The Silver Rectangle [1, 2nd reprint ed.]

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Models © John Cunliffe / adapted / traditional Diagrams © John Cunliffe BOS booklet #21 First published by British Origami Society, April 1983. Reprinted March 2008 Printed in the United Kingdom. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system, or transmitted, in any form or by any means, electronic, mechanical photocopying, recording or otherwise without the express written permission of the author and of the British Origami Society. The British Origami Society is a registered charity 293039 www.britishorigami.info

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THE SILVER RECTANGLE AN EXPLORATION OF THE FOLDING PROPERTIES OF 1: 2 SIZE PAPER, COMPILED BY

John Cunliffe

Silver is the king’s stamp...... John Webster, Northward Hoe.

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CONTENTS INTRODUCTION AND REFERENCES. SYMBOLS AND ABBREVIATIONS. CONVERSIONS - any sheet of paper to a silver rectangle by folding. THIRDS - how to divide a silver rectangle into three equal parts by folding. ARROW + CONE. Arrow for use as a direction indicator; cone as a marker, funnel or corner to hold documents. BOX - long or cubical. The long one makes a practical pencil box. Two the same size canbe slotted together to make a closed container. (Adapted from a design by G. Baggi, USA.) BOX - collapsible. This also makes a practical wallet when folded from A3 or size to suit requirements. BOX - banded. The bands add strength to legs. Narrow opening at top suggests use CASE, CAP, WRAPPER. In the initial stages, location points can be adjusted to requirements i.e. head size for forage cap, gift size to wrapper. (Adapted from a design by J. French, England.) CUBE. A container or improvised dice. FRAME. Method of folding corner slots for picture postcards etc. HEXICARTON. A practical six-sided container for pencils etc. Can be adapted to any rectangle and varying number of sides. Suggested alternative to the fold shown is A3 - size stiff paper with hem taken down to base line. Slotted card can be used for internal divisions. Calendar months can be pasted on the outside, two to each panel. If internal layers are required, closed hexagons from the next smaller A-size will fit more or less exactly. HEXAGON. A geometric figure with closed edges and closed sides. Many uses decorative also as table mats or improvised counters in board games. (Adapted from designs by T.Yenn, Denmark, and G. Andersson, Sweden.) LETTERFOLD. Saves an envelope! If two sheets are folded together a pocket is formed inside. (Adapted from a design by D. Shall, USA.) MITRES. Folding procedure for mitred corners. NOTECASE/PAD. (Adapted from a design by M.Yamaguchi, Japan.)

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NOTECASE, also square. Any rectangle can be used but A3 or A4 should meet most requirements. The square is a closed geometric figure see also Triangle. (Adapted from a ‘Pureland’ design by J. Smith, England.) PENTAGON. With closed edges, which can also be formed into a threedimensional decoration. (Adapted from a design by D. Brill, England.) PENTAGRAM. A five-pointed star without cutting. (Adapted from a design by S. Fujimoto, Japan.) SQUARE, PENTAGON, HEXAGON, EQUILATERAL TRIANGLE - quick outlines for use in geometry or as design aids. SEED PACKET. A secure container for small items such as seeds, beads, tablets, coins etc. Also useful for pointed objects - pens, scissors etc. Size to suit requirements. A5 is suggested. (Adapted from a design by Marlene Stroud, England.) SET SQUARE. A5 and 30/60 degree triangles as quick aids in draughtsmanship. The 45 degree fold is also known as a magician’s purse since it has two identical side openings and items placed in one seem to have disappeared when the other is opened. TETRAHEDRON, RHOMBUS. Mainly for use in the study of geometry but also decorative. TRIANGLE, equilateral. A geometric figure in which all the ‘raw’ edges have been folded inside so as to form closed edges. Similar examples of this technique occur in the 45 degree triangle, rhombus, hexagon, octagon and square. TABLE NAPKIN. For use where a napkin does not have a square base although a square can also be used. Z-ENVELOPE. An envelope which can be folded and closed without using adhesive. Any rectangle can be used as well as a silver, and an interesting feature is that a sheet of paper the same size as that from which the envelope was formed will, if quartered, fit into the envelope exactly. PROPORTION TABLES. A set of JZ multiplication tables. (In some cases the three figures following the decimal point have been rounded off for ease of reference.) CARTOON. One of a series of ‘Origags’ cartoons specially drawn for the British Origami Society magazine. BOS publications

