Thursday, February 24, 2011


As in my previous post I'll borrow a term from math and science to describe what is entirely art. The term, randomness, is an interesting way to consider asymmetrical tilings like some of the scaling girih tilings. If I'm clear that this is neither math nor science I think I can suggest that aspects of this art are random.

I'll use the example of the diagram, below, with its eight large pentagons each divided by a tessellation of tiles from a common tile set. The large pentagons are identical in their boundary tiles, but their interiors appear randomly ordered. The pentagons can be filled with tiles, no gaps, edge-to-edge, in a number of asymmetrical and random-like patterns. The process of arranging the interior tiles is almost entirely non-random. There are initial random choices in tile selection for each, but the rest of the process of filling in the pentagon is driven by those choices and is a tedious trial and error attempt to find tiles that fit edge-to-edge leaving no gaps.

Putting aside for a moment the fact that the tile patches within each pentagonal boundary were carefully selected, I can say that a little randomness is enough. Following the fixed arrangement of boundaries, tile selection began with an arbitrary choice. The tiles do appear somewhat randomly arranged within. The limit of their scale relative to the boundary suggests that there are few such solutions, though how many is hard to say. So, we have eight examples of a closed system in a fixed state that at least started with some randomness, analogous to a system with very small but not zero entropy.

This highly systematic process appears free form because of the asymmetry. It's mostly not. Trial and error are necessary to complete any one pentagonal tile patch because I choose not to catalog all possible arrangements. The final diagram is thoroughly asymmetrical. In this case there may be as few as eight possible states. I could only know how many possible arrangements might exist by somehow proving that I've diagrammed all of them. I think there must be more than eight, but probably a small and finite number.

Now consider that the scaling tile set I've used could be extended infinitely. Some of the tiles could be replaced by smaller tiles, and still maintain the tessellation with no gaps and no overlaps. The elongated hexagons can be replaced, but not by infinitely scaling patches. The decagons can be replaced by three elongated hexagons and a bow tie. There are other possibilities, as well. Most close the process. However, the decagons can be replaced by rings of scaling trapezoids and closed finally with a small decagon. Also, the decagons can become scaling kites indefinitely, and with the rhombus the center gap can be closed at any point, preventing a singularity gap. As long as the tile set is infinite, and a decagon is included, the number of possible tilings is infinite. When to stop and close the final gap could be randomly determined. This potential for large entropy applies if the initial shape is simply a decagon instead of a pentagon, and the tile set is only decagons with scaling trapezoids and/or kites plus a small rhombus.

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