′ m pC′ mm p4′ p4m′m′ p4′g′m p6′

p2/p2 pm/pm(m) pmm/pm pmm/cmm pmg/pgg cm/pm cmm/pmm p4/p2 p4m/p4 p4g/cmm p6/p3

pm′ c′m pm′m′ pmg′ pb′ mg pC′ g pC′ mg pC′ 4 pC′ 4mm p4g′m′ p6′mm′

pm/p1 pm/cm pmm/p2 pmg/pm pmg/pmg cm/pg cmm/pmg p4/p4 p4m/p4m p4g/p4 p6m/p3m1

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Fig. 4.25: Three different stacking variants of white and turquoise tiles in the Casa de los Pilatos, Sevilla, Spain. ‘A dichroic LEGO for sale’; dichroic groups pb′ mm, cm′m and pb′ m′m.

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Fig. 4.26: A colorful wall pattern of dichroic swastikas and lozenges (p4′). A small madrasa along the bazaar street in central Isfahan, Iran.

Fig. 4.27: The wellknown prosaic symbol of dichroic ornaments from the Alhambra, Granada, Spain: p4′g′m.

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131

Appendix: Dichroic groups

p1

pb’1

p2

p2’

c’mm

Two-dimensional dichroic groups of symmetry (also known as groups of antisymmetry) introduced by the uncolored “parent” plane groups (in the first column) in the oblique, orthorhombic, tetragonal (= square), and trigonal/hexagonal systems. Action of symmetry elements is expressed via general positions (oriented triangles) in a unit cell (or its multiple for the last category). Black and white coloring defines antisymmetry operations.

p b’2

pc’gm

pmg’

pg’g’

cm’m’

pm’g

pg’g

cm’m

pmg

pgg

cmm

Fig. 4.29

pm’m’ pm’m pmm

pc’mm

pc’g pc’m cm’ cm

pb’gg

pa’g pb’g pg’ pg

pm’g’

pa’m pb’m pm’ pm

pb’mm

c’m

pb’mg

pc’gg

pa’mg

Fig. 4.28

p3m’1

p6’

p6m’m’

p3m1

p6

p6mm

Fig. 4.31

p31m’

p31m

p3

p6’m’m

p6’mm’

p4’ gm’

p 4’ g ’ m

p4 gm

Fig. 4.30

p 4 ’m m ’

p 4 ’ m’ m

p 4 mm

pc4

p4’

p4

p 4 g ’m ’

p 4 m ’m ’

pc4gm

pc4mm

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5 Polychromatic patterns The tile mosaics of the Alhambra are renowned for their colors. White tiles alternate with green, honey yellow, brown, black and blue tiles; true red is missing or has been added by restorers (Acedo 1989). What about patterns in which more than two colors interchange by the action of symmetry operations? For the so-called “perfect” coloring of the plane group p4gm (something like that reproduced in Fig. 5.1), we can define a fourfold axis with four consecutive, different colors produced by partial rotations, e.g., a “red-yellow-green-blue” as a 1-2-3-4 sequence, which will be symbolized as 4(4) . The glide planes are dichroic, symbolized in color groups of symmetry

Fig. 5.1: A four-colored version of the plane group p4gm pattern. The four-colored sequence 1-2-3-4 and its implications (dichroic to tetrachroic operations of symmetry) are discussed in the text. Unit cell, mirror planes and diagonal glide-reflection planes are shown.

134 | 5 Polychromatic patterns by a symbol g (2) . We shall find that in this group a given reflection plane m changes two colors and preserves two other colors of the four-color set (a case of a so-called partial chromaticity), which is written as m(2,2) . The symbol for the perfectly colored pattern is (p4(4) g (2) m(2) )(8) , whereas for the nearly perfectly colored pattern in Fig. 5.1 it is (p4(4) g (2) m(2,2) )(4) . In the illustrated case, it is impossible to decide whether the positions on m are from the former or the latter group. However strange it may appear, less-than-perfectly colored patterns are the ones which are common. As indicated, the number of color permutations connected with each symmetry element in these symbols, and the resulting number of color permutations produced by the colored group, is given as a superscript (n). The superscript (n1 ,n2) indicates that the operation acts as a color permuting one on n1 colors but it leaves n2 colors unchanged. Each tile of the pattern has to obey coloring of the site in which its center is positioned, even if the body of this tile overlaps small portions of adjacent, differently colored fields. Threefold rotation axes allow trichroic coloring (1-2-3), denoted as 3(3) , while the sixfold axes allow the 1-2-3-4-5-6 color sequence, 6(6) , but also a trichroic 1-2-3-1-2-3 (i.e., 6(3) ) and a dichroic 1-2-1-2-1-2 sequence, 6(2) identical with 6′. It is not necessary that at least one individual symmetry operation is polychromatic, for example, in the tetrachroic group (p4(2) g (2) m(2) )(4) only dichroic operations are present but they result in four different colors. The most versatile operations are the translations. Besides the achromatic ones, dichroic, trichroic, etc. translations can be the defining element of the design. For example, they are three-colored in a color group pa(3) b (3) 3 with uncolored 3-fold rotation axes, as opposed to p3(3) , with uncolored translations and a trichroic threefold rotation axis. Only the tiles (fields) in general positions of any group have the fundamental colors; those in special positions, on sites of polychromatic symmetry operators, have “composite” colors – hypothetical combinations of two, three, four or six generalposition colors. Similar to the dichroic cases, there are also other systems of notation for the color groups but we find the one used here easiest to apply “in the field”. Was a “practical version” of this principle of color groups known to the old masters? If it was known, it was not frequently used: for example, both the Islamic artists (Makovicky & Fenoll 1999) and the Italian Romanesque artists (Makovicky 2015a) practiced the approach based on the (sometimes repeated) use of dichroic symmetry with different kinds of modifications. The patterns are characterized by the profusion of white tiles, which lighten the pattern especially in the not-so-well-lit portions of the buildings. Fifty percent of identical tiles are white (Figs. 5.2 and 5.3), whereas the other 50% (the “black tiles” of the first, dichroic coloring stage) are subject to subsequent recoloring according to one of three principles: 1. Superposition of (multi)color waves on the “black set” of tiles, with the wavepropagation direction parallel or diagonal to the periodicity of the underlying uncolored pattern. Glazed ceramic tiles allow true multicolored waves (Fig. 5.4). As a result of limited color scale of natural rocks used for floors, however, we may

5 Polychromatic patterns |

Fig. 5.2: Idealized coloring sequence from the Mexuar, the Alhambra, Granada. Coloring waves were applied after one stage of dichroic coloring. The majority of such patterns of this type are completely disordered in Mexuar.

2.

3.

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Fig. 5.3: Another idealized coloring sequence from Mexuar. Color periodicity of small elements in this pattern deviates by 90° from that of large tiles.

find several repetition periods of the underlying pattern with the same color, separated by a one or ½-period broad boundary with another color (Fig. 5.5). Construction of “supertiles” by assigning the same color to a geometric patch of original tiles and combining patches of different color into a large-scale pattern (Fig. 5.6). Most intriguing are the “sequentially dichroic patterns” (Makovicky & Fenoll 1999): the “black” (sometimes the “white”) tiles of the first dichroic stage are dichroically colored again; one of the subsets of this second dichroic stage is allowed to keep its own, new color, whereas the other subset again is dichroically colored, and so on, the same selection being performed at each stage. In this way we obtain color distributions with color frequencies of 50%, 25%, 12.5%, etc. of the tiles present, and very intriguing color patterns. The operations can be stopped at any stage, at the artist’s discretion (Figs. 5.2, 5.3, and 5.7).

Among the colored symmetry patterns which do not follow the above, initially or repeatedly dichroic scheme, the Pre-Columbian cultures of South America are

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Fig. 5.4: A “pajaro” tile pattern from the niche in the Patio de Comares N, with a diagonally oriented four-colored modulation wave on propellers and a diverging three-colored modulation wave on stars. Details are in the text.

Fig. 5.5: A mosaic floor in the Baptistery of San Giovanni, Firenze, Italy. A dichroic pC′ mm pattern which is color modulated by a wave with λ = 4b.

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Fig. 5.6: A “supertile” pattern of pajaritos in the NW niche of the Patio de Comares, the Alhambra, Granada, Spain. Before symmetry reduction by the p3 pajarito scheme, the color scheme is pb′ mm with a yellow-blue alternation. Green is the “gray” color on m′.

renowned for their polychromatic textile and ceramic art. As described, for example, by Makovicky (1986), especially among Paracas textiles simple sequences pa(n) 1 are common, in which n can be equal to 4, 5 or 6. They may be complicated by color reversals along the sequence. The Nazca and Tiahuanaco cultures used mostly dichroic symmetry. Ornamented temple facades drawn and painted in the Mayan Codex Vindobonensis offer a fascinating case where the morphology has a subperiod by one interval larger or smaller than the number of colors in the periodic color scheme, and as a result, the color scheme “glides over the morphological sequence”, each consecutive

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Fig. 5.7: A hueso pattern from the niches of the Sala del Trono, Torre de Comares, the Alhambra. Details of derivation are in the text and in Fig. 5.15.

subset being colored by a color sequence shifted by one color of the repeating spectrum (Makovicky 1986). The two South American examples shown here come from the exhibits in a museum in Madrid. The zoomorphic design (Fig. 5.8) shows a typical divergence of animal designs and coloring. If we define one colored rectangle occupied by an animal as a subcell with the a direction downward and the b direction to the right, a full scale of colors takes five rectangles with a given animal whereas the stacking of animal rows repeats (probably) every four rows. A complete large cell contains 20 rectangles. The coloring can be modeled as a 5-color wave with the wave front parallel to (110) of the subcell and the wave vector perpendicular to the [1-10] direction. As for the symmetry, the rows of right-facing birds, with a repetition period of four tiers, are interspaced by rows of left-facing birds of the same kind (Fig. 5.8). However, the resulting plane group of the uncolored rectangular array, pg, is invalidated by unequal occupation of rows between them (by dogs and monkeys, respectively). The distinct Huari (or Wari) empire ruled the Andean region in the years 700–1100 AD (or 600–900 AD by other sources). Huari textiles inherited the artistic excellence of Nazca and Tiwanaku cultures (Cáceres Macedo 2005). The typically abstract Huari (or Wari) design (Fig. 5.9) consists of cloth strips made on a backstrap loom and applied in parallel with red interspaces (resulting in a sleeveless tunic). It represents an array of asymmetrical quadrangles which contain a stepped key design with differently colored tips. It is a four-color design with a sophisticated color sequence without a dichroic precursor. The illustrated color sequence seems to be quite standardized on

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Fig. 5.8: A multicolored Pre-Columbian textile from the Andes, South America. The zoomorphic periodicity and the color periodicity diverge. A detailed treatment is in the text. Museo de las Americas, Madrid.

Fig. 5.9: A sleeveless Huari tunic from the Andean region of South America. Details of symmetry and color scheme are in the text. Museo de las Americas.

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Fig. 5.10: Excavations of Pompeii, Italy.A color modulated scale pattern cm converted into pb (4) 1 by a four-color diagonal modulation wave. Museum in Napoli.

Fig. 5.11: Excavations of Pompeii, Italy. Color modulated dichroic pattern cm′ changed into oblique four-colored pb (2) 1. Museum in Napoli.

a number of cloth pieces (e.g., consult Cáceres Macedo 2005 and Gheller Doig 2005). In a zigzag design each key-and-background color pair continues diagonally, passing symmetry-wise through a series of twofold axes which alternate with local glide steps (Fig. 5.9). Two isochromatic sequences are oriented toward the upper left-hand corner, the other two to the right-hand corner. They cross in the twofold axes. The vertical and the horizontal white lines are dichroic reflection planes and those of their intersections that are not occupied by twofold rotation axes are occupied by dichroic rotation axes. The color group is (pa(2) m(2) m(2) 2)(4) . The zigzag sequence of quadrangles was intended to be symmetrical but, probably for technical reasons of the weaving process, it comes as asymmetrical in most cases, from gently curved field boundaries and somewhat uneven areas seen in our example, to the cases of all quadrangles on one side of all adjacent strips reduced to slim wedges and their motifs extremely reduced but always present. This common phenomenon, which might have become a fashion, we can call “Huari asymmetry”. Other “simple” color sequences are the Roman wall mosaics from Pompeii, and the Renaissance copies inspired by them, shown in Figs. 5.10 and 5.11. The cm pattern of “overlapping” scales (Fig. 5.10) has been colored diagonally by a four-colored sequence, resulting in a pb (4) 1 pattern. A special case is a scale pattern in Fig. 5.11, originally dichroic cm′, recolored to pb′ 1 but preserving vestiges of the original m′.

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Fig. 5.12: Marble pavement from the Duomo of Siena, Italy. Trichroic symmetry, color group pa (3) g(2,1) m; details in the text.

Fig. 5.13: A “chessboard” pattern from the Duomo of Siena, Italy. Color group pa (3) g(2,1) g2(2,1) ; compare the action of g(2,1) , in Fig. 5.12 with that in Fig. 5.13.

The cathedral of Siena (Italy) was begun in the late twelfth century but finished only in the fourteenth century. However, the superb floor paving was made only in the fifteenth to sixteenth century. The large scale pgg pavements are three-colored, in white, black, and rich brown marble. Figure 5.12 shows a simple three-color sequence of colored zigzags, color group pa(3) g (2,1) m, whereas Fig. 5.13 has a three-colored pgg-based “chessboard” of three colors, pa (3) g (2,1) g2(2,1) . A three-colored sequence of tiles which all have one orientation, combined with g along that direction, and g (2,1) perpendicular to this direction produce this chessboard.

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We can ask what did the mosaic artisans of the Alhambra do? Answering this question we have to differentiate between simple patterns, patterns of intermediate complexity, and large-scale patterns with a number of different tile types. Small-scale simple patterns and some patterns with intermediate complexity in the Alhambra display a high percentage (50%) of white tiles whereas the nonwhite tiles show a scheme of 3–4 colors, in equal or in different proportions. This feature intrigued several mathematicians, e.g. Grünbaum et al. (1986) and Ruiz-Garrido and Pérez-Gómez (1995). The above explanation (sub (a)–(c)), which apparently is in keeping with the original methods, was offered by Makovicky (1986) and further developed by Makovicky and Fenoll (1999). After a dichroic first step, the “white” subset was left intact whereas the “black” subset underwent further color modifications. In the color-modulated “pajarito” pattern of threefold propellers (Fig. 5.4) (uncolored plane group p3), the “pajaritos” with nested white hexagons (not a true dichroism in this case!) were colored by a four-

Fig. 5.14: A triple-period color sequence based on coloring of the black subset of original p4gm tiling which was converted into dichroic p4′g′m. Patio de Mexuar, the Alhambra.