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INTRODUCTION AND REFERENCES What is a Silver Rectangle? In paperfolding, an oblong sheet of paper with sides in the proportion 1: √2 or 1: 1.414. When such a sheet is folded in half by bringing the two shorter edges together and creasing along the fold line, the resulting oblong is proportionately identical to the original. In geometry it may be described as a rectangle in which the longer side is equal to the diagonal of a square whose side is the same length as that of the rectangle’s shorter side. In paper manufacture, a group of these sizes are known as A, B and C size. Seven A-sizes can meet all stationery requirements. This booklet is A5 for example. The next larger size A4, is used for a whole range of printed forms, catalogues and documents.

HISTORICAL REFERENCES In China, where paper was invented, currency notes were at one time 330x230 mm. In France, paper sizes with √2 proportions were introduced in 1792. Germany followed with DIN sizes in 1870 and Great Britain with A-size in 1959 when poised to join the European community. (BOS Magazines 75 & 83.)

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ARCHITECTURE: Although a few noted buildings such as certain Gothic cathedrals and the Paris Opera House were designed using the diagonal of a square (√2) as a module, there is no evidence to suggest this was a widespread practice. The famous French architect, Le Corbusier, preferred the Golden Section when he devised his system (never adopted) of international measurement. (BOS Magazine 75; Dictionary of Architectural Science, H. S. Cowan.) NEWSPAPERS: Daily papers have long been printed on sheets of silver rectangle proportions, broadsheets such as The Times and Telegraph being 600x400 mm approximately with tabloids correspondingly smaller. It is no accident that the well known fold the Printer’s Hat has a silver rectangle base. Newsprint operators are said to make such hats from blank news sheets and wear them to protect their heads from printing ink. (Origami 1, R. Harbin.) WHY SILVER? To distinguish it from other rectangles such as the Golden which has sides in the proportion 1: 1.616; Harmonic (1.667); Halved (1.5); Tertian (1.33); Quartan (1.25) etc. Paper sizes in proportions known as quarto and foolscap (203x333 and 253 mm approximately) have been almost entirely superseded by A sizes during recent years. The name Silver was suggested by the science section of the Oxford Dictionaries organisation and was selected by the British Origami Society in 1979 following a search for a suitable name. (BOS Magazine 75.) OF WHAT IMPORTANCE IS THE SILVER RECTANGLE IN ORIGAMI? The number of designs or models which must start from such a base is small compared with, say, a square. In fact in many cases, pentagonal and hexagonal folds being exceptions, any oblong will suffice. However, the 1: √2 or Silver Rectangle is likely to be the most widely used shape in document paper for a long time to come. It thus has the not inconsiderable advantage of being in good supply and already trimmed to the base shape required. ADDITIONAL READING One of the books which is essential reading for anyone interested in the properties of the 1: √2 rectangle is Flying Origami by E. Nakamura (Japan Publication 1972). Here such a rectangle is called a ‘true’ rectangle. As well as folding instructions for flying objects from A4 paper, the introduction contains much geometrical information. Another Japanese who has investigated this rectangle is S. Fujimoto. Although his works are not published in English they have nevertheless reached British folders and his Pentagram and Icosehedron have been taught at conventions. (Bristol, Oxford Convention folders; BOS Magazine 89.)

