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There are tessellations and tessellations. The word comes from the Greek for a small quadrilateral tablet used as a token or a tally. It became applied to the small stones used in mosaics and from these became applied to tiles used for floors or to decorate walls of buildings. The most advanced tiling of this kind was done by the peoples of Islam, because their religion forbade them to use representations of people or living creatures for decoration; and the finest of all Islamic tessellations are generally accepted to be those in the Alhambra Palace at Granada in southern Spain. As the word is used today, a tessellation is a pattern made up of separate tiles, sometimes of one shape and sometimes of a mixture of shapes.

There has been an extensive study of tiling patterns in mathematics and numerous books and papers have been written. The bible of the subject is the magnificent "Tilings and Patterns" by Branco Grimbaum and G.C. Shepherd, (It may still be in print, although I was astonished to get mine as a remaindered copy) and the High Priest of the subject in Roger Penrose, the Oxford don and writer about artificial intelligence, who (in his spare time) was the discover of non-periodic tiling patterns. Penrose contributed to a big symposium on Escher at Rome in March 1985 and was one of the editors of the simply splendid book of Proceedings.

Tessellations in paperfolding are somewhat different. They are not made up out of separate tiles, but out of a single sheet of paper. There are two principal stages. First a pattern is drawn or directly creased on the sheet of paper. Although this is not made up of separate tiles, it can, nevertheless, regarded as the pattern of a composition of separate tiles. Secondly, the paper is creased in mountain and valley folds along the lines of the primary pattern, and when this has been done, the paper is "collapsed" into a second tiling pattern, which appears quite different from the first pattern. The second pattern is usually two-dimensional, but three-dimensional forms can be folded. This is what is generally regarded as the completed paperfolded tiling pattern or tessellation. It is remarkable, because it still retains hidden within its folds the original tiling pattern! There is another bonus, too. If the paper used is semi-transparent, a third rich shadow pattern is revealed when the paper is held up to the light.

The principal western folders investigating paperfolded tiling patterns today are Chris Palmer and Tom Hull of the United States and Alex Bateman of England. Chris is an artist, while Tom and Alex are mathematicians.

The originator of this kind of paperfolded tessellation was Shuzo Fujimoto of Japan. It appears to have been discovered by him in the course of his investigations into what he called "Twist Origami". (Although the "twist" itself in paperfolding appears to have been discovered by Yoshihide Momotani somewhat earlier and used by him in his earlier books on folding flowers.)

Fujimoto is a school mathematics teacher and a delightful and approachable person. His book, "Twist Origami" was home-made and reproduced on a copying machine in 1976. Many of the models are abstract and flower patterns and it is possible to see them verging towards true tessellations. In addition, however, there is one page of illustrations of tessellations of the fully-developed kind. More tilings, some of them three-dimensional, appear in Fujimoto's "Twist origami 2", dated apparently 1983. Yet more appear in "Solid Origami" a professionally printed paperback of 1976. Fujimoto's most extensive treatment of tessellations is in "Sozo Suru Origami Asobi", a hard-backed book of 1982. This is one of the small number of truly great origami books, but it's distribution in the West has been very limited and sadly, I understand it is now unobtainable, though I hope I'm wrong in this. All of Fujimoto's books are in Japanese. Fuimoto himself discovered how to disclose the hidden patterns in the folding by using semitransparent paper and holding the completed pattern up to the light.

Notwithstanding the prevalence of origami tessellations folded from a single sheet of paper, another kind of tiling has been used in paperfolding. In this individual shapes are folded and then arranged in a pattern in exactly the same way as tiles are fitted together to form a floor-surface. It may have been done before, but the example with which I am most familiar came about in this way.

In his younger days, before the Second World War, Maurits Escher, the Dutch graphic artist (1898 - 1972) visited the Alhambra Palace and was immediately fascinated by the Moorish tiling patterns which pervade the building. He studied them closely, and made copies. Subsequently he devised his own very imaginative pictorial tiling patterns, often building them into works of remarkable imagination. The vogue for Escher's work (which is still continuing) started in the early 1970s and in April 1975, the spring convention of the British Origami Society was held in London. Visiting Foyle's bookshop, I came across "The Graphic Work of M.C.Escher". I had never before seen anything like it and I promptly bought it. I was keen to show it to other members of the Society (at breakfast, I think!) and it was an immediate hit. Everyone who saw it was enthusiastic and Escher obviously appealed to the "origami mind". Mick Guy (now the president of the BOS), in particular was fascinated, especially by the periodic tiling patterns, and he began to reproduce similar patterns from folded paper. It was not easy to get the subtle angles right, but he achieved some success.

Then in 1976 the Arts Council of Great Britain held an important and impressive exhibition of Islamic Art in London and it featured charts showing the mathematical basis of Islamic tiling. About the time of the exhibition, the was a rush of remarkable books on the subject of Islamic patterns, some of which I bought. Mick Guy visited me and studied them closely to try to find new patterns which could be reproduced in paperfolding. So successful was Mick that he had several of his origami tessellations framed and hung them about his house in Birmingham. (I have one on my wall - Web Ed.) The became a regular feature of BOS exhibitions. So far as I know, nobody else took up this kind of folding with any great enthusiasm and eventually Mick, himself, tired of it.

Mick's tilings were, of course, very different from those of Fujimoto and his followers. But it would be a pity if this fascinating backwater of origami were to be forgotten. Someone might still like to take it up again. As for Escher, there have been numerous books. One of the best on his tiling patterns is "Visions of Symmetry - M.C, Escher" by Doris Schattschneider, published by Freeman in 1990.

David Lister Grimsby, England.

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