## Sunday, February 6, 2011

### Stochastic

Here's another diagram, below, continuing the asymmetrical girh tile drawings that I described in my last three posts (Asymmetry, More Asymmetry, and Systemic). It illustrates another aspect of this system that I think is in contrast to the order of math art. That is, I'm using a stochastic process, arbitrarily selecting from available options which then determines what must happen next. I design an overall shape with its boundary tiles, but within the boundary (or in this case, six patches) I more or less randomly select some of the inner tiles. To keep the tiling edge-to-edge, I am limited to several choices from the full tile set. Each selection often necessitates that subsequent options are limited or determined, but predicting the outcome is difficult to impossible. The initial choices determine the options available for the rest of the process — filling the tile patch.

This particular drawing, with a central asymmetrical decagon patch and five radiating congruent but otherwise differently patterned hexagonal areas emphasizes the random nature of the process. I started with three scaled sizes of the five girih tiles, a narrow rhombus, and two scaling tiles. As tiles are added it becomes clear that there will always be a solution for filling the patches without gaps, but each selection limits the possibilities. Some choices that preserve the edge-to-edge force gaps that can't be filled, necessitating back tracking. The variety seen in the five hexagonal areas is a visual mapping of the possibilities. The common motifs around the decagons is conversely a clue to the limits of choice.