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color modulation wave with a wavelength λ and the wave vector [uv] = [24] in terms of the trigonal asubmesh parameter. A tetrachroic unit cell can be defined as {a, 4a} pa(4) 1, not only as p1. A separate feature of this mosaic is the horizontal rows of differently colored stars which form the centers of white “pajaritos”. These stars alone form a p6mm submotif. A three-color wave for these stars, with a color sequence “greenhoney yellow-red”, produces a colored cell ca (3) m(2,1) m (Fig. 5.4). Combinations of a “major” motif composed of large tiles with a “minor” motif produced by small tiles in interspaces of the large ones and with a different chromaticity and periodicity is typical for the Alhambra. The visually conspicuous mosaics on the walls of Mexuar (uncolored motif p4gm; probably created or repositioned after post-reconquest building modifications) show four-color and three-color variants for the modulation waves applied to one-step dichroic “black” tiles (50% tiles are a white subset) (Figs. 5.2 and 5.3). On most panels, however, the distribution of colored tiles on “black sites” is disordered. A more complicated color sequence “oscillating around black nodes” was used in the Patio de Mexuar (Fig. 5.14). As already mentioned in sub(3), in the sequentially dichroic patterns, dichroic symmetry has been applied again and again, but only to the black subset of the previous stage. The sequence of applied dichroic groups is the function of a pattern. Thus, the much discussed pattern of “huesos” in the Tower of Comares (Fig. 5.7) has a sequence p4gm → p4′g′m → pC ′mm → c′mm (pa′ mm would be another possibility to choose in the last stage) (Figs. 5.15a–d). Several patterns of intermediate complexitywere mostly colored in a similar way to the simple patterns. The very worndown threshold between the Sala de Dos Hermanas and Mirador de Lindaraja is covered by a three-tile mosaic of “propellers” (p4gm) which was first colored in a dichroic manner (pC ′4) but subsequently its white tiles were modified by a three-colored modulation wave with a wave front (100) (Fig. 5.16). An inconspicuous wall pattern (Fig. 5.17) with uncolored symmetry p4mm was colored by means of a green-yellow-blue color wave with a wave vector [010] and λ = 3/2b. Each color is a strip only ½b broad. Surprisingly (or not), the largest, colorful complex patterns in the Alhambra are dichroic. The p4mm pattern of 16- and 8-fold rosettes in the complex of the Patio de los Leones has the 8-fold rosettes two-colored in a pC ′4gm scheme (Fig. 4.9). The green and yellow rosettes are on reflection planes whereas the 16-fold rosettes are on fourfold axes – both in special positions. In the museum, the same p4mm pattern is colored in a dichroic group pC ′4mm, with the 16-fold rosettes in two colors and 8-fold rosettes which are “gray” in special positions. Interestingly, coloring “decomposition” of these rosettes (complex (disjoint) tiles) into differently colored rays was not attempted. Sala de los Reyes has a complex p6mm pattern with 12-fold rosettes which was first colored by a black and blue trellis as cmm, and then it was converted into pC ′mm. It is described among the dichroic patterns (Fig. 4.10). Other examples of beautifully

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colorful large-scale patterns are uncolored; a prime example is the p422 interlace pattern, colored as p4mm on the rear wall of the Mexuar, apparently with a long history of mounting and dismounting (Fig. 6.19). An especially complicated case is a floor pavement from antique Villa Adriana, Tivoli, Central Italy (Fig. 5.18). It is a marble floor (uncolored plane group p6mm) with a detailed tetrachroic mosaic (pb (2) m(2) m2(2) )(4) . When the a axis points downward, the reflection planes m(2) , parallel to (100), turn white tesserae into black ones, and purecolored tesserae into pastel-colored ones. The (010) m, perpendicular to the b axis, is positioned on the colored tiles. The interstitial (010) m(2) is between these mirror planes and between the tiles of any color shade, so that it is positioned on the black and white tile rows. Tiles in these rows are “gray” for the (010) mirror planes but they are not gray for the (100) system of mirror planes. It means that the tiles which are gray for one system of reflections are not gray for the perpendicular system of reflections. It

b

o a1

a a2

b

2' a

2'

m' 2'

m m' m

m' m

m'

Fig. 5.15: The process of sequential dichroic coloring for the “hueso” pattern in the Sala de Trono, Torre de Comares. Starting in upper left corner: (a) first-stage dichroic p4′g′m pattern; (b) secondstage dichroic pC ′mm pattern obtained by coloring the blue tile subset in (a); (c) third stage dichroic c′mm pattern based on the blue subset in (b); (d) the resulting sequentially dichroic “hueso” pattern with the elements of the overall dichroic plane group c′mm indicated.

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Fig. 5.16: A very worndown threshold between the Sala de Dos Hermanas and Mirador de Lindaraja. A three-tile mosaic of “propellers” (p4gm) was first colored in a dichroic manner (pC ′4) but subsequently its white tiles were modified by a three-colored modulation wave with a wave front (100).

Fig. 5.17: A wall pattern from the Alhambra with uncolored symmetry p4mm. It was colored by means of a green-yellow-blue color wave with a wave vector [010] and λ = 3/2b. Therefore, each color is a strip only ½b broad.

is a tetrachroic group which contains only dichroic individual operations. The mosaic from Pompeii (Fig. 5.19) is simpler, pb (4) 1 whereas the other Italian example (Fig. 5.20) is a sequentially dichroic pattern.

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Fig. 5.18: An especially complicated case of color symmetry is a floor pavement from antique Villa Adriana, Tivoli, Central Italy, with uncolored plane group p6mm resulting in a tetrachroic group (pb (2)m(2) m2(2) )(4) .

Fig. 5.19: Mosaic of colored glass tesserae from Pompeii. Tetrachroic coloring of a scaly cm pattern as pb (4) 1. These mosaics are not based on sequential application of dichroism.

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Fig. 5.20: A marble floor pattern from the Baptistery of San Giovanni in Firenze, Italy. A sequentially dichroic coloring pa′ gm in the first stage, then the black subset dichroically colored to black-and-red in the same dichroic group with one translation doubled (“white” became the “gray” element at this stage).

6 Beyond 2D groups: Hypersymmetry, superstructures, two symmetries in one pattern, the “order-disorder” patterns, homothety and similarity, inversion and nonlinear patterns This chapter deals with topics which complicate a relatively straightforward picture of different types of symmetry groups which we meet in ornamental art. On the one hand, it is an array of phenomena which require a description/explanation other than a classical symmetry group description. On the other hand, it is symmetries other than the shape and size preserving congruence. They will be dealt with one by one here.

6.1 Hypersymmetry When an element of the ornament has its own (local) symmetry higher than the symmetry determined by the symmetry group for the site in which it resides (the so-called Wyckoff position), it is hypersymmetrical with respect to the “prescribed” local sym-

Fig. 6.1: Tympanum of the Nadir Divanbegi Khanaka (seventeenth century) in central Bukhara. A p6mm pattern with hypersymmetric occupation of sixfold sites by 12-fold rosettes and of threefold sites by 9-fold rosettes. The pattern is worked out in the pentagon-lozenge-rosette Kond style, which is discussed in Chapter 9.

6.1 Hypersymmetry |

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Fig. 6.2: A wooden house door in central Bukhara with 12-fold and 4-fold design elements (dodecagons and ditetragons); plane group of symmetry p4mm. Old reparations of desiccation cracks.

metry. This does not exist in crystal structures, in which it will be – even if slightly but definitely – distorted from the ideal shape (e.g., the so-called “coordination octahedra” in general position). In the artistic one- or two-dimensional designs, however, the ideal shape, and the excess local symmetry it generates, is preserved for both constructional and aesthetic reasons. Examples of such hypersymmetry are countless, ranging from four- or six-petalled flowers on twofold rotation axes to regular octagons centered on fourfold axes, wreaths of regular pentagons or of 5-fold stars arranged around N-fold rosettes, etc. In many Islamic mosaics, hypersymmetry is the principal ornamentist’s tool, the site symmetry of the plane group being too low and too repetitious for an impressive design. For example, combinations of two kinds of hypersymmetric elements, such as 12-fold and 9-fold rosettes, are common in two-dimensional Islamic patterns (Fig. 6.1); implications of such interesting combinations are mentioned in the final discussion. In some cases, regular arrays of hypersymmetrical elements may form boundaries on which they generate ambiguity in the direction of pattern propagation, causing either intergrowths of two different structures or generation of so-called order-disorder structures which will be dealt with below. If we take the tetragonal pattern of 12-fold rosettes which we have seen at several localities in Uzbekistan (Fig. 6.2), only four out of twelve rays of the 12-fold rosette will be in a special position (for example, on a mirror plane set) of this pattern. However, six rays of the same rosette can be in a special position in a hexagonal pattern which has these rosettes on sixfold rotation axes. When the intervening design allows it, this can lead to a seamless transition from the

150 | 6 Categories beyond 2D groups p4m

p6m

Fig. 6.3: A seamless transition from a p4mm pattern from Fig. 6.2 into a p6mm pattern with the same 12-fold rosettes.

tetragonal to a hexagonal pattern, as in Fig. 6.3. Both patterns are included as separate patterns in Bourgoin (1973). This seamless transition did not escape attention of the old artists and was used, e.g., in the mosaics of the Friday Mosque in Isfahan.

6.2 Superstructures and designs with several levels A large-scale structure with spacings which are multiples of translation periods of an underlying periodic small-scale structure (motif) is an m × n superstructure of this motif. Although the occurrence of such combinations is not frequent, a nice example is given here from the Friday Mosque in Isfahan (Fig. 6.4). The substructure is a tetragonal motif with four light blue flowers forming squares around fourfold rotation axes in the corners of a unit cell. The other set of fourfold axes is in poised squares of blue flowers (visible under the large fourfold star) which are nested in octagons of such flowers (Fig. 6.4). The superstructure is a periodic array of protruding fourfold stars, solid octagons and paired pentagons with a mesh equal to 2 × 2 periods of the submotif cell. In this case, both have the plane group p4mm. This ornament style is well established in Isfahan, for example in the Masjide-Hakim and Medresa Chahar Bagh (the latter illustrated in Castéra 2016). Another example, based on a hexagonal pattern, comes from Abdul Aziz Khan Madrassah in

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151

Fig. 6.4: A 2 × 2 superstructure created by disposition of large elements on an underlying p4mm pattern of flowers placed in Salomon stars. Courtyard of the Friday Mosque, Isfahan, Iran.

central Bukhara, Uzbekistan (Fig. 6.5). Not all two-level patterns made in style close to these have an immutable substructure. In some cases with a fourfold deltoid “kitelike” superstructure, the substructure was substantially varied in order to accommodate the geometry of the large kites of deltoid shape.

6.3 Two or more symmetries in one pattern When observing the high-quality stucco patterns on the walls of the Alhambra (Spain), on many of them we can separate a principal motif (expressed most strongly as a primary formline) and one or two levels of finer motifs, interlaced with or overlapping upon this primary motif as more complicated, ornamentally knotted or interlaced secondary (and tertiary) forms (Fig. 6.6).

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Fig. 6.5: A coarse superstructure of large hexagons and triangles anchored on sixfold rotation axes of an underlying p6mm pattern which has threefold rotation axes surrounded by hypersymmetrical ninefold rings of small blue tiles. The substrate can be interpreted as a star-shaped agglomeration of partly coalesced ninefold rings around sixfold symmetry sites. Abdul Aziz Khan Madrassah, Bukhara, Uzbekistan.

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Fig. 6.6: Plasterwork from the Alhambra, Granada, Spain, with three levels of design. Primary formline forms a strong “gable-like” grid, secondary formline is a thin finely lobed and interlaced cable, partly submerged under the primary formline. The scallop-holding arms form a tertiary formline.

Taken separately, the plane-group symmetry, or more often the two-sided layer-group symmetry of each of these formlines may be different from those of the other formlines in the same pattern …. The choice of individual pattern symmetries is “endless” although the treatment appears more standardized. For example, in the lateral spaces of the Salon de Comares, a pattern with prominent dividers (Fig. 6.7) consists of a primary (framing) formline with a layer group p2an (note the orientations of pattern nodes). The second-order intertwined pattern has a quite different scheme, c211, whereas the subdued tertiary pattern of leaves is cm. The combination p2an – cm is also present in Fig. 6.8. Most stucco patterns in the Alhambra have a primary formline which outlines strongly the individual partitions of the pattern, with symmetry cm. The incised pattern in Fig. 6.9 has the secondary formline again cm, whereas all of the plasterwork in

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Fig. 6.7: A plaster pattern from the complex of Comares, the Alhambra. Primary formline (“a frame”) is p2an, the secondary formline is c211, and the subdued leaves are cm.

Fig. 6.8: Although different from Fig. 6.7, this pattern has the same symmetries, p2an and cm, for the lobed and arched formlines. However, the latter is reduced to pm by alternating infill of the panels under arches.

Fig. 6.9: An incised stucco pattern in the Alhambra, with a cm – cm combination of the lobed and arched formlines.

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Figs. 6.6, 6.10, and 6.11 has the secondary formline in c211 (minus the inscribed texts, which try to approximate this symmetry but obviously have to break it in agreement with the character of Arabic writing). Figure 6.10 has a prominent tertiary motif (as buds and inscriptions in separate columns); when the text asymmetry is ignored, its symmetry is pm; Fig. 6.6 has a tertiary motif cm. Two different plane and/or layer groups employed in the same pattern are very transparent in selected examples of the marble pavement from the Baptistery of San Giovanni in Firenze. Figure 6.12 represents a particularly complicated case which combines a two-sided layer symmetry and dichroic symmetry. The background is a chessboard pC ′4mm whereas the layer group of all twisting cables, put together and disregarding the color, is p222. The dichroic plane group of “tops” of the twisting ca-

Fig. 6.10: A stucco pattern from the area of the Oratorio de Partal, the Alhambra, Granada. Enhanced shading allows individualization of all elements. For the primary, secondary and tertiary forms, the symmetry groups cm, c211, and pm were used (when the asymmetry of writing is ignored).

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Fig. 6.11: Stucco from the Sala de Comares, the Alhambra, Granada. For the primary to tertiary forms, the symmetry groups cm, c211, and pm were used.

bles is pb′ 1; diagonal to a rectangular mesh. A combined dichroic-and-layer group will preserve the c222 framework in which uncolored – dichroic – uncolored – dichroic twofold rotation axes alternate along both arrays of horizontal axes, respectively parallel to the a and b axis. A similar alternation is present for the vertical axes, perpendicular to the pattern plane. Simpler but effective is the pattern in Fig. 6.13, in which the background pattern of circles has symmetry p4mm whereas the pattern of interlaced circles is p422; they are not unifiable without a loss of properties (Makovicky 2015a). An apparently postconquest cmm floor mosaic of decagonal rosettes (Fig. 6.14) was restored from a fragment preserved in the Museum of the Alhambra. A 4-coloured modulation wave that colors the petals of decagonal rosettes is perpendicular to the dichroic wave valid for rosette cores. The former can be described without loss of symmetry information as two dichroic lattices shifted against one another by ½ of the diagonal of the uncolored cmm cell, each of them being pb′ mm. The latter is simpler: pa′ mm with black rosettes on m′ planes (gray positions). As another example we can take the subrecent bark paintings of Australian Aborigines from Arnhem Land with diamond grid (plane group cmm) as an ornamental background. The grid is divided into two subsets of alternating tiles which are colored in different ways (e.g., flat color versus colored cross-hatching); in most combinations the dichroic or polychromatic symmetries of the subsets are independent from one another (Makovicky 2003).

6.4 Twinning

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Fig. 6.12: A marble floor mosaic from the Baptistery of San Giovanni in Firenze, Italy. A pattern with two motifs with different, incompatible symmetries. Details in the text.

A lone example of such a combination might be dismissed and described by the symmetry elements common to both (or more) submotifs or subunits but when they represent a systematic and repeated canon of a certain culture or period, they cannot be ignored without consequences for understanding its ornamental arts and their rules.

6.4 Twinning Two copies of a periodic ornamental pattern can be cut and joined on a rational plane through their underlying periodical lattice in such a way that they are not a simple translation of one into another but they are related by reflection, glide reflection, or rotation, which is not present in the original symmetry group of the pattern. By analogy to natural crystals we shall call this a process of twinning. The two adjacent patterns in a twin relationship are equally inclined to the composition line, but from opposite sides. Pattern elements present on the boundary are cut and their portions respectively present in the two copies are joined into new elements in agreement with the above

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Fig. 6.13: Another example of a composite floor pattern from the Baptistery of San Giovanni, Firenze, with two different tetragonal symmetries, p4mm and p422.

listed twin operations. Twinning has been used to enliven panels of otherwise rather simple patterns both by reorientation of their portions (Fig. 6.15) and by introduction of ornamental boundaries inside the ornament field. Especially it has been used as a nonplanar twinning of a planar pattern for construction of N-sided ornamental domes or ceilings (e.g., in the Hispano-Islamic and Mudejar architecture of Spain and Central Asian Islamic vaults). The three-dimensional 8-fold twin of the periodic c222 pattern of decagonal rosettes in Fig. 6.16 comes from the Reales Alcazares in Sevilla and is a smaller edition of a similar ceiling now installed in Villandry, Loire, France (Makovicky et al. 1998). Accidentally or not, the same fragments of a decagonal pattern and the same twinning mechanism were used for a small dome in a Bukhara madrassah. Twinning possibilities of a decagonal pattern were analyzed by Makovicky et al. (1998). The famous dome “La Media Naranja” in the throne room of the Reales Alcazares in Fig. 6.17 (constructed by Diego Roiz in 1427) is composed of a twelvefold rosette at the zenith and 12 radial strips of the decagonal cmm pattern, which partially overlap at the top and become separated by ladder-like interspaces at the bottom; it is not a twin construction. Octagonal twinning of another orthorhombic pattern is shown in Fig. 6.18.

6.4 Twinning

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Fig. 6.14: A reconstructed cmm floor pattern of decagonal rosettes, with four colors for the rosette rays and two colors for their cores. The complex color symmetry is described in the text.

Fig. 6.15: Twinning of stripe-colored motif in a corner of an oblong field of the originally monochromatic p4gm pattern. The Alhambra, Granada. A twin composition plane acts on the color scheme but not on the underlying pattern which is continuous.

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Fig. 6.16: A dome ceiling from the Reales Alcazares, Sevilla, Spain. A three-dimensional 8-fold twin of a periodic c222 pattern of decagonal rosettes.

Fig. 6.17: The “La Media Naranja” dome from the Reales Alcazares composed of 12 radial strips of the decagonal cmm pattern with overlaps in the upper portions and interspace-filling in the lower parts of the dome.

6.5 The “order-disorder” patterns |

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Fig. 6.18: Twinning of an orthorhombic cmm pattern on (110). Vault of the vestibule of the Abdullah Khan Madrassah, Bukhara, Uzbekistan.