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Martin Gardner Puzzle Books (Dover Publications) These have several references to the √2 rectangle, mentioning that it can be found in the works of Leonardo da Vinci and Albrecht Durer the painter. A playing card trick is also illustrated whereby a sheet of thick paper or card is divided into five sections (see paper sizes) with the face of the same playing card being printed on each. The card is then folded even smaller and smaller or larger and larger. Theorems and observations about the √2 rectangle were presented in a two page paper by P. G. Scopes of Avery Hill College in 1959 or thereabouts. In it, the rectangle was called an M-rectangle since the folds were produced from ‘metric paper’. P. G. Scopes concluded: There is obviously much more mathematics still to come along these lines. (BOS Library.) THE FOLDS The folding diagrams which follow show how to make a range of practical objects using a sheet of A4 paper or any Silver rectangle. Most are, for want of a better word, simple in the sense that they can be completed in twenty-one or less folding sequences without a great deal of intricate creasing. (Twenty-one is approximately the number of folds required to form the traditional symbol of Origami - the Flapping Bird.) As to the origin of the designs, so often it is impossible to establish just when and by whom a fold was created. Every attempt has been made to identify sources and apologies are expressed to anyone who may feel that his or her contribution has not been properly documented. This will be rectified in any subsequent edition. There are of course many more folds other than those included here which require a silver rectangle base. Some are in the BOS Library and additions are made periodically. It is hoped that readers, too, will contribute folds, comment on any new information and those coming to this booklet and the folding art for the first time may like to know that further details about the British Origami Society and its facilities can be obtained from the Secretary, contact details are on our website www.britishorigami.info ACKNOWLEDGEMENTS The compiler thanks all those who have so kindly assisted in the preparation of this booklet, especially Dr. Robert Morassi, who unsparingly provided much valuable reference material.

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BOS publications 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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History of Origami Origami and Mathematics History of Origami in Britain Origami Instruction Language Teaching Origami Geometric Division Chess Sets of Wall, Hulme & Elias Napkin Folds Origami and Magic Neal Elias: Selected Works 1964 - 1973 Flexagons Martin Wall: Early Works 1970 - 1979 Orikata Pureland Origami 1 Max Hulme: Selected Works 1973 - 1978 18 Of My Paperfolds Origami Games

John S. Smith John S. Smith David Lister John S. Smith John S. Smith Mick Guy Mick Guy & Dave Venables John Cunliffe Ray Bolt Dave Venables Paul Jackson Martin Wall Thoki Yenn John S. Smith Dave Venables Paul Jackson Mick Guy and Paul Jackson

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Philip Shen: Selected Geometric Paperfolds Tony O’Hare: Selected Works 1973 - 1982 Origami Christmas Tree Decorations The Silver Rectangle In Praise of the Bird Base Index: BOS Magazines 1 - 100 Index: BOS Magazines 101 - 120 Envelope and Letter Folds Animal Origami Origami Structural Reconstruction Pureland Origami 2 Paper People and Other Pointers Mor’igami Patterns in Paper Animal Origami 2 Neal Elias: Miscellaneous Folds I Neal Elias: Miscellaneous Folds II Neal Elias: Faces and Busts Jeffori’ 3 Four ‘igami Animal Origami 3 Genius of Jan Willem Derksen Index: BOS Convention Packs Larry Hart: Selected Works (1971 - 1991) Pureland 3:Smith Multiplication The Origami of Stephen Palmer Modular Construction and Twists ABC of Origami Making Faces World of Fred Rohm I World of Fred Rohm II World of Fred Rohm III Origami Models Folded from Rectangles Origami Models Folded from Silver Rectangles More Origami Models from Silver Rectangles Sink or Swim with Ted Norminton Petal Folds and More Pureland 4 Owrigami 10 Pop-Ups A Medieval Court in Origami Decorative Boxes from Single Squares Animals and Birds Ship-shape and Bristol Fashion Faces’n’Fings Planar Modulars (CD-rom) The Nativity - An Origami Scene Selected Works of Quentin Trollip Which Came First? Selected Works : A Second Selection My Fundamentals

Paul Jackson Daniel Mason Paul Jackson John Cunliffe John S. Smith John Cunliffe John Cunliffe John Cunliffe Edwin Corrie Jeff Beynon Ricky Wong John S. Smith David Petty Jeff Beynon John S. Smith Edwin Corrie Dave Venables Dave Venables Dave Venables Jeff Beynon Jeff Beynon Edwin Corrie David Petty David Petty Larry Hart John S. Smith Jeff Beynon David Petty David Petty Eric Kenneway David Petty Pete Ford Pete Ford Pete Ford John Morgan John Morgan John Morgan Ted Norminton Ted Norminton John S. Smith Francis Ow John S. Smith Julia Pálffy Arnold Tubis and Leon Brown Tony O’Hare Tony O’Hare Tony O’Hare David Petty Lore Schirokauer Quentin Trollip Robert Neale Max Hulme Kuni Kasahara

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