I have long been fascinated by tessellations as metaphors for ways of knowing about and being in the world. A set of prototiles that can cover a world in an exhaustive and mutually exclusive way constitutes something like a theory of that world. The rules of the tiling are the rules of the world. The set of prototiles is the ontology underlying the theory. The size of that set is a measure of the efficiency of your understanding. Recognitions of repeating patterns in the emergent tiling are understandings of specific aspects of the phenomenology of the world. Actually creating a specific tiling by placing tiles on a smooth surface, to create navigable striations, is the praxis of the way of knowing.
To state it in terms of my new favorite frame, tessellations are something like metaphors for protocols of knowing and being. Given the right set of tiles, you can know the world and be in it, in a powerful way. Perhaps this is one way to understand the story of Robert Moses, architect of New York. He tessellated his world with tiles of his choosing.
Ideally, you want the richest, most complex tiling possible to cover a “blank” world, such as the 2d Euclidean plane, to maximally reveal the possibilities latent within it. Yes, you can cover the Euclidean plane with a boring regular grid of square tiles, but you can also cover it with strange aperiodic tilings, and in some ways, the latter constitute a truer “theory of the plane.” The intuitively appealing principle that you should look for the richest possible tessellation is a kind of dual to Occam’s razor. Instead of choosing the simplest explanation that covers a given world of facts, you choose the covering that produces the most complex world of facts. Ideally, the maximally complex set of facts. Instead of solving for explanatory parsimony, you solve for generative profligacy.
One proxy for such maximality is Turing-completeness, and at least some (all?) aperiodic tilings, like Wang tilings, are known to be Turing complete. Jed Yang published a PhD thesis in 2013 about the computational aspects of tessellations, and also connected tiling-based computation to Turing-complete cellular automata, such as Wolfram’s Rule 110. Googling around, I also found this fascinating presentation by Kathleen Lindsay about playing Conway’s Game of Life on an aperiodic tiling. It seems like tilings and cellular automata are equivalent ways of understanding universal computation, and at least to me, these spatial processes seem more intuitively appealing than infinite tape machines or the lambda calculus. And of the two, I think I prefer tessellations over automata, since the computational process is embedded in the texture of the space itself, as opposed to a 0/1 switching process playing out on it.
As you may have guessed, aperiodic tessellations have been on my mind lately because last week the first aperiodic monotile, the “hat” (an “Einstein” tile, named for the German ein stein, or one stone, rather than the physicist) was discovered. It is not quite a monotile since you have to use it along with its mirror image to aperiodically tile the plane (the blue vs. yellow instances in the picture below), but still, this is a fascinating leap. The last best attempt, the class of two-tile solutions known as Penrose tiles, seemed like the End of History of Tessellations to me, but apparently we had a chapter left. I suspect this is the end though. I somehow doubt we’ll get it down to a single kind of tile without the mirroring cheat (I wonder if anyone has proved that a single tile, without mirroring, cannot tile the plane aperiodically).
Recreational mathematicians are going a bit nuts with this discovery, and I’m 3d-printing a set to play with (I’m using this model) as we speak.
I’m especially intrigued by the idea of painted kites. The hat can be decomposed into 8 kite shapes that can be “decorated” in a way that the resulting hat tilings create strange aperiodic maps. These feel exciting in the same way the original pictures of the Mandelbrot set felt exciting in the 1980s. Unfortunately, my 3d printer is single-nozzle, so I can’t print these easily.
While I’m nerdsniped by these tilings and painted tilings, I’m not enough of a mathematician to truly explore them in any technically deep way. But I suspect that after the late J. G. Ballard, whose complete short stories I just finished, I’m quite possibly the person who has spent the most time thinking about tessellations as world-narrative metaphors, so let me talk about that instead.
The goal: get to post-Ballardian ways of thinking about our End of History condition, via aperiodic tessellations (plus noise).
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