6.5 The “order-disorder” patterns Every periodic pattern can be cut up into a set of parallel strips of desired width. For the majority of patterns, the position and orientation of the adjacent strip is unambiguously determined by the continuation of the motif across the strip boundary. There exist patterns, however, with continuation ambiguity in the position and/or orientation of the adjacent strip, when their boundaries have been properly selected. This excludes the patterns in which disorder is introduced just by substituting some elements by other compatible elements. Such a substitution derivative from the hall of Mexuar, the Alhambra, Granada is shown in Fig. 6.19. The OD (order-disorder) theory of patterns with stacking ambiguities discerns intra-strip and inter-strip operations of symmetry. The former are one-dimensional frieze groups or two-sided frieze groups of the strip itself, whereas the latter can be twofold glide-rotation axes and glide planes in the strip boundary or one-step symmetry operations with a gliding component, which are perpendicular to the strips and recast the nth strip into the (n + 1)st strip of the pattern. We shall not try to enumerate the possible (and mutually compatible) combinations of inter- and intra-strip symmetries. The OD concept can be best understood analyzing an example in Fig. 6.20. If the strip symmetry is the frieze group p2mm with the translation period t and a perpendicular mirror plane running through the origin,

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Fig. 6.19: A p422 pattern with uniform p4mm coloring and with substitutions of selected large 16fold rosettes by 822 elements. Sala de Mexuar, the Alhambra, Granada. A case of substitutional disorder.

(a)

(b)

Fig. 6.20: A schematic OD structure with p2mm strips and interlayer shift equal to a fraction of strip periodicity. (a): All interstrip shifts from the nth to the (n + 1)st strip have the same sense; (b): interstrip shifts with alternating shift direction. There are two equivalent positional choices for the (n + 1)st strip.

6.5 The “order-disorder” patterns

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and the adjacent strip is displaced parallel to the infinite strip direction by a translation period equal to a general fraction of t (1/2 t is excluded), this displacement can happen in the + a and −a directions with equal probability. Addition of strips results either in an oblique +++++ (or −−) sequence of them, with the twofold axes of the frieze group acting as generalized symmetry operations or, alternatively, in a rectangular sequence +−+−+− in which the reflection planes of the frieze group parallel to the strip become universal operations together with the glide planes perpendicular to them.

Fig. 6.21: An interlaced zigzag pattern, with a trichroic layer group pa (3) 2. A wall pattern in the Reales Alcazares in Sevilla, Spain with alternating strips of “single-flame” and “double-flame” tiles. For double-flame columns, coloring introduces up-up, down-down, and down-up positions of equally colored double-flame neighbors, which form a basis for potential OD phenomena. The pattern is described in this chapter as a potential order-disorder pattern on the colored level.

164 | 6 Categories beyond 2D groups

The wall mosaic from Sevilla (Fig. 6.21) is an example where double-sided layer symmetry of the strip is lower, p121 (or p11m, when the interlacing is ignored), and there are alternating vertical strips of single and double “flames”. Stacking changes occur on double flames, so that we shall consider the single flame as subsidiary – the single flames always have one double flame neighbor shifted half-a-period “up” (U) and one, just on the opposite side of the “single-flame” row, shifted “down” (D). A given “double–flame” row can have both adjacent “double–flame” rows shifted “up” by one uncolored period (1/3 of the three-color period), both “down” by the same shift, or one neighbor “up” and one “down”. The artist preferred a regular sequence UU – DU – DD – UD – UU – (repeat) but a sequence UU – DD – UU – DD – …is equally possible (markings are for double-flame rows only) as also are any more complicated or disordered sequences. The shift of adjacent “double–flame” rows is along an in-plane screw axis (it is a “quantum corkscrew” with parallel shifts by discrete intervals) parallel to and situated between the double-flame strips, with a one-step operation + 1/3 of the three-color period or −1/3 of the three-color period (or such a glide-reflection plane when interlacing is ignored). Is there an order-disorder design in which such phenomena occur for an array of blocks instead of strips? The Shipibo Indians living in Peru might have created a unique, purely geometric ornamental style which corresponds to such a criterion. It

Fig. 6.22: A ceramic pot and a painted table cloth made by Shipibo Indians in Peru (Makovicky 2011a). A pronounced set of primary lines (bold) is accompanied by thinner parallel secondary lines and the compartments are filled by sinuous tertiary lines; the latter are the topic of the orderdisorder principle discussed in the text and in Fig. 6.24.

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165

b

a

Fig. 6.23: A detail of the Shipibo pattern, painted Fig. 6.24: A theoretical reconstruction of the on a table cloth. Shipibo tertiary design by means of randomly oriented Truchet tiles with two corners marked. A tile has symmetry mm2 but on its boundaries the curves are perpendicular to the boundary so that the curvature in the adjacent tile can freely face either to the right or to the left, without regard to its orientation in the first tile.

is a tertiary design which fills the ornamentally shaped compartments determined by the bold lines of the primary large-scale design, and by the accompanying secondary thinner lines (Figs. 6.22 and 6.23). This sinuous design varies freely within simple rules that can be understood as a variation in the orientation of underlying square tiles with two vertices marked by quarter-circles (Fig. 6.24), known under the name of “Truchet tiles” in a form derived by Smith and Boucher (1987). Such an underlying pattern, however, is invisible in Shipibo creations. All lines are correlated pairwise so that they create “bead-like” interspaces; there are no individualized line pairs with a special role in the tertiary design. Changes of direction appear arbitrary (by a 90°, rotation of the imaginary “square tile”) and any periodicity that can be found in the design is often only a short-range periodicity. If they do not form closed loops, the lines almost always disappear into the frame of secondary lines. Simple variations of this pattern have plane groups of symmetry pgm and two distinct cases of cmm. With two kinds of twin formation, when the orientation of the sinusoids is altered, they account for the entire spectrum of pattern varieties observed in Shipibo art. No firm evidence on the origin of these patterns exists. The Shipibo Indians are Panoan speakers residing on the middle Ucayali River in the low jungle environment of the Peruvian Montaña (Roe 1980, Jørgensen 1992). Archeological research ascribes them a residence period of about 1,000 years in this region (quoted in Roe 1980) but

166 | 6 Categories beyond 2D groups

the painting technique probably is only about 130 years old. The Shipibo-Conibo style is a complex geometrical style applied equally to pottery, wood carving and textiles. It is fairly conservative. A more detailed analysis of the geometric Shipibo art, based on the analysis of typical examples of pottery and painted textiles, is in Makovicky (2011a). Methods of their preparation were concisely described by Jørgensen (1992).

6.6 Homologous series It is not unusual to find a series of patterns with the same plane or layer group at the same locality or region; they often reveal common construction features and kinship beyond the simple symmetry criterion. Closer analysis can show that on the one hand they contain a series of the same elements regularly distributed over the pattern or forming distinct pattern zones, which are identical in all members of the pattern family, and on the other hand they contain common elements forming blocks or zones of regular structure, which occur in different amounts in different patterns, resulting in different thickness of such zones. The multiplicity of the latter elements across the diameter of these zones determines the homologue order N of the pattern in question. In simple cases it does not alter the pattern symmetry. As an example, I here illustrate three patterns from the Mosque of Cordoba (Spain) where the elements with increasing multiplicity are the bricks in the frames around swastikas and squares, which occur in regular increments from one to three (Fig. 2.65). These accretional homologous series are the simplest examples of textural analysis of pattern families. In other cases, patterns have been modified by local changes against a more or less copied example, or by insertion of frills, of complete elements, or of entire strips (zones) of elements not present in the parent version (intercalation series). Elements can also be substituted, omitted or reduced in size (or expanded), etc.; in general, a spectrum of these changes reflects all aspects of human creativity and it happens nearly in all cases when complicated older patterns are copied and applied by the followers. The interested reader can seek more details in Makovicky (1989). An interesting problem is typified by the tower of the cathedral in Siena, Italy (Fig. 6.25) where the multiplicity of narrow window openings increases upwards in a very spectacular way. This is not a homologous series. Perhaps, it can be defined as a multiplicative catamorphy, a term used by Weyl (1952) for similar phenomena.

6.7 Inversion in a circle The circles and cusps of the ornamental tracery in the windows of Gothic cathedrals are examples of ruler-and-compass constructions of the old stonemasons (Sutton 2007, 2009, Chamoso et al. 2009). However, Jablan (1995) demonstrated that they equally well are a result of inversion in a circle, which converts a circumscribed square

6.7 Inversion in a circle

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167

Fig. 6.25: Tower of Siena Cathedral, Siena, Italy.

into a fourfold pattern of overlapping circles, or of ornamental cusps surrounding an empty space. In a similar way, inversion transforms a circumscribed triangle into analogous threefold configurations (Fig. 6.26). If the circle with the radius r is positioned on the origin O, in the process of inversion, the point A of the vector OA will be inverted into point A′ such that OA. OA′ = r2 . These results, however, can be obtained for each pair of points in a purely constructional way, based on two similar triangles based on the unit circle as described by Brannan et al. (2004). This might have been exactly the way the medieval masters operated. What I find especially intriguing is that oriented half-tangents to the unit circle are inverted into boundaries of “fish-bladders” (vesica) typical in Gothic architecture (Fig. 6.26). Two half-tangents placed on the opposing points of a circle generate a yin-yang scheme of two “bladders” (Fig. 6.27); three half-tangents at 120°, generate a three-bladder configuration or full tangents a trefoil (Fig. 6.28), whereas a four-full-tangent scheme generates a quatrefoil (Fig. 6.29).

168 | 6 Categories beyond 2D groups P r O

t A′

A

t t′

Fig. 6.26: Examples of inversion in a circle: the principle (top), action on half-tangents (center) and on full tangents (bottom). Redrawn from Jablan (1995).

Fig. 6.27: Gothic window tracery with two “fish” fields in each circular field. Red “Buntsandstein” sandstone. Frankfurt, Germany

6.8 Homothety and similarity | 169

Fig. 6.28: A window of a Gothic palace with trefoils. Firenze, Italy.

Fig. 6.29: Gothic windows and trellis with 3-, 4-, and 6-foils. Apsis of the Cathedral of Notre Dame, Paris.

6.8 Homothety and similarity When we relax the criterion of congruence by allowing different sizes of otherwise identical geometric figures, which originally represented a regular translational sequence of figures between two parallels, by altering them into a sequence of figures placed between two intersecting lines, these will increase in size and spacing according to the law A′B′ = k AB of homothety (Fig. 6.30). In art, this simple model is often complicated by varying the dilatation coefficient k in the equation and/or by simplifying the shape and contents of figures when they approach the lower limit of usable size. Similarity and its modified editions (which deviate in some way from the above equation) are often applied to the domes, floors and sometimes parietal discs of sometimes impressive size. In these cases, each step in the homothety operation is mostly connected with a partial rotation around the center of the disc (keystone of the dome) resulting in an alternating or spiral packing of ever increasing figures.

170 | 6 Categories beyond 2D groups

a

b

c

d

e

f

Fig. 6.30: Linear sequence of half-squares (Mezquita Alhama de Cordoba). Comparison of (a) congruence, (b) affinity, (c–d) “pseudohomothety” and (e,f) homothety applied to a half-square and half-lozenge, respectively.

As just said, the fundamental operation in similarity symmetry is homothety, which is a central dilatation K with a dilatation coefficient k. In this transformation, every line segment AB is expanded in the homothetic image of the original according to the above equation; this is valid for every consecutive step of this operation. This ensures equiformity and equiangularity of the images and parallelity of their orientation. There is a central, origin point to such a sequence and the above equation is valid also for spacing between consecutive images. A simple but powerful case of homothety is presented by the wall inscription in the Great Mosque of Bursa (Turkey). This large mosque was adorned by a series of calligraphic inscriptions (quotes from Quran) on all square pillars and much of the wall space in the nineteenth century. Large and crisp, black-on-white, with rare red additions, they changed the interior into a unique architectural experience (Chapter 1). In the rare homothetic construction, the powerful “wa” (“and”) repeatedly connects the words written in angular, fully vocalized Kufic script with minimal adornments (Fig. 6.31). Equivalent points in all wa letters are on straight lines originating from a point inside the smallest letter. Other similarity operations are dilative rotation L2n (k,π/n), which combines every dilatation step (coefficient k) with a partial rotation π/n given in the symbol, and dilative reflection M in which a dilatation step is connected with reflection (a superficial similarity to a glide-reflection plane). When we use Schoenfliess’ notation (Cn and Dn , symbols signify pure-rotation point groups N and rotation-reflection point groups Nmm, respectively) (Jablan 1995), the similarity symmetry groups of rosettes are CnK, Cn L, Cn M, Dn K, Dn L, and Dn M. The subscript n describes a multiplicity of the rotation

6.8 Homothety and similarity |

171

Fig. 6.31: Great Mosque of Bursa.

axis. Coincidences between resulting configurations eliminate DnM, leaving only DnL. The Dn L2n (k,π/n) case builds an alternating pattern; other cases produce wonderful logarithmic spirals which preserve equiangularity and equiformity of motifs. In practice, genuine similarity is often not maintained: the logarithmic spiral (Fig. 6.32) (difficult to construct by the artisans, as suggested by Jablan 1995) is replaced by an equidistant one, violating the constancy of the k coefficient, and the equiangularity and equiformity of the motifs (Figs. 6.33, 6.34).

172 | 6 Categories beyond 2D groups

Fig. 6.32: A spiral disc in the floor pavement of the Duomo di Firenze, combining homothety, central rotation axis (n = 12) and incremental rotations (π/12) so that the fields (note trompe l’oeil) preserve their shape along the dilation direction and alternate along the radius direction. A simple interlace was introduced for the white bands so that the radial mirror reflection planes which result from the above symmetry operations are transformed into horizontal, radial twofold axes.

Fig. 6.33: Circular pseudosimilarity with regular radius increments instead of homothety and rotation by a half of 1/16 of the circle. Rhodos, Castle Museum.

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Fig. 6.34: A pseudohomothetical floor design in Duomo di Firenze, Italy. Eightfold rotation axis, with three affinely related rays of rectangles as a rotated unit but no homothetic relationship along the rays. Note how the rectangles change side ratios along each ray, preserving the “thickness” when translated outwards.

Fig. 6.35: A floor pattern from Pompeii, Italy. A circular pattern with a 12-fold rotation axis combined with homothety and partial rotations by ½-period.

174 | 6 Categories beyond 2D groups

Figure 6.35 shows a Roman mosaic with 12-fold symmetry (except for the fourfold core of the design), 12mm or D12 and dilative rotation π/12. In terms of dilative reflection (see above), the M plane has to be placed asymmetrically over the black (or, alternatively white) elements, in an analogous way as a glide-reflection plane. The second alternative is probably how the ancient artists were designing such a pattern, using radial lines. A unique collection of forty Roman similarity rosettes is in Balmelle et al. (2002b). Their cluttered cores were usually avoided by placing instead a completely different motif in a circle. A revealing fact is that 29 of them have the Dn construction principle (before coloring), and only 11 are Cn in geometric design. Apparently, Dn is a construction preference. Out of forty, nine are Dn K and six Cn K. Dichroic rotation which enlivens the design is present in seven cases; 3-, 4- and hexachroic rotation operations are illustrated as single cases. Overall coloring is present in such a similarity disc either as “windswept” arches or as “petals” from counter-oriented arches. Ten cases of dichroic dilative reflection and a complicated case of hexachroic M operation are present in the body of examples. Two interesting examples of dichroic reflection on concentric circles, defined as C8 L and D10 K, were found. Two cases of C48 L which result in a set of eccentrically anchored straight “pseudo-radial” boundary lines between tesserae were constructed. Preferred multiplicities of rotation axes are n = 16, 24, 64, a bit less frequent are 32, 8, 10, 48; the rest are infrequent 18, 28, 40, 44, 56, 60, 80, 84, and a monumental 130.

6.9 Chirality, enantiomorphs, hands Two (or more) objects related by a rotation operation are identical both in shape, size and orientation in space (the latter is denoted as their “hand”, by analogy to the right and left hand of Homo sapiens). They are easily transformed into one another by this rotation operation. However, objects related by reflection symmetry can be identical in shape and size but they display irreducible difference in orientation (“hand”) of their subunits in the cases when they do not have reflection or inversion symmetry themselves. Just like our hands, they come in two editions, “right” and “left” enantiomorphs, and translations of each of them alone can produce right and left copies of a two-dimensional pattern. We speak of chirality (enantiomorphism). Combination of both enantiomorphs in one pattern gives a “racemic” design, different from the design produced using one enantiomorph alone. An interesting ordered combination of such a design is offered by the Iznik tiles in Fig. 6.36. Adjacent tiles are enantiomorphs and if only one subset were used, a nice p4 pattern would result, in a right- or left-handed version, according to which tiles are selected. The racemic combination, however, results in p4gm with unit cell covering the whole tile of one enantiomorph and side-oriented quarters of four surrounding tiles of the other enantiomorph. The unit cell of the pattern composed of one enantiomorph alone is small; there are four of them in each tile.

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175

Fig. 6.36: A tetragonal Iznik tiling from the Rüstem Pasha Camii in Istanbul, composed of two enantiomorphous versions of the same p4 pattern with tiles alternating in a chessboard manner. Plane group p4gm with a large unit cell oriented diagonally to the ceramic tiles. Mirror planes of the plane group coincide with tile edges, glide reflection planes run diagonally through quarter-edges of the tiles. The fourfold rotation axes of the p4 submotif situated on the # configurations are out of action in the p4gm pattern and only the fourfold axis in the center of ceramic tile remains as such in the new group.

