Diagramming tessellations

Dorigami asks (20/21 August 97) just exactly what is meant by "tessellations" and asks if what is meant is Escher-like constructions.

Escher-like constructions have, indeed, been created in Origami, especially by Mick Guy in the 1970s. Escher got his inspiration about tiling patterns from the Moorish tiling patterns in the Alhambra Palace in Grenada in Southern Spain. But tiling patterns made from sepate tiles are not what is meant in the present context of Shuzo Fujimoto, Chris Palmer, Tom Hull, Alex Bateman, Paulo Taborda Barreto et al.. It is necessary to distinguish between first, actual physical tiling laid on a floor or wall and second, the total abstract tiling PATTERN which can be analysed mathematically. A tiling pattern acts as a blueprint for laying out tiles on the floor. But in this context, the tiling pattern is studied in its own right by mathematicians and also used itself as a basis of folding by the Origami Tessallators.

To enlarge on this: in the Escher kind of tessellation, many separate paperfolds are made, like pottery tiles, (but usually in more complex shapes: sometimes abstract, sometimes in the shape of animals or fishes) and they are then fitted together physically to make a repetative pattern, just as you would lay tiles of one shape or different shapes on a floor.

In the case of Chris Palmer et al. the word "tessellation" does not relate to separate folded "tiles". It is used in the first instance for a regular PATTERN in the paper which resembles possibly a simple, but more often, a more elaborate tiling pattern. Quite apart from their interest to paperfolders, tiling patterns today form a major branch of recreational mathematics and, indeed, of more serious mathematics concerned with such subjects as that of symmetry, which pervades everything.. Such tiling patterns are usually regular, in which the same pattern repeats itself a periodical intervals over the whole pattern and indeed to infinity if the paper had no edges and could be extended without limit. But in some instances of tiling patterns, such as those discovered by Roger Penrose, the patterns are irregular and do not repeat periodically, (I may add that I, personally, have yet to see an irregular tessellation pattern in Origami. I doubt if it's possible, but I certainly wouldn't be dogmatic about it. Perhaps it's a challenge to the experts).

In this kind of paperfolded tessellation, the crease pattern of maountain and valley folds may first be drawn with a pencil on the paper. Or a network of parallel creases in several directions may made as preliminary guidelines. Then mountain and valley folds are made along the lines in the particular order required by the pattern of the tessellation being folded. Some people prefer to make their creases without preliminary drawn lines, but, except in very simple instances, this is much more difficult. Again, some people, like Paulo Taborda Barreto have discovered how to create their crease patterns by computer.

Having made creases across the whole of the sheet of paper, the pattern is then "collapsed", by folding in the respective valley and mountain crease, doubling the paper, making twist folds and generally cajoling the paper until it is reduced to a flat pattern. I think that "cajoling" is an apt word in this context! Nothing happens quickly. It is usually necessary to work inwards from the edges of the paper, but in my experience, one does whatever seems easiest and most appropriate in the circumstances. I can't pretend that I have ever found it easy, but I expect that practice makes perfect and I have never had the time to practise. Chris Palmer has certainly mastered the art.

Having managed to "collapse" the paper, you will have created your finished work of art. It will be seen that the creases you have made form a pattern of a NEW or secondary tessellation, a sort of bas relief which arises out of, but is different, often unexpectedly different, from the original pattern drawn on the paper before it was creased. Some folded tessellations are three dimensional, with such phenomena as boxes or mountains rising above the tessellated plain at regular intervals.

Then, if you have used semi-transparent paper, you can hold your creased pattern up to the light and you will see yet another hidden pattern painted in light and shade accoring to where the paper is doubled or trebled in thickness.

For more about tessellations, please see my posting to Origami-L dated 16th July, 97, where I give more general information and also say something about Maurits Escher and also Mick Guy's Escher-like tessellations. I added a little more, (mainly about Paolo Taborda Bareto) in my posting dated 19th July.

Tessellations are a long way from mainstream paperfolding, but they are a glorious aspect of the wonderfully diverse kaleidoscope that is Origami.

David Lister Grimsby, England.

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