Here are two more diagrams, below, continuing the asymmetrical girh tile drawings that I described in my last two posts (Asymmetry, and More Asymmetry). I've stated that they defy the math art penchant for symmetry and quantification; and, that these drawings were the result of a search for ways to vary scale and density with tiles. I should also add that the math involved is relatively simple. I've needed little more than Euclidean geometry, basic trigonometry, and an extremely simple application of self-similarity. It's convenient in the programming to use matrix mathematics, but it's not essential. I regularly ignore practices that mathematicians would prefer, like designing tilings to fill the plane and eliminating gaps.
With this project I'm searching for knowledge, not so much refining objects. As the project is underway, I don't spend a lot of time adjusting or working with the minutia. I record the project as a web page, the de facto communication tool for our times. Eventually I'll turn the files over to production, in glass and possibly with computer numerical control. In one case the refinement will be accomplished by another artist, and in the other, a machine.
So, a key aspect of this project is that it's a system that could be reproduced by anyone with the capability to create slightly complex tile sets. I've suggested everything needed to generate endless variety. I happen to have borrowed from Islamic art the girih system. My contribution and evidence that something worthwhile is happening here is that I modified girih tiles to get the scale and density changes that I felt were needed. But we're not limited to girih style decorations. Given edge-to-edge tilings, with defined vertices, we could create other patterning methods. Besides being a search for knowledge, this can be an open-ended path for further investigation.
The two diagrams below, with their underlying tilings, emphasize the projects systemic capability with their asymmetry. The underlying tilings are based on the proposition that we could arrange tiles in endless ways, but still create planned, ordered overall shapes. If the system can do this with asymmetry, then it's flexible and forgiving.