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6.10 The nonlinear art of Hans Hinterreiter Hans Hinterreiter (1902–1989) and M.C. Escher (1898–1972) are two geometric artists from the recent past who contributed substantially to art based on a tiling of plane and symmetry. Both discovered and systematized color symmetry groups in their own ways before the crystallographers did. There exists a plentiful literature on Escher and on his drawings animated by fanciful creatures (e.g., Schattschneider 2004; see

Fig. 6.37: Hans Hinterreiter (redrawn). Original symmetry pb′ 2.

Fig. 6.38: Hans Hinterreiter: Opus E45. Original symmetry is pb′ 2 mapped on a curvilinear net.

Fig. 6.39: Hans Hinterreiter: Opus E73. A p4 motif mapped onto a curvilinear net.

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177

Fig. 6.40: Façade of a house in central Madrid, Spain. A p4′ copy of Escher’s Metamorphosis III, made in 1967/68, including a termination by the gray figures on the top. Created from a pC′ 4mm chess-board at the bottom of the facade. It illustrates a change of the (dichroic) plane group via a less definite transitional pattern in the embryonal stage of lizards.

Fig. 6.40) whereas that on the strictly geometric art of Hinterreiter is much smaller (e.g., Makovicky 1979, 2011b; Albrecht and Koella 1982). In order to conform to the expectations of (his) modern times, Hinterreiter mapped his two-dimensionally periodic designs onto affine, projective and variously curvilinear nets, introduced singularity points and/or elaborate twinning schemes. In this way, and with the help of color symmetry, he produced highly dynamic representations. His working methods are illustrated by Figs. 6.37, 6.38 and 6.39. What resulted from his efforts, which predate the computer age, are the most detailed and elaborate pictures of constructivist art – unfortunately without known followers and with only a handful of admirers. Concerning computer art – sorry, I talk about old masters here (Fig. 6.41)

178 | 6 Categories beyond 2D groups

Fig. 6.41: Combination of techniques, materials and symmetries. Médersa Bou Inania, Meknés, Morocco

7 Quasiperiodic patterns Plane groups of symmetry lead to parallelograms, rectangles, squares, triangles and hexagons as the plane-filling motifs. Pentagons, octagons and dodecagons are missing from this list, together with heptagons, etc. In 1974 Penrose demonstrated that an agglomerate of pentagons and intervening 36° lozenges, the latter isolated or forming clusters of 3 or 5 as half-stars and stars, will yield aperiodic tiling when furnished with appropriate edge-matching markings. A subsequent version of Penrose tiling had only two tile types, which are called “darts” and “kites” according to their shape, both with a 72° vertex. Another version of Penrose tiling with only two prototiles is composed of two Penrose rhombs, with 36° and 72° vertices, respectively. Both versions are furnished with appropriate edge- and vertex-matching conditions, which are not usually reproduced in popular articles, and all three versions are easily accessible in literature or on internet. In 1977, Ammann discovered octagonal aperiodic tilings. Ammann’s A5 set is a set of 45° lozenges and squares with vertex and edge markings, in order to enforce aperiodicity. Mathematical details of the octagonal tiling and of the dodecagonal tiling, which was recently discovered in Fez, Morocco (Makovicky & Makovicky 2011), were worked out especially by Socolar (1989). A very important property of these aperiodic tilings is self-similarity, which can also serve as a proof of aperiodicity. In the self-similarity operation, several tiles of the aperiodic pattern are combined into larger tiles with the same form and disposition as the set of original tiles. This can be repeated, obtaining larger and larger patterns, identical with the starting one. Periodicity of the plane patterns is conveniently described by a 2D lattice based on two divergent translation vectors. Ammann (1977) discovered that the decagonal and octagonal quasiperiodic tilings can be described by a quasilattice, which consists of a quasiperiodic sequence of two distinct intervals: a unit interval of the width equal to “1”, and another interval with the width specific for the symmetry of the tiling (symmetry of an aperiodic tiling is the point symmetry of its “cartwheel” edition). “The other interval” is equal to (1 + √5)/2, also known as τ (tau) for the decagonal Penrose tiling and it is √2 for the octagonal tiling. Typical quasiperiodic sequences are, e.g., 1, √2, 1, √2, 1,1, √2, 1,√2, … and 1,τ, 1, τ,τ, 1, τ,1, τ, … (notice the “opposite” principles of “interval pairing” in these two sequences). The lines of quasilattice cut (or mark) each tile of a given type in exactly the same way. The decagonal quasilattice has five equivalent lattice directions and the octagonal one has four equivalent directions, unlike the two (resp. three) equivalent directions of the tetragonal and hexagonal lattices. Besides this primary “Ammann quasilattice”, a denser secondary Ammann quasilattice can be drawn, the lines of which cut each tile type in a limited number of allowed ways. In the octagonal tiling, they run at 22.5° to the primary ones. After this minimalistic general introduction we can search for historic examples. The Alhambra, the Royal Alcazar of Seville and some Moroccan madrasas tell us that

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“rediscovered” and “defined in modern mathematical sense”, is a better definition for the achievements of the twentieth century in quasiperiodicity. Still, when comparing the concepts of the fourteenth and twentieth centuries, we must keep in mind that the Hispano-Islamic artisans were artists, not theoretical scientists. They concentrated on the aesthetic aspect of the mosaic pattern so that their creations lie between rather strict, mathematically “rather clean” patterns on the one hand, and heavily modified patterns with fanciful but beautiful modifications on the other hand. The HispanoIslamic artists, however, were preceded by the Seljuk artisans of eastern Iran, who constructed in the town of Maragha the first known cartwheel aperiodic decagonal pattern already in the years 1196–1197, based on principles different from those of fourteenth century Andalusia. The Maragha story has been told by Makovicky (1992), dealt with again by Lu and Steinhardt (2007) and recounted by Makovicky (2007, 2008) based on new analysis. In current geometry treatises, a start from a core with a decagon- or octagon-like outline and expansion of the pattern as a so-called “cartwheel pavement” became popular means of illustration for quasiperiodic tilings. Let us recall the first Penrose tiling, which was an assembly of edge-sharing pentagons with interspaces ranging from a simple 36° lozenge through a “half-star” to a fivefold star. In 1992 Makovicky demonstrated that the 800-year-old large-scale tiling (Fig. 7.1) on the tomb tower at Maragha, NW Iran, is a decagonal cartwheel tiling with a quasiperiodic character, the tiles of which can be derived in a very simple way from the first variety of Penrose tiling. One has to inscribe a pentagon in each of Penrose’s pentagons and accept the resulting modification of Penrose’s lozenges-to-stars into “butterflies” and 72° lozenges (Fig. 7.2). Distribution of the two latter elements over the pattern, and distribution of pentagon- and decagon-shaped aggregations of these three tile types (composite polygons in Fig. 7.3) reflects the unusual freedom with which a given tiling can be modified by means of tile flipping and partial rotations of tile aggregates. This is an inherent property of the decagonal quasiperiodic tiling. The old masters knew it and, as shown in Fig. 7.3, they replaced an aggregate pentagon, which originally was filled asymmetrically by two lozenges, three pentagons and a butterfly (they can still be traced in the pattern), by an aggregate pentagon with a simplified, rotation-symmetrical fivefold infill, as a token average of five possible orientations of the asymmetric fill. The same holds about a 10-fold star outline with asymmetric contents, which they filled with a lovely symmetrical fivefold group (Fig. 7.3) (Makovicky 1992 and 2008, Lu & Steinhardt 2007). For periodic patterns of this tiling type this was illustrated by Castéra (2016). Al Ajlouni’s (2012) theoretical derivation models incorporate some of these variations. The Maragha pattern (AD 1196–1197) has been paralleled (probably directly copied, although with small creative modifications, which was usual in “inherited” Islamic patterns) at several localities of the Islamic world (Makovicky 2015b). I shall illustrate the tympanum of the Darb-e-Imam shrine in Isfahan (AD 1453) which contains decagonal cartwheel patterns of this type composed of small ceramic tesserae

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181

Fig. 7.1: Gunbad-e-Qabud (Blue Tower), a tomb tower from 1196–1197 in Maragha, NW Iran (from Makovicky 2008). Its walls contain the first known decagonal quasiperiodic tiling (large “tiles” on the walls).

(Figs. 7.4 and 7.5). Masjed-e-Jameh (The Friday Mosque) in Isfahan contains the same motifs in different settings (Fig. 7.6). In both, the tenfold star elements filled by tiles (Fig. 7.3) were replaced by void (white/yellow painted) tenfold stars. Karatay Madrasa in Konya (Turkey; AD 1251–1252; Rigby 2005) has a barrel vaulted iwan covered by a periodic cmm pattern composed of rectangular fields specified in Fig. 7.7; each of these rectangular fields corresponds to a single wall of the Maragha tomb tower. Among profoundly modified patterns is the remarkable tympanum pattern from Masjid-iHakim in Isfahan (AD 1656–1662), in which the pentagon and butterfly elements have

182 | 7 Quasiperiodic patterns

– – 0 0 – – 0 0 0 – 0 – 1 1 – 11 0 22 0 – 2 – 1 – 0 – 1 0 – 0 – 0

– 1 – 0

1 2 – 1 2– – – 2 00 0 1 – 0 0 0 – – 1 – – 1 0 11 1 – 1 0 0 – – 22 2 2 0 – 2 – 1 0 1 – – – 0 0 1 – 1 0 – 0 1 0 – – – 0 –1 0 1 1 0 – – 1 2 0

– 1 – 0

0

1

– 2 2 2– 2 1 0

– 2

0

1 1 – 2

0

Fig. 7.2: A Maragha-type tiling on a background of the original Penrose tiling (upper right-hand corner) and with Penrose’s aperiodicity markings transcribed onto Maragha tiles. Figure from Makovicky (1992, 2008).

Fig. 7.3: Fundamental tiles of the Maragha-style tiling: small pentagons, butterflies and decorated lozenges. Light gray: derived composite tiles used at Maragha and afterwards: asymmetrically filled large pentagons and 10-fold stars and their artistically symmetricized versions with fivefold rotational symmetry.

7 Quasiperiodic patterns |

183

Fig. 7.4: Tympanum of the Darb-e-Imam shrine in Isfahan, Iran. A large-scale skeletal pattern (blue lines), with plane group cmm is filled by Maragha-style tiles, with the small ornamental 10-fold stars emptied and colored yellow for ornamental purposes. The quasiperiodic pattern fills the large 10fold stars of the cmm pattern and reaches only to the centers of pentagons which surround them.

Fig. 7.5: Analysis of the mosaic-laying procedures by mapping out small-scale details of the pattern in Fig. 7.4. All decorated lozenges have been colored red. The principal cartwheel-like quasiperiodic area at the bottom appears flawless while its peripheral parts, the remaining quasiperiodic bits, and the periodic positions display tile-laying variations in keeping with the freedom allowed by the decagonal tiling.

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Fig. 7.6: A detail of the Maragha-style tiling in the Masjed-e-Jameh (Friday Mosque) in Isfahan, Iran. A part of the large tenfold star is at the right-hand edge of the figure.

been modified by combining their basic shape with multiple ‘Ali tiles overwritten on them (Fig. 7.8). The vertical and horizontal axes of the Maragha pattern are exchanged in this pattern and the more distal portions are modified as well. Details of these complicated arrangements are in Makovicky (2015b). Decagonal patterns (i.e. patterns with 10-fold elements, especially rosettes, and 5-fold elements, usually stars) are frequent in Islamic art and sometimes the quasiperiodic and the strictly periodic cases were confused in literature. Especially the basic cmm pattern with 10-fold rosettes in the lattice nodes and the edges in the ratio 1 : 1.376, and with numerous small 5-fold stars surrounding and separating the

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185

Fig. 7.7: A glazed mosaic of the barrel vault in the iwan of the Karatay Madrassa in Konya, Turkey. A periodic pattern composed of aperiodic rectangles, each comprising 1/2a × 1/2b of the centered cmm pattern seen in the figure. A complex decagon surrounding the origin of the Maragha pattern was replaced by a rosette. In spite of the early age (AD 1251–1252) the blue glaze is of astounding quality.

Fig. 7.8: A Maragha-style quasiperiodic pattern with tiles “overwritten” by a twinned kūf¯ı ‘Ali calligraphy. Masjid-e-Hakim, Isfahan, Iran (1656–1662). A detailed analysis of this process is in Makovicky (2015b).

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rosettes, is common all over the Islamic world, and often modified into a c222 interlaced version (Figs. 3.1 and 3.2). We find it in the Alhambra, Granada, in the Reales Alcazares in Sevilla (Fig. 3.3), Cairo (Fig. 24 in Makovicky et al. 1998), Samarkand etc. It is also a pattern from which panels of Mudejar artesonado ceilings in various Spanish churches and monasteries were cut out and joined in 3D constructions (Fig. 6.16) (more in Makovicky et al. 1998). When prominent, the unit and τ bars are periodically repeating in these patterns. The most powerful example, however, is a linear cut-out in the court of the Friday Mosque in Yazd (Fig. 7.9), where this pattern was combined with a twinned Kufic “Ali” tiles, similar to Masjid-i-Hakim in Isfahan (Fig. 7.8). In this operation, its symmetry has been reduced to c1. The same principle was used in a “blind window” panel in the Masjid-e-Imam (Isfahan) where the Kufic elements are three-colored (with a rather erratic color distribution) (Fig. 7.10). Inspection reveals that there are two different patterns “intergrown” in this panel, a bottom one and a top one with different periodicities. None of these is a quasiperiodic pattern, however. As already mentioned in the chapter on plane groups, there is a number of periodic patterns composed of Maragha or Kond style tiles. The extreme is a fundamentally chaotic tiling of pentagons and butterflies seen on some tympana, with only locally ordered orthorhombic portions.

Fig. 7.9: A frieze cut-out from a decagonal cmm pattern, altered into c1 by introduction of the same twinned Kufic ‘Ali calligraphy as in Fig. 7.8. This outstanding pattern lines a courtyard of the Friday Mosque in Yazd.

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Fig. 7.10: A blind window panel in the entrance portions of the Masjid-e-Imam in Isfahan, Iran, with a combination of multicolored calligraphic elements (twinned Kufic ‘Ali name) in an intergrowth of two related patterns.

In the fourteenth century Islamic architecture of Andalusia (the Alhambra in Granada, Reales Alcazares in Sevilla) and contemporaneous madrassas in Fes, Morocco, decagonal and octagonal quasiperiodic panels were constructed in which quasilattices of the type described above are the leading construction and aesthetic principle. The boundaries of intervals have mostly been enhanced by thin white lines whereas the panels were often colored in agreement with the point group symmetry of the cartwheel pattern (Fig. 7.11). Decagonal panels are adorned with tenfold rosettes,

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Fig. 7.11: A central portion of an octagonal quasiperiodic cartwheel panel illustrating the tiles in the unit and √2 strips of the quasilattice and the white ceramic lines which separate them. The alicatado tiles and the white ceramic rods were clipped out of ceramic plates instead of being fired individually.

fitting well into the quasilattice. The same holds for the octagonal patterns in which, however, the outline of the octagonal rosettes has been modified to fit the quasilattice. Two interlaced decagonal quasiperiodic cartwheel panels are exhibited in the Museum of the Alhambra, under nos. 4584 and 4589. They were partly reconstructed (with variable success) from large fragments excavated from the refuse at the Plaza de los Alquibes, adjacent to the Mexuar. Allegedly they come from the Sala de las Aleyas, destroyed when the palace of Carlos V was built. Both have sufficiently large original fragments allowing reconstruction of a cartwheel pattern. Number 4584 is larger (Figs. 7.12 and 7.13) and allows investigation of quasiperiodicity. Close analogies of these are found in Morocco (Madrasa Al Attarin (1323–1325) in Fez (Fig. 7.14) and as a smaller pattern, Madrassa Al Buinanyiah, 1350–1357 in Fez) in a much better state of preservation (Makovicky et al. 1998). Copies of these patterns continued to be constructed until modern times and they are illustrated, e.g., in Castéra (1996) as “l’étoile à 10 en zellij” (zellij are Moroccan ceramic tiles used in mosaics). These cartwheel patterns, as a whole, have point group symmetry 10mm, altered by interlacing into a two-sided layer symmetry without mirror planes. They contain two rings of 10-fold rosettes whereas the 5-fold rosettes based on 5-fold stars are imperfect in shape. Five systems of white dividers, orientated 72° apart, outline and inter-

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Fig. 7.12: A decagonal quasiperiodic mosaic panel in the Museum of the Alhambra, reconstructed from the material preserved in the refuse from destruction of the Alhambra, probably accumulated by the architects of Carlos V. Dark portions are original.

