Nonperiodic Tilings

I’ve developed quite an interest in both the math and the aesthetics of non-periodic tilings of the Euclidean plane. In particular, I find the substitution tilings of mathematicians Dirk Frettlöh and Lorenzo A. Sadun to be especially engaging. Frettlöh discovered the Pythagoras and Pythia tilings, and Sadun discovered a family of Pinwheel variants. After learning how to write a bit of Python code (thanks to the help of my friend Noah Weaver), I created some imagery in which I attempt to artfully illuminate the geometry of these tilings.

For example, here is an image of one of Sadun’s Pinwheel variants with parameters p=8 and q=9. I’ve applied the substitution rule 43 times to obtain this image in which triangles are colored according to their sizes (so there are a total of nine colors).

pinwheel(8,9,43)_final

As another example, here is an image of Frettlöh’s Pythagoras tiling with parameters m=20, j=1. I’ve applied the substitution rule 375 times, and colored the resulting triangles according to their sizes.

pythagorean(20,1,375)_finalsized

Here’s another Pinwheel tiling (p=1, q=3) with a color scheme that I really like.

pinwheel(1,3,9)_redorange copy

If you want to see a bunch of other nonperiodic tilings, visit the Tilings Encyclopedia.

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