Sunday, February 13, 2011

Filling the Plane

The drawing below has a orderly border that could be extended to fill the plane. It requires just three girih tiles at the same scale to continue the pattern up-down and left-right. Yet, this pattern converges in the center to asymmetrically arranged girih tiles (plus three) scaled down, twice. This is the closest I've come to demonstrating that the scaling girih tiles are capable of filling the plane asymmetrically. The only method I know of to create such a drawing is by trial and error. The possibility that any of these patterns might fill the plane defies math — you'd have to complete an infinity of trials and corrections for error to accomplish the task. There's no simple or recursive definition at work.

It's interesting that this system of scaling girih tiles can support a creative system of trial and error. It doesn't depend on math nor symmetry, but allows for arbitrary choice without precluding resolution. This drawing began with ten decagons surrounded by common patterns, like seed crystals, but arranged by whim. The only limitations were the fixed tile set of eight tiles times three scales, and the requirement for the result to be edge-to-edge, no gaps, no overlaps — a tessellation. These ten tile patches were surrounded by two levels of scaling tiles then girih tiles to the grid-like border decagons — the solution.

It's mildly interesting that the three scales of girih tiles in this drawing have sides in the ratio of 1.6180339. . . , the golden ratio. This of course falls out from the fact that we're dealing with pentagons.

I'm more interested in what I can create with a system of scaling tiles than the math, which is fairly simple. Still, it occurs to me that there should be some mathematical interest in such a system that accomplishes a task like filling the plane, but requires endless trial and error. If it's extended to three dimensions is it a model of crystal discontinuity? Are there any natural systems that continue even infinitely through a constant correction like trial and error?

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