connect the rosettes. In any one of these identical systems of lines, there is a quasiperiodic sequence of spacings, 1 and τ, denoted as short and long, respectively, which reads as follows: S(central), L, L, S, L, S, L, S, L, L, S, L, when counted on a line starting from the center of the pattern. The same is observed in the best Moroccan pattern quoted above (Fig. 7.14). The center of the cartwheel in the Alhambra is masked by a large 20fold rosette, absent in the Moroccan counterpart, in which the interior portions of the quasiperiodic pattern are fully developed. The quasiperiodic sequence of Ammann bars denoted by the thin white dividers shows clearly that it is a quasiperiodic pattern as far as the construction reaches. The ultimate proof of quasiperiodicity is that the bars (between white lines) correspond to lattice rows of a weighted reciprocal lattice of a real decagonal quasicrystal (Janssen et al. 2007; their Fig. 2.5); prominent reflections of this reciprocal lattice correspond to the most conspicuous elements of the Alhambra pattern (the rosettes) and the entire geometry of these two patterns is identical. It should be noted, however, that the outer portions and corners of the Alhambra panels were, in part, wrongly reconstructed from the loose material. For the short and long intervals of a given bar

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Fig. 7.13: Reconstruction of the decagonal quasiperiodic mosaic panel in the Museum of the Alhambra. Note five orientations of bars of unit and τ width (for definition see text), local exchanges of the unit strip – τ strip and τ – unit strip sequences and the overall quasiperiodic sequence of unit and τ intervals.

sequence, written as S and L, the Ammann bar sequence allows easy phason flips of a type S, L, S, L, L, S → S, L, L, S, L, S, i.e. occasional exchange of the S-L pair by an L-S pair along the length of the bars. We can see that this was sometimes done by the artisans who introduced phason flipping along the bar length when it was necessary to position a desired ornamental element (e.g. a star) (Fig. 7.13). Again, analogies occur in the reciprocal lattice of a real quasicrystal. These observations lead us to suggest that the quasilattice was the primary construction element of these mosaics although it is not known how they arrived at this principle. When you look for other affinities, consult Makovicky et al. (1998). A very fine octagonal quasiperiodic pattern (Figs. 7.15 and 7.16) is situated in the Mirador de Lindaraja of the Alhambra (Makovicky & Fenoll 1996). This mosaic, with minute tile size and a fine spectrum of colors selected for it, adorns side walls of the broad entrance to the throne room of Muhammad V. Hardly noticed by ordinary tourists, who are always commanded to look at the last stained glass preserved in the Alhambra, and are seduced by the splendid plaster ornaments adorning the head of the room, it represents the culmination of ornamental art in this palace. The oblong panel contains a central cartwheel pattern which is partly copied in the corners (Fig. 7.15). The dense Ammann quasilattice delineated by the white ceramic spacers

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Fig. 7.14: A decagonal quasiperiodic mosaic panel in the Madrasa Al-Attarin in Fez, Morocco (1323– 1325). Upper half of a twinned panel of two cartwheel presentations. There are slight modifications at some points of the upper, semicircular boundary.

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Fig. 7.15: A panel of octagonal quasiperiodic pattern with similar configurations planted in the corners of the rectangle. The finest mosaic of the entire Alhambra is situated in side panels of the Mirador de Lindaraja, adjoint to the Patio de los Leones. The alicatado tiles and the resulting unitwidth and √2-broad bars are separated by white line spacers. Details of pattern anatomy are in the text.

hides the contacts of these five separate areas; many intervals continue from one area into another. Octagonal 8mm point group coloring accentuates the cartwheel character of each of these areas. The dados of the Patio de las Doncellas in the Reales Alcazares in Seville are covered by numerous larger and square-shaped panels of octagonal quasiperiodic patterns. In each of them only the central cartwheel pattern is present (Figs. 7.17 and 7.18). They display colorful variants of the octagonal 8mm point group coloring. The octago-

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nal quasilattice has four fully equivalent directions, 45° apart. As in all quasiperiodic Hispano-Islamic and Moroccan patterns, the origin of the cartwheel pattern is in a narrow unit interval and not on a dividing line. There are a number of phason flips present, used by the artisans to accommodate some of the ornamental stars. The radial sequence of intervals, starting in the origin, is S, L, S, (S, L), S, (L, S), (L, S), (L, S), S, L, S in the Alhambra panel. The through-going intervals are outside the round brackets in this sequence, whereas the pairs in which phason flipping of a type LS ↔ SL takes place somewhere along their length are enclosed inside the brackets (Fig. 7.15). The same bar sequence and those with small variations, e.g., S, L, S, (S, L), (S, S, L) are at the Alcazar. The quasilattice of the pattern from the Mirador de Lindaraja has a geometric counterpart in a diffraction pattern of an octagonal quasicrystal. The best fit was found with the weighted reciprocal lattice of Mn4 (Si,Al) published by Kuo (1990) (Makovicky & Fenoll 1996).

Fig. 7.16: The interlace scheme for unit-width and √2-broad bars in the octagonal quasiperiodic panel from Mirador de Lindaraja, the Alhambra.

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Fig. 7.17: An octagonal quasiperiodic mosaic panel from the Patio de las Doncellas, Reales Alcazares, Sevilla, Spain. One of many quasiperiodic panels on the walls of this patio. The quasiperiodic sequence of unit and √2 bars and their flips are described in the text.

Fig. 7.18: Another octagonal quasiperiodic mosaic panel from the Patio de las Doncellas. A different point-group coloring from Fig. 7.17.

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In their entirety the octagonal patterns show variations in interval sequences but always those on x and y lines are identical to those on the two “diagonals” at 45°. There are several panels present, however, in which the diagonal sequences differ because the panel has been purposely altered to a “tetragonal” one (Fig. 7.19). Details are to be found in Makovicky and Fenoll (1996), Makovicky et al. (1998), Makovicky (2011b), among others. Drawings of some Patio panels were published by González-Ramírez (1995).

Fig. 7.19: A tetragonal pattern in the Patio de las Doncellas, Reales Alcazares, Sevilla obtained by a change of S and L sequence on diagonals, which became different from the sequence on the axes.

A possible objection by mathematicians who state that quasiperiodicity (aperiodicity) of a pattern can be proven only when it is examined to infinity can be answered as follows: a real-world definition (Makovicky 2007, Saltzman 2008) is that a pattern is considered quasiperiodic when, for a sufficiently large patch, all its local isomorphies coincide with those of a strictly derived quasiperiodic pattern. Accidentally, the extent of the quasiperiodic Alhambra patches and that of the quasiperiodic patches usually illustrated in the books on geometry, practically coincide. What happens if we intro-

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Fig. 7.20: A square mosaic from Torre de la Cautiva, the Alhambra, Granada. In this member of the Fig. 7.20–Fig. 7.21 pair, the differences between the unit and √2 bars are preserved.

Fig. 7.21: A square mosaic from Torre de la Cautiva, the Alhambra, Granada. In this member of the Fig. 7.20–Fig. 7.21 pair, the differences between some sets of unit and √2 bars were eliminated by the process of “averaging” described in the text.

duce a periodic sequence SLSLSL …? For two such sequences at 90°, to one another, the diagonal sequence is SSSLSLSSSLSL (Makovicky & Fenoll 1996). The quasiperiodic decagonal pattern and a quasiperiodic octagonal pattern in the Alhambra were produced in the fourteenth century, presumably during the reign of Muhammad V. The madrasas in Morocco were constructed in the same century. The sublime and engaging beauty of the “mathematical” quasiperiodic patterns was not enough to stop the artisans from further speculations. The Hall of Ambassadors (Salon de Comares) in the Tower of Comares and Torre de la Cautiva, both in the Alhambra, host outstanding examples of their efforts. The original dating of both buildings is under Yusuf I (before 1354) but we know now that Muhammad V refurbished the Tower of Comares when rebuilding Patio de los Arrayanes. The panels to be described might belong to his period. Exact line drawings of some of these panels are in Pavón Maldonado (1989). Dados of the Salon de Comares contain six varieties of these brightly colored panels. The S/L sequences of these patterns are variable; they vary in length and are cut from different portions of a theoretical quasiperiodic sequence. Phason flipping is very frequent; it may even become mosaic-like. Local SSS triplets, violating the quasiperiodicity principle, may be present. Another observed variation requires a different understanding of the quasiperiodic octagonal pattern than we used before: any two systems of Ammann bars which are perpendicular to one another carve out “L-squares”, i.e., (1 + √2) ×(1 + √ 2) square

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units containing √2 × √2 squares in one corner and an L-shaped area of unit width along two opposite edges of the L-square. This concept is close to the original A4 tiling of Ammann (Grünbaum & Shephard 1987). The L-squares either assemble edge to edge in such a way that unit strips are formed, interspersed by √2 strips, or there is one more unit strip inserted between their rows. In some panels selected L-squares are “symmetrized” by their division into four equal (but pairwise colored) segments, each with a width of (√2 + 1)/2. In the Torre de la Cautiva we find, side by side, a panel with well-developed L-squares and an almost identical panel with practically all L-squares “symmetrized” (into prominent brown-and-blue “crown” motifs) (Figs. 7.20 and 7.21). The large panels close to the entrance into the Salon de Comares (Fig. 7.22), called “trompe-l’oeil” squares by Makovicky and Fenoll (1996) display eight dark-colored 8fold rosettes surrounding a central one. What is fascinating is that these rosettes are the intersecting portions of dark strips with a quasiperiodic spacing of 1 and √2 on a large scale, and at 22.5° to the underlying small-scale quasilattice. Geometric relationships suggest that the dark strips and the white small-scale quasilattice are in the relationship of primary and secondary quasilattices, perhaps separated by one inflation stage on the former lattice. A more detailed discussion of this problem is in Makovicky and Fenoll (1996).

Fig. 7.22: A large “trompe-l’oeil” panel in the Salon de Comares, the Alhambra. Dark strips obey the rules of octagonal quasilattice and are a secondary lattice to the small-scale quasilattice created by tesserae.

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Fig. 7.23: A “radiant disc” from the Salon de Comares, the Alhambra, with the scale of the central disc different from that of the square background. Heavy modifications of a quasiperiodic pattern.

Another, very effective modification is the introduction of a central disc with spacing different from the bulk of the panel. The best example, a “radiant disc” pattern from the Salon de Comares (Fig. 7.23) has a central rosette again composed of intervals with the ratio 1 : √2. They are on a smaller scale than the rest of the pattern. Contrary to the expected simple situation corresponding to the nearest deflation, which would yield 1/√2 and 1 (= √2/√2), the inner disc is enlarged about 1.2 times in order to produce pleasingly narrow boundaries, close to (2 − √2) in width. A system of narrower stripes at the above deflation rate is used in another pattern which has a complicated 8-fold star enclosed in two orange squares whereas the cases with central rosettes rotated by 22.5° operate with unchanged interval widths. Adjustments along the margins and in the corners of square panels occur in the majority of these panels. They vary from insignificant to substantial. Aesthetically these panels enliven the vast throne room of Comares, with less light today than originally, because the building stability problems after the explosion of the gunpowder factory in the valley below required the walling up of some windows. While enjoying their bright color combinations, try to analyze the underlying quasicrystalline background and then the modifications introduced by the artisans.

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Fig. 7.24: A fountain mosaic from a street in Fez, Morocco, with 8mm symmetry although some parts resemble/approximate a 16-fold rotation axis. A typical example of a large family of such fountains.

Fig. 7.25: A small but famous street fountain of Nejjarin in central Fez. Outstanding 8mm mosaic work is accompanied by equally outstanding plasterwork.

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Fig. 7.26: An undated side altar in the Grand Mosque of Cordoba with a quasiperiodic pattern very similar to that in the Alhambra but without white linear spacers. The width of unit and √2 bars is determined by parallel and diagonal orientation, respectively, of elongated hexagonal tiles. Stylewise akin to the Moroccan mosaics.

Fig. 7.27: A large wall panel in the mausoleum of Moulay Ismail in Meknes, Morocco, with a transition from the small-tile quasiperiodic arrangement (mainly left side) to the 8mm arrangement of ornamental “flowers” and “suns” (right side).

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Fig. 7.28: A dodecagonal quasiperiodic cartwheel pattern in the gate construction of the Zaouïa Moulay Idris II complex in Fez.

The octagonal panels of Moroccan madrasas, souks and fountains have a somewhat altered character; they were studied particularly by Castéra (1996). The most frequent type of small tiles has a shape of elongate hexagons, and halved hexagons; small rosettes of fitting size should be mentioned as well. Larger, composite ornamental elements host a number of further tile types – darts, rosettes, half-rosettes, pencils, etc. The majority of panels have large elements in an 8mm arrangement (Figs. 7.24 and 7.25), in which the large 8-, 16-, and 24-fold rosettes visually predominate even

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if their boundaries are a matter of geometric compromise. The other extreme, seen in the pure form on the church side altar in the Grand Mosque of Cordoba, is an octagonal quasiperiodic pattern with the thin white spacers omitted and the thin, unit strips (Fig. 7.26) formed by length-wise arranged small tiles whereas the √2 strips are composed of diagonally oriented elongated-hexagon tiles, interspersed with variously oriented darts and small tiles. This system works perfectly, and the panel from Cordoba (Fig. 7.26) copies perfectly the octagonal quasiperiodic panels with white spacers (Makovicky & Fenoll 1996). The system of small tiles is reduced in panels with the above mentioned large elements, arranged as 8mm, to mere boundaries of the latter. Nevertheless, they present magnificent examples, such as the fountain in Fig. 7.24 and the Fountain of Nejjarin in Fez (Fig. 7.25). A rare example of a very instructive smooth transition between the two principles is a panel from the mausoleum of Moulay Ismail in Meknes (Fig. 7.27). The left-hand side has a traditional small-tile pattern with small red octagonal rosettes fitted around a central rosette, whereas the right-hand side is an agglomerate of “suns”, octagons and Salomon stars defined by thin tile boundaries. The old, finely structured dodecagonal quasiperiodic panel from the Zaouïa Moulay Idriss II in Fez (Fig. 7.28) is a lonely example of its category, without firm dating because of many rebuildings of the object. Again, Makovicky and Makovicky (2011) found a corresponding electron diffraction pattern in literature, which confirms its quasiperiodicity. In the recent reconstruction of the Zaouïa, this panel situated high above a narrow alley was painted screaming red.

8 Fractals and fractal character A fractal is a geometrical figure in which an identical motif repeats itself on an ever diminishing scale, i.e., with a built-in self-similarity (Lauwerier 1991). Some fractals have strict self-similarity, the same motif exactly persisting on every scale. Other fractals are more “realistic” (both in reality and in art) in the sense that the self-similarity of their form is statistical, and is statistical at every level of the fractal scale. In addition, they may display affine transformations and more complex distortions. This leads to realistic models for trees, corals, etc., and should be the way to evaluate fractals in art.

Fig. 8.1: A Cosmatesque floor mosaic composed of red Imperial Porphyry and green Greek Porphyry in the Church Santa Maria in Cosmedin, Rome, Italy. Plane group pm modified to dichroic pa′ m; pattern with a fractal character.

Two simple exactly self-similar fractal models (produced by deterministic algorithms) are directly applicable to art: Cantor’s fractal and Sierpinski triangle. The Cantor set is produced when a line segment 0–1 is divided into three equal portions and the 1/3–2/3 portion is omitted. The 0–1/3 and 2/3–1 portions are again divided into three equal portions, the central ones are omitted, the remaining ones again divided, and so on …until unequally distributed “Cantor dust” is produced in infinity. The existing segments can be volumes in the direction perpendicular to the Cantor’s line. The Sierpinski triangle

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Fig. 8.2: A dichroic Cosmatesque floor mosaic in the Church Santa Maria in Cosmedin, Rome, Italy. Plane group cm modified to dichroic pa′ 1; a pattern with a fractal character.

Fig. 8.3: A hexagonal 6mm configuration with a partial fractal character. Sierpinski triangles are dichroic, point group 6′mm′. A Cosmatesque floor mosaic in the Church Santa Maria in Trastevere, Rome, Italy.

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Fig. 8.4: A hexagonal 6mm configuration with a fractal filling of interspaces in form of “hyperbolic triangles” and of the circular perimeter. Four fractal stages; nos. 3 and 4 are partly truncated by the inflated petals. A Cosmatesque floor mosaic in the Church Santa Maria in Trastevere, Rome, Italy.

is produced by dividing a triangle into four equal triangles sharing vertices in the midpoint of triangle sides. Then, the central triangle of this second generation remains void whereas the other three triangles are divided up according to the same scheme, their central triangles remain void while the three “outer” ones are divided again, etc., to infinity. Sierpinski triangles of the Cosmatic mosaics can be equilateral, in the illustrated example (Fig. 8.1) arranged in parallel rows instead of a hexagonal array (as in the ideal Sierpinski case) but they also can be affinely distorted (Pajares-Ayuela 2002), as the right-angle version of tesserae in Fig. 8.2, or can be embedded in a hexagon (Fig. 8.3) and finally, and visually very effective, in a flaring (and not strictly triangular and trigonal from the point of symmetry) white field of a hexagonal point-group pattern (Fig. 8.4). These illustrations show that dichroic modification of the fractal pattern is common. Insertion of few tesserae of this kind in sectioned squares of tetragonal patterns is frequent. It makes the patterns more sparkling in the dim church light. A circular application of equilateral triangles is shown in Fig. 8.5. A small templette, adjoint to the famous temple of Isis on the island of Philae in the River Nile (all saved from the waters of the Aswan Dam by relocation), has capitals of lotus leaves which fairly strictly follow the Cantor fractal principle in four orders. They are hidden under the four uppermost “cover leaves” of the capital (Figs. 8.6 and

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Fig. 8.5: A fractal fill of the circular perimeter of a Cosmatesque floor pattern. From Cosmatesque floor mosaics in the Church Santa Maria in Trastevere, Rome, Italy.

