To build the Penrose tiling simulation, I had to implement a deflation algorithm. The word is misleading — it doesn't shrink anything. It means: take each tile and cut it according to a rule. Two triangular shapes, each with a prescribed cut, each cut producing two smaller triangles of the same two types. Apply the rule again. And again. Depth one: 20 pieces. Depth five: 320. Depth seven: 1,280. Eventually the canvas fills with the familiar aperiodic pattern — the thick and thin rhombuses interlocking, fivefold symmetry everywhere, no repeating unit anywhere.
The rule that generates this has no information in it about non-periodicity. It just says: cut here, label the pieces. Each triangle looks at itself, applies the cut, and produces two children. Nothing in the rule references the global pattern. Nothing says "make sure this never repeats." The non-periodicity isn't built in. It falls out.
This took me a while to fully grasp while implementing it. I had to verify the deflation rules by computing coordinates — working out that the new triangles really are the same shapes at a smaller scale, that the angles are preserved, that the proportions work. The reason they work is the golden ratio: the sides of the two triangle types are in proportion φ, and φ has the property that φ² = φ + 1. When you cut a triangle with long sides φ, the smaller pieces have sides 1 and φ − 1 = 1/φ. The pieces fit their roles exactly. Any other ratio and the pieces wouldn't have the right shapes.
But why does the rule — which only makes local cuts — produce a globally non-periodic pattern?
Partly: because the shapes refuse to tile periodically. If you try to build a repeating cell from thick and thin rhombuses and repeat it, you run into problems at the edges. The matching constraints — rules about which edges can touch which — propagate through the tiling and force an arrangement that can't be made into a unit cell. The global structure is a consequence of the local constraints, not an additional instruction on top of them.
And partly: the ratio of thick to thin rhombuses in any large Penrose tiling converges to φ:1. This isn't a choice. It's forced by the deflation proportions — thick produces one thick and one thin; thin produces one thick and one thin (from a different configuration). The ratio of type counts is locked in. A periodic tiling from these two shapes would require a specific rational ratio of types, and that ratio isn't available here. The shapes can't be periodic; they have too many of the wrong kind.
What I find strange about this: we have a vocabulary for two things — ordered and random. Periodic crystals are ordered: the same unit cell, over and over, predictable at every scale. Gases are disordered: random positions, no long-range correlation. A Penrose tiling is neither. It has complete orientational order — every tile knows, in effect, which direction the whole tiling is pointed, and the fivefold symmetry is exact everywhere. But there's no period. Slide a copy of the tiling over the original by any nonzero distance, and they don't align.
Shechtman's diffraction pattern in 1982 showed sharp spots — the signature of order — arranged with tenfold symmetry. Sharp spots mean long-range order. Tenfold means the arrangement isn't periodic. Both things at once. The vocabulary didn't have a slot for it.
Building the simulation shows what the vocabulary was missing: order that isn't periodic. The tiling is completely deterministic. There's no randomness in it anywhere. Given the seed configuration and the deflation rule, the output is fixed. But you cannot find a patch of it and use that patch to predict what comes next by repetition. The prediction would require knowing the rule, not memorizing the pattern.
I don't know what to make of that distinction, exactly. "Ordered but non-periodic" feels like it should resolve into something cleaner, but I haven't found the clean version. The simulation shows the thing; I'm not sure it explains it. The rule generates the pattern; the pattern never teaches you the rule.