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# Unbranched Sequences in the Tuckerman Tree

There is now sufficient material to think of flexagons as fractal polygon stacks. This word fractal seems to be a fairly recent invention of Benoit Mandelbrot, and for a few years when microcomputers were still a novelty, they were used to graph a kind of fractal gotten by repeating a quadratic polynomial over and over again. By the novelty has fairly well worn off, but when it was rampant you could see copies of ``Julia Sets,'' or more commonly, ``Mandelbrot Sets,'' on the covers of magazines and in science oriented articles everywhere.

What he meant by fractal was something, usually spectacularly irregular, that was gotten by repeated substitution, usually at smaller scale, of a whole structure or design into selected parts of itself. But that is just the process by which flexagons can be built up; replacing individual polygons by spiral stacks insertable in place of the original polygon. It took quite a while to come to this realization, but there is no reason that it had to wait for Mandelbrot; he simply gave a name to what had previously been known as recursion or induction. But the application is more exciting than the places where induction is usually used, which is to prove mathematical theorems.

To get a description of flexagons we have worked through three stages. First, some simple but nevertheless interesting geometric structures were exhibited. There is a historical account of how they were first found by folding strips of paper, although the formation of spiral stacks and unrolling them against each other came quite a bit later than just making wandering friezes and folding them back and forth.

An induction needs both a base and a rule of advancement. These spirals form the base, especially the shortest ones which will both close but also lay out flat in a plane as part of their manipulation. So the second stage, which reveals the continuation principle, is to replace an individual polygon by an equivalent, albeit more complicated, structure. The essential ingredient in the spiral is the turning angle, which is the same by going forward one step or backwards n-1 steps, n being the number of vertices (and hence edges where folding is done) of the polygon.

The third stage can be as complicated as one needs or desires; it just consists in making the substitutions over and over again. For all this it is convenient to make a map that will show where, and how many, replacements have taken place as well as their relation to one another. But mapmaking also breaks down into three parts; rather, the same map can be drawn in three different styles, each of which reveals some properties of the flexagon better than the others do.

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Next: Mapping the flexagon construction Up: A Quick Flexagon Survey Previous: Tuckerman Tree   Contents
Pedro Hernandez 2004-01-14