Fig. 8.6: An upward view of the capital at the Philae templette, illustrating its Cantor-type fractal character.

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Fig. 8.7: Overall view of an ornamental column and capital from the templette next to the temple of Isis, the Island of Philae on the river Nile, Egypt. The capital has a fractal structure.

8.7). The visible damage is probably due to natural weathering of the most exposed elements of this almost mathematical design. It is a late Ptolemaic construction and the same kind of capitals, although more damaged, appear in the temple at Kom Ombo (80–50 BC). Quasiperiodic patterns sensu stricto, have a fractal character as well, repeating themselves on ever-larger scales, as illustrated here by Fig. 8.8 in which several inflation and deflation stages of the Maragha pattern are illustrated (from Makovicky 2008).

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Fig. 8.8: Inflation and deflation stages of the quasiperiodic Seljuk pattern from the Blue tomb tower of Maragha (Iran).The original pattern is a thin line drawing which is colored only in the upper ring that surrounds the deflated patterns (Makovicky 2008).

An artistically important body of traditional Chinese ornamental art preserved in the utilitarian buildings of higher classes displays a clear fractal character, which is more obvious in the case of “flat-surface” patterns (illustrated here by ornamental panels in Figs. 8.9–8.11). The large, primarily wooden houseblock from where these examples originate is known under the name “Jiu Tou Ma”, and is located at Qi Yang village, He Shang town, Changle city, Fujian province, China. The fractal character is more subdued or sometimes problematic for the lattice patterns composed of line segments (Fig. 8.12). Under “lattice” we understand here the traditional Chinese window lattice (e.g., Dye 1974) and not the crystallographic lattice. These are the patterns which display certain repetitiveness of the motifs, but on different scales and maximally with a point group symmetry which itself does not describe sufficiently the complete nature of the pattern. To the best of our knowledge this topic has not yet been treated in this way. In a good approximation, the fractal art of the Cosmati and that seen at Philae in Egypt can be described as a (fairly) strict application of a straightforward deflation principle (by a deterministic algorithm), once the starting shape has been established, whereas the Chinese art also employs random/statistical iteration, affine

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Fig. 8.9: A wooden-and-stucco panel in a traditional wooden building complex “Jiu Tou Ma”, located at Qi Yang village, He Shang town, Changle city, Fujian province, China, with a clear fractal character.

transformations (described, e.g., by Barnsley 1988), etc., resulting in a much more variable result. The fascinating art of Chinese bronzes (Fong 1980, Zhou and Wang 1993) (Fig. 8.13) spans several centuries. Already in the Xia period (traditionally 2205–1766 BC) bronze vessels and other objects were adorned with very characteristic complicated scroll patterns which, when taken liberally (e.g., including affinity), display signs of fractal arrangement (Fig. 8.14a). Under the ensuing Shang dynasty (fifteenth to eleventh century BC; the traditional count, however, gives dates from 1766 to 1122 BC), this fractal-type style crystalized to complicated, densely ornamented panels on the walls of ritual vessels. The overall composition of the vessel is supposedly a face with round button eyes. The large, first-order features of these patterns, and the space between them, are covered with larger and smaller scrolls, the latter often branching from the former which can be elongated to accommodate more than one smaller branched scroll. Progressive miniaturization of the small scrolls, using a dense set of lines, turns the latter into a fine background of large, principal elements (Fig. 8.14b), separating these two patterns with different orders of magnitude from one another in the later art. The Western Zhou conquerors continued in this artistic tradition with a number of splendid vessels (1122–771 BC). After the movement of the capital away from the attacking nomads, the so-called Eastern Zhou period sees a change in the ornamental style (Fong 1980, Zhou and Wang 1993) which, when of fractal character, gravitates to the style we see on relatively recent artefacts (Fig. 8.14c). A rich field for a critical application of fractal geometry!

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Fig. 8.10: A wooden panel in a traditional wooden building complex “Jiu Tou Ma”. The flat panel displays a clear fractal character and occasional affine modification of the rectangular spirals.

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Fig. 8.11: A typical Chinese wooden openwork – an aperiodic pattern with only one vertical reflection plane. Application of fractal principle as a propagation of “L-motifs” on increasingly smaller scale. The houseblock “Jiu Tou Ma” at Qi Yang village.

Fig. 8.12: A typical Chinese lattice window, with the application of fractal principle. The houseblock “Jiu Tou Ma” at Qi Yang village.

Some of the Chinese plant scroll patterns can be described as fractals as well. The plant/dragon from an unknown locality suggests this relation (Fig. 8.16). This is true also for selected scroll ornaments of other nations (e.g., those I saw in the Islamic art in Turkey and Iran) when the scroll pattern decreases in size and branches repeatedly

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Fig. 8.13: A detail of the Chinese bronze vessel. The Art Institute of Chicago. Compare with Fig. 8.14a and 8.14b.

when progressing to its apex (Fig. 8.15). Complications of various kinds are usual in the process of pattern propagations. A pattern of waves and wave crests in a Chinese marble relief shows a deliberate fractal development (Fig. 8.17). Finally, selected examples of Celtic brooches (Müller 2009, Farley et al. 2015) appear to contain a fractal approach as well. A potential Art Deco candidate is illustrated by a screen in Fig. 8.18, with two or more orders of (in principle) similar elements, and mirror reflection in the central part balanced by dilative reflection in the lateral panel. Motif in Fig. 8.19 is a real surprise.

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(a)

(b)

(c) Fig. 8.14: Rubbings of outer surfaces of three typical ornamented ritual bronze vessels from Early Bronze Age China. (a) The early vessel with one order of magnitude for all lines and the most fractal character. (b) A later vessel with a “coarse” set of larger elements and their scroll systems separated by orders of magnitude from the dense set of fine scrolls in the “background”. Both vessels have eyes and the coarse set in (b) mimics animal features. (c) Ornamentation of a bronze vessel from the Han dynasty. Reproduced with permission from Zhou and Wang (1993).

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Fig. 8.15: A branched ornamental scroll. A wall mosaic from the Friday Mosque, Isfahan, Iran. Colors have been inverted on a computer in order to enhance the scroll structure.

Fig. 8.16: A combined Chinese plant–dragon scroll from an unknown architectural object.

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Fig. 8.17: A Chinese marble panel with a fractal development of a traditional Chinese wave pattern. Beijing, Forbidden City.

Fig. 8.18: An Art Deco screen with a general fractal character of ornamentation (with arrays of interconnected rectangles and squares of different sizes). Museum of Modern Arts, Paris.

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Fig. 8.19: An unnamed Uzbek lady’s way to fractals. A subrecent wall hanging at the Art Museum, Tashkent, Uzbekistan.

9 Style and symmetry – symmetry and style The great majority of art is produced in a certain style, determined by the time and location of its origin. This is doubly true for ornamental art. A consideration and explanation of material culture style has been the topic of dozens if not hundreds of studies, both applied to a given culture and about style in a general sense. Hegmon (1992) writes: “the notion of style is one of the undiscussed self-evident concepts upon which our historical consciousness is based”. In spite of it, a number of definitions of style have been proposed by archeologists and art theoreticians and “it sometimes appears that archeologists have as many approaches to style as we have works on the topic” (Hegmon 1992). A number of these are highly theoretical, nearly impossible to comprehend without the entire background theory. Our simple working definition is as follows: A style of material culture is a “vocabulary” of forms and objects used together with a “grammar” of their combination.

I use the word “vocabulary” instead of “inventory” and “grammar” instead of “rules” in order to stress the innermost human aspect of creative work by the artist. Our experience confirms that there are certain connections between style and technology, that style allows for individual variations, that there can be a category of emblemic style for ethnic groups and social casts, expressing social standing and/or belonging, that elements of style may have symbolic meanings, and also that style undergoes a process of standardization during the process of learning and mass production, as mentioned by Hegmon (1992) and others. None of these factors, however, is absolute or all-determining, and the schools of interpretation which absolutize one aspect, or a narrow choice of aspects, are simply wrong. We agree that certain aspects of style are pervasive, valid for all technologies and groups, whereas other aspects are partitive, limited to certain products and/or groups. An indelible part of style, rarely mentioned explicitly or defined exactly, is the symmetry of the patterns produced. For an overview of the plane groups of symmetry used in certain context, we constructed a half-circular plot (Fig. 9.1), based in an arbitrary logarithmic scale (in order to include the small contributions which would disappear in the linear scale). Plane groups are distributed according the unit rotation angles of their rotation axes – 0°, 60°, 90°, 120°, and 180°. The lowest groups always start first, at the inherent angle of rotation. Markings for quasicrystalline compositions have also been added at appropriate angles. If we want to compare symmetries of a collection of different ornaments from a particular locality, we have to convert all intertwined (interlaced) layer patterns, as well as dichroically and polychromatically colored patterns, defined and analyzed in previous chapters, into uncolored and noninterlaced plane patterns (Makovicky and Makovicky 1977). Colors of tiles must be mentally subtracted, so that only pure tile

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shapes remain (plane groups do not deal with nongeometric properties such as color). For an intertwined pattern this means that we “paint all the intertwining strands black” so that we cannot see what is above and what is below when the strands cross (as, for example, did Wade (1976)). Effectively we reduce the intertwined, two-sided patterns into a one-level plane pattern. Escher

Total number of classified patterns: 163 Fig. 9.1: A graph for plotting frequency of plane groups of symmetry and quasiperiodic patterns in a body of data collected from a given culture/object. Numbers are percentages; the sample illustrated is the distribution of plane group types in the material produced by M.C. Escher as collected by Schattschneider (2004).

As (perhaps) the most studied example, a statistical evaluation of 200 patterns collected and redrawn by Bourgoin (reprinted 1973) reflects the symmetry preferences and taboos of classical Islamic design. In full agreement with our own observations, his pattern collection shows that the bulk of Islamic patterns has been designed in p6mm and p4mm groups, when we forget interlacing (as Bourgoin did). The reflectionfree p6 group and the trigonal p31m group are minor contributors. The second most used tetragonal group, p4gm, has only one-sixth of the number of cases that we counted for p4mm. Among lower symmetries, the cmm symmetry leads, pmm occurs with 50% frequency compared to cmm, whereas pmg and p2 are minor contributors. Characteristically, plane groups without twofold rotations (cm, pm, pg, p1) and pgg are missing from Bourgoin’s list. Abbas and Salman (2007) published graphical statistics of their own for the preferred symmetry in Islamic cultures. Again, p6mm and p4mm (without color, interlacing and similar complications, as they state) occupy almost two-thirds of the patterns, cmm, pmm and p6 represent the bulk of the remaining patterns in a decreasing order,

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whereas p4, p31m, pm, p3m1 are rare. The remaining plane-group symmetries are almost negligible. When we look at early Islamic art, which heavily depended on brick designs, the statistics of plane groups for the art of the Grand Mosque of Cordoba and for the Seljuk tomb towers and minarets differ from each other but both are different from the above general statistics of Islamic art. On the walls of the Kharraqan towers (Figs. 2.10–2.15), the p4mm group is most frequent, p4gm less so, and they are followed by a much rarer p4 symmetry. The plane group cm is frequent, pm much less common, while cmm clearly prevails over p2, pgg and pmg. If interlacing is ignored, p6mm is important (introducing a complicated masonry problem, Figs. 3.24–3.27), less so p31m and p6. The Cordoba Grand Mosque (Figs. 2.59–2.68) differs from the Kharraqan and from the general statistics, especially in the lack of hexagonal and trigonal brick patterns. The plane group p4mm is most frequent but only by a narrow margin, because the plane group p4, closely followed by p4gm, is almost as frequent as the p4mm patterns. Half as frequent are cmm, pmm, pgm and p2; a rare case of pm was registered (Makovicky & Fenoll 1997a). Tetragonal patterns mostly show intermediate complexity, while the rest are a mixed group concerning their complexity. Only marble trellis windows of the Mosque display hexagonal character (Fig. 3.19). Among frieze patterns of the Mosque, those with pmm2 symmetry heavily predominate over a mixed group of frieze symmetries from which only p1 is missing. Ornamental art of central Seljuk Turkey, approximately from the years 1220–1280, has been chiefly sculpted in young basalt rocks occurring in this region. Its bulk are friezes of very variable width with geometry closely related to the two-dimensional panels of the Blue Tower of Maragha in NW Iran and the Karatay Madrasa in Konya, central Turkey (see Chapter 7, Figs. 7.1 and 7.7). They use a set of “Maragha-type tiles” of Makovicky (1992) (here sculpted as outlines in solid basalt) which in Persian are called “Kond tiles” (Mofid & Raeeszadeh 1995). This set has a small pentagon as a fundamental element, with a composite lozenge and a butterfly-shaped element as the other fundamental elements. The two derived elements, a filled larger pentagon (that contains two lozenges, one butterfly and three small pentagons) and a filled tenfold rosette (star), as well as their symmetrized versions (see Fig. 7.3), are the other principal elements of this style. The symmetrized larger pentagons contain one central small pentagon and five kite-like hexagons, and are rather conspicuous in a pattern when used. The 10-fold rosettes, however, are not so outstanding, and only the core pentagons of the symmetrized version, with small lozenges attached to the apices, catch the eye immediately (Fig. 7.3). A frequent type of narrow strip present on the Anatolian Seljuk buildings is composed of the latter ornamented pentagons alone; they are interlocked via their apical lozenges; the resulting rod group of such interlaced patterns is p222 (Fig. 9.2). Several more types of narrow strips are present, based on the basic tile types (Fig. 9.3). Broad strips (Fig. 9.4) are symmetrically bordered by alternation of the filled and symmetrized pentagons and ornamented symmetrized pentagons with attachments, situated in 10-fold stars (as illustrated in Fig. 7.3), whereas the central

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Fig. 9.2: Two types of sculpted friezes. Haci Kiliç Cami in Kayseri, Turkey (thirteenth century). Both contain selections of simple (left) or composite (right) tiles from Fig. 7.3, both obey two-sided ribbon groups p222.

strip is asymmetric with ornamented pentagons in alternation with vertically oriented lozenges. Thus, the p222 symmetry dictated by the two marginal strips is reduced to p211 for the total width, with the remaining twofold rotation axes perpendicular to the elongation. The interior of the central symmetrized rosettes can be reoriented freely by rotating them around a twofold axis which runs along the centerline of the total strip, through the intervening lozenges, and parallel to the length of the frieze. This reverts the orientation of the frieze but does not change the frieze itself. Besides the illustrated

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Fig. 9.3: Center and right: Friezes composed of basic Maragha/Kond tiles (pentagons, butterflies and composite lozenges), both in two-sided frieze group p221 2. Mahperi Hunat Hatun Complex, Kayseri, Turkey (about 1238).

frieze with the symmetrized side rosettes “pointing outwards”, a frieze is present at the same locality with the same configuration but with its rosettes pointing “inwards”. The extremely complicated friezes of this unified style appear unintelligible until they are deciphered and interpreted in the above Kond–Maragha sense. They are com-

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mon in, and typical for, a unified artistic region between (at least) Maragha to the east and Konya to the west. It was a definite style type in the midyears of the thirteenth century. However, interspersed with these examples, among the ornaments of Kayseri and beyond, are examples of another style, with small stars instead of the small pentagons (Tond and Shol tilings in Persian, Mofid & Raeeszadeh 1995, see Chapter 2) which apparently coexisted with the Kond-style tiles in the studied period, sometimes producing mixed panels. Should we differentiate between these three versions, as I am inclined to do, or should we consider them variants of a “superstyle”? They appeared to be used and to coexist in artistically outstanding examples over several hundreds of years (e.g., Mofid & Raeeszadeh 1995, Makovicky 2015b, Castéra 2016). Although it appears to be designed with the decagonal style in mind, an interesting feature of designs with the Kond style (i.e., the Maragha style of Makovicky (1992)) of tiles is that it did not remain limited to the decagonal (pentagonal) sets of tiles which contain regular pentagons, “butterflies” and composite lozenges (Figs. 7.2 and 7.3) as fundamental tiles. Importantly, the latter two tile types have apex angles adjusted to those of the pentagons. Within the Kond style, there is a broad trend towards using the system of regular (or close-to-regular) pentagons and lozenges in the patterns of other symmetries, e.g., tetragonal patterns that are based on combinations of 8- and 4-fold stars or of 8- and 12-fold stars (Makovicky 2015b, also Figs. 9.5 and 9.6), and hexagonal patterns composed of 9- and 6-fold stars (Fig. 9.7). Another such variety are hexagonal patterns with sixfold stars which contain butterflies instead of lozenges (Fig. 9.8). The apical angles of the larger high-symmetry stars in these patterns vary from a case to a case, because variation of the angles at the star tips and recesses allows to form a ring of “near-regular” pentagons of the “Kond” style around them. The “Maragha-type” lozenges in the nondecagonal pattern types do not preserve the length-to-width ratio of the decagonal tiling and there is a slight irregularity of pentagons. Patterns which are composed of small ceramic tesserae/tiles do not allow evaluation of this irregularity because the tesserae have been hand-fashioned by clipping material from large ceramic tiles. Therefore, I reconstructed the pattern construction procedures from the patterns composed of large ceramic tiles. The procedure starts with a desired packing scheme of large regular polygons which share selected edges. This is followed by a number of further construction steps, too complicated to be described here in adequate detail. These patterns with symmetries other than decagonal are found, for example, in central Turkey, Bukhara, Samarkand, and Isfahan (Figs. 9.9 and 9.10) and within a time span between early thirteenth century and late sixteenth century. Entirely different is the outer ornamentation of the dome of the Aramgah-e-Shah Ne’matollah Vali in Mahan, E Iran (early fifteenth century) which reminds of “classical” Kond tiles, but employs a scheme reminiscent of cmm, modified by changing the star order (multiplicity) and by adjustments to a curved surface (Fig. 9.11). The common characteristics of art creations in the Kond style are the abundance of small pentagons in corner contact, presence of composite lozenges and/or “butter-

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Fig. 9.4: A broad frieze composed primarily of the secondary composite tiles from Fig. 7.3. Marginal thirds of the frieze are related by a twofold rotation axis situated in the axis of the frieze, the central one is asymmetric but rotatable about the same axis. Remnants of ornamental Kufic inscription along the right-hand side. Yeni Cami, Kayseri.

Fig. 9.5: A Kond frieze with fourfold and eightfold rosettes. Bazar street madrasa, Isfahan.

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Fig. 9.6: A fourfold deltoid panel with a combination of eightfold and twelvefold rosettes in the background pattern. Structure of the background changes freely to fit the deltoid panels: two types of tetragonal patterns and a hexagonal pattern. Either of the rosette types can have a wreath of small pentagons, interconnections are composite lozenges or “butterflies”. There is a seamless transition between one type of tetragonal pattern and the hexagonal pattern. Friday Mosque, Isfahan.

flies”, a usual presence of larger star discs of different order, absence of rosettes and small fivefold stars (except for mixed-style products), and tile flipping and rotational freedom of composite tiles observed in decagonal variants. This results in a very definite overall effect that is different from the Iranian Tond and Shol styles which appear to be less widespread. All this determines the character of the Kond–Maragha canon as an art style, comparable for example with European Rococo style and era. Similar to

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other art styles, it was spread over a considerable area of Turkey, Iran and Central Asia, and for some uses it survived for a considerable time. The decagonal quasiperiodic patterns (Chapter 7), which owe their existence to the special geometric properties of the decagonal tile set, were the most important subfamily of the Kond (Maragha) style set (Makovicky 1992, 2008, 2015b with references). Friezes of the Turkish Seljuk mosques and madrasas carved in basalt or composed in ceramics form the other large group of examples from the early period of the Kond family.

Fig. 9.7: Nested ornamental panels with the eightfold and fourfold Kond combination (outer frame) and a ninefold and sixfold mostly Kond combination (inner panel). Kalian mosque, Bukhara, Uzbekistan.

Art Nouveau and Art Deco were short-lived exuberant styles, with some local variations. Among other things, the representatives of these styles designed wallpaper patterns and fabrics (Figs. 2.41–2.44). Besides having such material in hand directly, prominent artists also produced pattern books to be copied and applied in practice. These books can give us a good picture of the two-dimensional symmetry canon valid for the given time, style and movement. A 1988 reproduction of Dessins d’Ornementation Plane en couleurs Art Nouveau, published originally in 1900 by R. Beauclair, contains 47 examples of the p1 patterns, 37 of the cm patterns, 21 pg patterns, and about six pC ′m patterns (some combinations are overcomplicated). The remaining symmetries, p4, pg′, pa ′1, pm, pmm, cmm, p2, pC ′4, pb ′2, pa ′m, and pb ′g, occur once or twice. The contrast with the examples given in the early portions of

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Fig. 9.8: A sixfold interlaced panel with a Kond-related tiling of m-symmetrical pentagons and butterflies combined with sixfold stars and minute triangles. Shifaiye Medresa, Sivas, Turkey.

this chapter cannot be greater. However, there is a similarity of these results with the ornamented stone-paste tiles from Iznik, western Turkey, produced for the adornment of mosques and palaces of the Ottoman empire around 1510–1620 (Figs. 2.30–2.39). Actually, they have a common subject – a floral and vegetal decoration. Of course, because of the square shape and corresponding application to the walls, the Iznik tiles show a duality – an additional prominence of tetragonal motifs, which occur in competition with the “lower symmetries”. Thus, cm, p4mm, cmm, p1, pmm and pm are the principal symmetries, p4, p2, pgm, c′m and p6 are in the second row, whereas pg (popular in Art Nouveau), p4gm and c′mm are rare. Asymmetry can also be a popular topic in ornamental art (Fig. 9.12). One such period was Rococo art which, with its rocaille decoration, overtook the symmetrical baroque ornamentation (Coffin et al. 2008). The other period was Art Deco, or some portions of it, here represented by Fig. 3.28. In the (textile) Art Deco pattern book by A. and M.P. Verneuil (1925), reprinted in 1988, in which 98% of patterns are in p1, only two are in pg. The delicate symmetry-breaking in many patterns of Art Nouveau, in which one flower (Fig. 9.13) or only one blade violates the overall mirror symmetry, is another

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Fig. 9.9: A Kond-type tiling of twelvefold and ninefold stars surrounded by wreaths of pentagons mixed with composite lozenges (i.e. pairs of partly overlapping pentagons). Tympanum of the entrance to Kalian mosque, Bukhara, Uzbekistan.

Fig. 9.10: An ornamental panel partitioned into deltoids with the Kond-like combination of eightfold and twelvefold rosettes. Friday Mosque, Isfahan.

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Fig. 9.11: Aramgah-e-Shah Ne’matollah Vali in Mahan, E Iran. The magnificent dome ornamented in a cmm-like scheme with tiling which contains a number of Kond-like elements.

example. In this epoch, whole buildings were designed to be asymmetric, following the thesis proposed by Viollet-le-Duc in 1877: “La symétrie n’est nullement une condition de l’art …c’est une habitude des yeux, pas autre chose”. A distinguished example is Castel Béranger in Paris, built by H.G. Guimard in the years 1896–1897 (Thiébaut 1992) (Figs. 4 and 1.4). The same attitude permeates many examples of nineteenth century calligraphy from the Ulu Cami, Bursa, Turkey, where the intercrossing of stems and blades is often solved as one set of them piercing the other set, leading to the point group “1” (Figs. 1.17–1.18). This asymmetry peaks with a large toughra (official signature) of a sultan in the Ulu Cami (Fig. 1.23), which is based on a combination of approximate motif repetition and asymmetry. See also Fig. 9.14. A monumental effort to produce a scientifically substantiated asymmetry was done by Hinterreiter in his book “Die Kunst der reinen Form” (1978), which collects his partial contributions from 1937 onwards. In principle, he remapped ornamental designs made on rectilinear nets onto affine, projective and curvilinear nets, and selected subsequently off-center rectangular or (hemi)circular portions as the picture field (Figs. 6.37–6.39). His work, which preceded and heralded the computer age, did not generate followers. Although dichroic designs were produced by many cultures, the question of color groups or, at least, polychromatic patterns is much more problematic. Many brightly

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colored patterns do not display colored symmetry operations at all. Color patterns are often subject to a set of rules– obeying a given technique or style of the day. The patterns produced in the Islamic and Mediterranean areas very frequently started with a dichroic stage, after which the tiles of the white set were left unmodified, as a color diluent, whereas those of the black set were further colored in different ways (Chapter 5). Interestingly, the tessera patterns of ancient Rome (as seen, e.g., in Pompeii, Fig. 5.19) and especially the polychromatic patterns of the South American Indian cultures (Figs. 5.8 and 5.9) do not appear to use this dichroic approach. They obey true, albeit simple, color groups of symmetry. Among them, a use of N-colored translations is a typical feature. The advanced color groups used by Hinterreiter (1948) and Escher (1958) in their mature creations were not used by the ancient artists. Similar to Washburn and Crowe (1998), who with their rich collection of examples from the whole world (mostly concentrating upon areas from which I show scarce examples) did not try to build a theory of symmetry in art, I prefer a view of regional and temporal canons, both in the style and symmetry, to any “developmental” theory. Each canon of material art blossomed in its own domain, determined by natural conditions, type of society, type of material production and their set of beliefs and traditions. Jablan (1995) has a somewhat different opinion: he suggests that the most important concept is that of maximal visual entropy which he defines as “maximal construc-

Fig. 9.12: Ornamental Art Nouveau plate with an asymmetric poppy motif. Probably made in Kremnica, Slovakia. Private collection.

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Fig. 9.13: Art Nouveau house in Firenze, Italy: gentle asymmetry

Fig. 9.14: Ornamental panel from the Friday Mosque in Isfahan, Iran with angular Kufic texts, different in different deltoids – subdued asymmetry in a fourfold frame. The frieze is a composite of ribbons in different orientations.

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tional and visual simplicity and maximal symmetry”. As an answer to the question of more complicated patterns, Jablan suggests that such patterns (interlace etc.) went first through a fairly simple stage and were later subjected to a desymmetrization. Still, he has not “lived” in the virtual world of Islamic ornamental art …. Makovicky (2008) describes the Maragha tiling as analogous to the high-entropy structure of synthetic quasicrystals with phason flips, i.e. a completely different notion of entropy to that of Jablan (1995). Another approach is given by Shubnikov and Koptsik (1974) who discern four structural sublevels, which in thematic pictures are followed by a thematic and an ideological level (Soviet times!). The structural sublevels are: a level of geometric composition; a level of graphical outline (i.e., a more detailed analysis of the distribution of masses, volumes, articulation), a level of light and shade and a coloring level. As for myself, in some cases I have given here a sequence of symmetries, starting from the symmetry of pure outlines to a final one, which takes into account interlacing, coloring, affinity, etc. This roughly reminds of Shubnikov and Koptsik’s divisions. What is important in the concept of Shubnikov and Koptsik is that they allow coexistence and combination of several symmetries in the same creation. As we have seen, such a coexistence of symmetries is especially important in a part on Islamic plaster ornaments (Chapter 6). Finally we should mention that Washburn et al. (2010) attempt to

Fig. 9.15: A detail of the ornamental combination of 12-fold and 8-fold stars with wreaths of pentagons in the Kond style. The small spiral elements show high orientation variability. Shah-i-Zindeh Necropolis, Samarkand, Uzbekistan.

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Fig. 9.16: A pattern of squares and triangles (try to trace!) with local sevenfold symmetry in points of contact. Predominant contacts are of the 4.3.4.3.3 type (4 denotes squares, 3 triangles), with local 4.4.3.3.3 contacts. Angles in triangles are modified to match the 51.4° angle of the sevenfold rotation. The panel is too small for restoring a possible 2D symmetry group; only a 222 point group transpires. Tomb of Hodja Ahmad (early fourteenth century), Shah-i-Zindeh Necropolis, Samarkand, Uzbekistan.

tie the symmetry preferences of the ornamental design to the social organization of the society which created the objects. They also quote a rich array of other researchers who shared this concept. If you intend to explore symmetry and style yourself, first prepare yourself by achieving a good overview of relevant groups of symmetry. Some cases may be rather complex (Figs. 9.15–9.17), especially when having a pattern with “tiles” only in special positions as given by the symmetry groups. Sources of practical symmetrology are quoted in the Introduction whereas the majority of books, especially on Islamic ornamental art (e.g., Broug 2013), concentrate heavily on the geometrical problems of construction. Train yourself on scanned copies of published examples – from this book and elsewhere. In general, the most interesting problems might be those which were overlooked by others, even by specialists in the field. Leave social or belief-bound interpretations until after a proper symmetrological analysis has been performed. Do not start from somebody’s theories, instead build your own set of facts and interpretations and check afterwards against published material. You might end up disappointed – it is very difficult to be original – but you might also have

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Fig. 9.17: A periodic arrangement of Kond tiles with fourfold lozenge groups forming “superlozenges”, and with ordered overlapping arrangements of pentagons and black “butterflies” as a background for a two-tier inscription. Bazar-street medresa in Isfahan.

discovered something the others have not noticed. And beware: the names given to the elements of the ornament might be very far from the original names, and merely a local parlance, so do not overly trust any meaning or interpretation based on present culture. Good luck and lots of pleasure!

References Abad-Zapatero, C. (2014): Notes of a protein crystallographer: The beauty of rose windows and the different meanings of symmetry. Acta Crystallographica Section D: Biological Crystallography, D70, 907–911. Abbas, S.J. & Salman, A.S. (2007): Symmetries of Islamic geometrical patterns. 2nd edition. World Scientific Publishing Co., Singapore. Acedo, A.C. (1989): The Alhambra in Detail. Granada. Al Ajlouni, R.A. (2012): The global long-range order of quasi-periodic patterns in Islamic architecture. Acta Crystallographica A. Foundations of Crystallography, A68, 235–243. Albrecht, H.J. & Koella, R. (1982): Hans Hinterreiter, a Swiss Exponent of Constructive Art. Waser Verlag, Zürich. Ammann, R. (1977) Letters to B. Grünbaum, unpublished. Anker, P. (1970): The Art of Scandinavia I–II. Paul Hamlyn, London – New York. Bain, G. (1973): Celtic Art – the Methods of Construction, 2nd edition. Dover Publications, New York. Balmelle, C., Blanchard-Lemée, M., Christophe, J., Darmon, J.-P., Guimier-Sorbet, A.-M., Lavagne, H., Prudhomme, R. & Stern, H. (2002a): Le Décor Géométrique de la Mosaïque Romaine. I. Répertoire graphique et descriptif des compositions linéares et isotropes. éditions A. et J. Picard, Paris. Balmelle, C., Blanchard-Lemée, M., Darmon, Gozlan, S. & Raynaud, M.-P. (2002b): Le Décor Géométrique de la Mosaïque Romaine. II. Répertoire graphique et descriptif des décors centrés. éditions A. et J. Picard, Paris. Barnsley, M. (1988): Fractals Everywhere. Academic Press, Inc., San Diego. Binding, G. (2000): Was ist Gotik? Primus Verlag, Darmstadt. Beauclair, R. (1900): Dessins d’Ornementation Plane en couleurs Art Nouveau. Paris. Reproduction (1988) by Dover. Belov, N.V. & Tarkhova, T.M. (1956): Groups of colour symmetry. Crystallography, 1, 4–13. Blazquez, J.M. (1975): Arte y sociedad en los mosaicos Hispanos del Bajo Imperio. Bellas Artes, 6, 18–25. Bourgoin, J. (1973): Arabic Geometrical Pattern & Design. 200 Plates. Dover Publications, New York. Brannan, D.A., Esplen, M.F. & Gray, J.J. (2004): Geometry. Cambridge University Press. Broug, E. (2013): Islamic Geometric Design. Thames & Hudson. Butler, C. (2006): Christ Church: A Portrait of the House. Oxford. Cáceres Macedo, J. (2005): Tejidos del Perú Prehispánico. Textiles of Prehispanic Perú. Lima-Perú. Castéra, J.-M. (1996): Arabesques. Art Décoratif au Maroc. ACR édition, Paris. Castéra, J.-M. (2016): Persian variations. Nexus Network Journal DOI 10.0007/s00004-015-0281-5; 1–49. Chamoso, J., Fernandez, I. & Reyes, E. (2009): Burbujas de Arte y Matemáticas. Diálogos de Matemáticas 6. Nivola Publ., Tres Cantos, Spain. Cimok, F., (ed.) (1998): The Book of Rüstem Pasha Tiles. A Turizm Yayinlari, Istanbul. Clevenot, D. & Degeorge, G. (2000): Ornament and Decoration in Islamic Architecture. Thames & Hudson. Coffin, S.D., Davidson, G.S., Lupton, E. & Hunter-Stiebel, P. (2008): Rococo – The Continuing Curve, 1730–2008. Smithsonian Institution. Cowen, P. (2005): The Rose Window: Splendor and Symbol. Thames and Hudson, New York. Coxeter, H.S.M. (1987): A simple introduction to colour symmetry. International Journal of Quantum Chemistry, 31, 455–461.

236 | References

Crowe, D.W. (1986): The mosaic patterns of H.J. Woods. Computers and Mathematics with Applications. 12B, 402–411. Dalgalidere, H. A. (2012): The Tiles of Rustem Pasha Mosque. TFM Yayincilik, Istanbul. Dye, D.S. (1974): Chinese Lattice Designs. Dover Publications, Inc., New York. Escher, M.C. (1958): Regelmatige Vlakvertdeling. Utrecht. Farioli, R. (1975): Pavimenti Musivi di Ravenna Paleocristiana. Longo Editore, Ravenna. Farley, J., Goldberg, M., Hunter, F. & Leins, I. (2015): Celts – Art and Identity. The British Museum Press. Fong, W. (1980): The Great Bronze Age of China. The Metropolitan Museum of Art & Alfred A. Knopf, Inc., New York. Garrido R.C. & Pérez-Gómez, R. (1995): Visiones matemáticas de la Alhambra. El color. La Alhambra. Epsilon, 2nd edition, S.A.E.M. Thales, Granada, 51–60. Gerdes, P. (1989): Reconstruction and extension of lost symmetries: Examples from the Tamil of South India. Computers and Mathematics with Applications, 17, 791–813. Gheller Doig, R. (2005): Textiles of Ancient Peru. Tejidos del Perú Antiguo. Serinsa, Perú. González-Ramírez, M.I. (1995): El Trazado Geométrico en la Ornamentación del Alcázar de Sevilla. Colección Kora, Universidad de Sevilla y Junta de Andalucía. Grünbaum, B. Grünbaum, Z. & Shephard, G.C. (1986): Symmetry in Moorish and other ornaments. Symmetry – Unifying Human Understanding. Hargittai, I., (ed.). Pergamon Press, New York. Grünbaum, B. & Shephard, G.C. (1987): Tiling and Patterns. Freeman and Company, New York. Hann, M. (2014): Symbol, Pattern and Symmetry: The Cultural Significance of Structure. Bloomsbury, London and New York. Hegmon, M. (1992): Archaeological research on style. Annual Reviews of Anthropology, 21, 517–536. Henderson, J. (2000): The Science and Archeology of Materials. Routledge, London and New York. Hinterreiter, H. (1948/1978): Die Kunst der reinen Form. Ediciones Ebusus, Ibiza – Amsterdam. Hondasu Ichiro (1992): A Large Book of Kamon. Goto Shoin, Japan. Horne, C. E. (2000): Geometric Symmetry in Patterns and Tilings. Cambridge. Hutt, A. & Hurrow, L. (1977): Islamic Architecture. Iran 1. Scorpion Publications. London. Jablan, S.V. (1995): Theory of Symmetry and Ornament. Matematički Institut, Beograd. Janssen, T., Chapuis, G. & de Boissieu, M. (2007): Aperiodic Crystals. From Modulated Phases to Quasicrystals. IUCr and Oxford Science Publications. Jørgensen, M. (1992): Peru–indianernes labyrint–kunst. Håndarbejde, 1992, 35–39. Kapraff, J. (1991): Connections: The Geometric Bridge between Arts and Science. New York. Kavaler, E. M. (2012): Renaissance Gothic. Yale University Press, New Haven and London. Kızıltepe, F. (2005): Analyzing the plane symmetry of Ottoman tiles as an architectural element between the 15th & 16th century. Symmetry: Culture and Science, 21, 413–432. Kopsky, V. & Litvin, D.B. (1990): Nomenclature, symbols and classification of subperiodic groups. Report of Working Group of IUCr Commission on Crystallographic Nomenclature. Kuo, K.H. (1990) Octagonal quasi-crystals. In: Quasi-Crystals, Jaric, M.V. & Lundquist, S., (eds.). World Scientific, Singapore, 92–108. Lauwerier, H. (1991): Fractals – Images of Chaos. Penguin Books, London – New York. Lazzarini, L. (2004): La diffusione e il riuso dei più importanti marmi romani nelle provincie imperiali. In: Lazzarini, L. (ed.): Pietre e Marmi Antichi. CEDAM, Padova: 101–134. Liu, Y. & Toussaint, G. (2010): Unraveling Roman mosaic meander patterns: A simple algorithm for their generation. Journal of Mathematics and the Arts, 4, 1–11. Loeb, A.L. (1971): Color and Symmetry. Wiley Interscience. New York. Lu, P.J. & Steinhardt, P.J. (2007): Decagonal and quasicrystalline tilings in medieval Islamic architecture. Science, 315, 1106–1110.

References

|

237

Ma, R. (1996): Color-paintings in ancient Chinese constructions. Publishing House of Cultural Relics, Beijing. Makovicky, E. (1979): The crystallographic art of Hans Hinterreiter. Z. Kristallographie, 150, 13–21. Makovicky, E. (1986): Symmetrology of art: Colored and generalized symmetries. Mathematics and Computers with Applications, 12 B, 949–980; also as: Symmetry – Unifying Human Understanding (I. Hargittai, editor), Internat. Ser. in Modern Appl. Maths. and Computer Sci., 10, 949–980 (1986), Pergamon Press Ltd., New York-Oxford-Toronto-Sydney-Frankfurt. Makovicky, E. (1989): Ornamental brickwork. Theoretical and applied symmetrology and classification of patterns. Computers & Mathematics with Applications, 17, 955–999. Makovicky, E. (1992): 800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: Fivefold Symmetry, I. Hargittai (ed.). World Scientific, 67–86. Makovicky E. (2003): Ornamental diamond patterns in aboriginal bark paintings from N.E. Arnhem Land, Australia. Records of the South Australian Museum 36 (1), 1–20. Makovicky, E. (2007): Comment on “Decagonal and quasi-crystalline tilings in medieval Islamic architecture”. Science, 318, 10.1126/science.1146262 Makovicky, E. (2008): Another look at the Blue Tomb of Maragha, a site of the first quasicrystalline Islamic pattern. Symmetry: Culture and Science, 19, 127–151. Makovicky, E. (2011a): Shipibo geometric designs. Symmetry – Culture and Science, 22, 373–389. Makovicky, E. (2011b): Crystallographer’s Alhambra: Beauty of Symmetry – Symmetry of Beauty. Book published jointly by Universities of Copenhagen and Granada. Makovicky, E. (2015a): Crystallographic symmetries of ornamental floors in the Baptisteries of San Giovanni in Pisa and Firenze. Rendiconti Fis. Accad. Lincei, 26, 235–250. Makovicky, E. (2015b): In the footsteps of Maragha: ornamental panels in the madrasas and mosques of Esfahan, Konya, Agra, Sivas and Yazd. Symmetry – Culture and Science, 26,421– 441. Makovicky, E. & Fenoll Hach-Alí, P. (1994): The O-D pattern from "Puerta del Vino", Alhambra, Granada. Bol. Soc. Española de Mineralogía, 17, 1–16. Makovicky, E. & Fenoll Hach-Ali, P. (1996): Mirador de Lindaraja: Islamic ornamental patterns based on quasi-periodic octagonal lattices in Alhambra, Granada, and Alcazar, Sevilla, Spain. Bol. Soc. Española de Mineralogia, 19, 1–26. Makovicky, E. & Fenoll Hach-Ali, P. (1997a): Brick-and-marble ornamental patterns from the Great Mosque and the Madinat al-Zahra palace in Cordoba, Spain. I. Symmetry, structural analysis and classification of two-dimensional ornaments. Bol. Soc. española de Mineralogia, 20, 1–40. Makovicky, E. & Fenoll Hach-Alí, P. (1997b): Brick and marble ornamental patterns from the Great Mosque and the Madinat al-Zahra palace in Cordoba, Spain. II. Symmetry and classification of one-dimensional patterns. Boletín de la Soc. Española de Mineralogía , 20, 115–134. Makovicky, E. & Fenoll Hach-Alí, P. (1999): Coloured symmetry in the mosaics of Alhambra, Granada, Spain. Boletín de la Sociedad Española de Mineralogía, 22, 143–183. Makovicky, E., Frei, R., Karup-Møller, S. & Bailey, J. C. (2015): Imperial Porphyry from Gebel Abu Dokhan, the Red Sea Mountains, Egypt. Part I. Mineralogy, petrology and occurrence. Neues Jahrbuch für Mineralogie, Abhandlungen (J.Min. Geochem.), 193/1, 1–27. Makovicky, E. & Makovicky, M. (1977): Arabic geometrical patterns – a treasury for crystallographic teaching. N. Jb. Mineral., Mh., 1977, 56–68. Makovicky, E. & Makovicky, N.M. (2011): The first find of dodecagonal quasiperiodic tiling in historical Islamic architecture. Journal of Applied Crystallography, 44, 569–573. Makovicky, E., Rull Pérez, F. & Fenoll Hach–Alí, P. (1998): Decagonal patterns in the Islamic ornamental art of Spain and Morocco. Boletín de la Soc. Española de Mineralogía, 21, 107–127. Massoudy, H. & Massoudy, I. (2003): l’ABCdaire de la Calligraphie Arabe. Flammarion, Paris. Meehan, A. (1995a): Celtic Design. Animal Patterns. Thames and Hudson, London.

238 | References

Meehan, A. (1995b): Celtic Design. Spiral Patterns. Thames and Hudson, London. Mencl, V. (1974): České středověké klenby (Czech Medieval Vaults). Publishing House Orbis, Prague. Michell, G., (ed.) (1978): Architecture of the Islamic World. Thames & Hudson, London. Mofid, H. & Raeeszadeh, M. (1995): Reviving the Forgotten Arts: Foundations of the Traditional Architecture in Iran narrated by Hossein Lorzadeh. Mola Publication, Tehran (in Persian). Müller, F. (2009): Art of the Celts. Historisches Museum Bern. Nakamura, S. (2009): Pattern Sourcebook: Nature. Rockport Publishers, Beverly, Massachusetts. Pajares–Ayuela, P. (2002): Cosmatesque Ornament. Thames & Hudson, London. Pavón Maldonado, B. (1989): El Arte Hispano–Musulmán en su Decoración Geométrica, 2nd . Ed. Instituto de Cooperación con el Mundo árabe, Madrid. Porter, V. (1999) Islamic Tiles, London. Rigby, J. (2005): A Turkish interlacing pattern and the golden ratio. Whirling dervishes and a geometry lecture in Konya. Mathematics in School, 34, 16–24. Roe, P.G. (1980) Art and residence among the Shipibo Indians of Peru — Study in micro-acculturation. American Anthropologist, 82 (1), 42–71. Ruiz-Garrido, C. & Pérez-Gómez, R. (1995): Visiones matemáticas de la Alhambra. El color. La Alhambra – Epsilon, 2nd ed. SAEM Thales, Granada, 51–60. Saltzman, P. (2008): Quasiperiodicity in Islamic ornamental design. In: Nexus VII: Architecture and Mathematics Conference Series, 153–168. Kim Williams Books, Torino. Schattschneider, D. (2004): M.C. Escher. Visions of Symmetry. Thames & Hudson. Schneider, G. (1980): Geometrische Bauornamente der Seldchuken in Kleinasien. Ludwig Reichert Verlag, Wiesbaden. Shubnikov, A.V. & Koptsik, V.A. (1974): Symmetry in Science and Art. Plenum Press, New York. Smith, C.S. & Boucher P. (1987): The tiling patterns of Sebastian Truchet and the topology of structural hierarchy. Leonardo, 20. Socolar, J.E.S. (1989): Simple octagonal and dodecagonal quasi-crystals. Physics Review, B39, 10519–10551. Stierlin, H. (2002): Islamic Art and Architecture. Thames & Hudson Ltd., London. Sutton, D. (2007): Islamic Design. A Genius for Geometry. Walker Publishing Company, New York. Sutton, A. (2009): Ruler & Compass. Practical Geometric Constructions. Wooden Books Ltd. Glastonbury. Thiébaut, P. (1992): Guimard, L’Art Nouveau. Decouvertes Gallimard. Van Roojen, P. & Navarro, M.A.H., (eds.) (2008): Art Nouveau Tile Designs. The Pepin Press, Amsterdam. Verneuil, A. and Verneuil, M.P. (1988): Abstract Art. Patterns and Designs. Bracken Books, London. Wade, D. (1976): Pattern in Islamic Art. The Overlord Press, Woodstock. Washburn, D.K. & Crowe, D.W. (1998): Symmetries of culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press. Seattle and London. Washburn, D.K. & Crowe, D.W. (2004): Symmetry Comes of Age: The role of Pattern in Culture. University of Washington Press, Seattle. Washburn, D.K., Crowe, D.W. & Ahlstrom, V.N. (2010): A symmetry analysis of design structure: 1,000 years of continuity and change in Puebloan ceramic design. American Antiquity, 75, 743–772. Weyl, H. (1952): Symmetry. Princeton Univ. Press, Princeton. Yerasimos, Y. (2000): Konstantinopel. Istanbuls Historisches Erbe. Könemann. Köln. Zhou, S. & Wang, S. (1993): Ornament Pattern Collections of Chinese Bronzes (in Chinese). Shangai Book Store, Shangai

Index Alhambra – colour symmetry 118, 142 Alhambra – Salon de Comares 196, Torre de la Cautiva 197 Alhambra – stucco patterns 50, 151–155 Appendix 131–132 Art Deco 110, 215–229 Art Nouveau XIII, 58, 120, 225 Art style 217, 224, 229 Asymmetry 226 Australian Aborigines 156 Baptistery of San Giovanni (Firenze) 124,155 Bukhara (Uzbekistan) 103, 148,152 Bursa (Turkey) 11, 170 Cantor’s fractal 203 Capella Palatina (Palermo,Sicily) 94 Chicago 110 Chinese architectural ornaments 9 Chinese cloud patterns 57 Chinese art with fractal character 208–211 Chinese bronzes 209 Chinese marble panels 31, 215 Chinese swastika patterns 79 Chirality 174 Color symmetry 133, (practical) 134–135, 229 Cordoba: the Great Mosque 71, 103, 219 Cosmatesque art (Italy) 13, 24, 115–118,205 Coxeter notation for dichroic groups 112 Decagonal patterns 184 Decagonal quasiperiodic pattern 179–182, 188–190 Dichroic symmetry 111–115, 128, 131–132 Dodecagonal quasiperiodic pattern 202 Egypt (Philae) 205 Enantiomorphy 174 Fez 124, 188, 199–202 Firenze 124, 155, 172–173 Fractals 203 Fractal character 203 Frieze groups of symmetry 20–21

General positions 35, 112 Gothic vesica windows 167 Gothic rose windows 2 Gothic church vaults 22 Groups (symmetry) XVII–XVIII Homologous series 166 Homothety 169–173 Huari 138 Hypersymmetry 149 I-beam swastika patterns 67 Interlace 95 Intertwined patterns 92, (Viking and Celtic) 98, (Tamil) 99, Roman 102 Inversion in a circle 166 Isfahan 48–49, 181–184 Islamic calligraphy (Bursa) 11, 170 Iznik tiles 53–61 Jugendstil

58, 120

Kamon (Japan) 5 Kharraqan tomb towers 42, 106, 219 Kond art style 222–225 Kond tiles 48 Konya (Turkey) 78, 181 Layer groups of symmetry (two-sided layers) 92, Table 3.1 Manji swastika patterns (Japan) 78 Maragha (Iran) 75, (quasiperiodic) 180–182, 207 Meanders 58–70 Mudéjar domes 158 Multicolor waves (modulation) 134, 142–143 Multiple symmetries in one pattern 151 Muqarnas 34 Non-linear art 176 North-west Coast Indian art 123 Octagonal quasiperiodic pattern 179, 190–195, (variations) 195–200 Order-disorder patterns 161 Ornamental art XII

240 | Index

Partial chromaticity 134 Phason flipping 180 Plane groups of symmetry 35–41 Point groups of symmetry 1 Polychromatic groups of symmetry 133, 229 Practical advice 232 Precolumbian textiles 138 Pueblo ceramics 31 Purpose of the book XVIII Quasiperiodic patterns 179, (Andalusia) 187, (Alhambra) 188, 190, 196, (Fez) 191, 202, (Reales Alcazares) 192–195 Quasilattice 179

Seljuk Turkey 219 Sevilla 94, 127, 158, 163 Sequentially dichroic patterns 135 Shifaiye Medresesi (Sivas) 75 Shipibo Indian Art (Peru) 164 Sierpinski triangle 204 Similarity operations 169–170 Shol tile style 48 Siena Cathedral (Italy) 141, 166 Silk Road caravanserai 82 Special positions 35, 112 Superstructures and pattern levels 150 Swastikas 59–71 Symmetry operations XVI Symmetries of finite objects 1

Reales Alcazares (Sevilla) 94–97, 158, 163 Roman antiquities 29, 140, 144 Roman rosettes 173 Romanesque friezes 30 Renaissance frieze 29

Theories of symmetry in art Tivoli (Italy) 144 Tond tile style 48 Twinning 157

Samarkand (Uzbekistan) Seljuk minarets 44

Variations of the quasicrystalline theme (Alhambra) 196, (Morocco) 201

76

217, 229