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cs.uwaterloo.ca

A follow-up to the “hat” aperiodic monotile: the “spectre” can tile without reflections

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I still find this stuff amazingly cool. It’s been Penrose tiles (kites and darts) for my entire life, and suddenly we have all these discoveries in aperiodic monotiles, all in a rush.

I have the sense that perhaps not everybody enjoys it so much, but ah well. I’m glad it’s around for those who do.

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Definitely, I had seen Penrose tiles like 20 years ago … but I didn’t realize there was even an open problem to “do better” once, let alone twice!

I’m not sure if I have the expertise to understand how that happened, but I’ll take a look

I took a quick look, and one interesting thing is that there are no convex aperiodic monotiles, but that wasn’t known until 2017! There was an exhaustive search published then

Little is known about limits on what sorts of shapes could potentially be aperiodic monotiles. Rao [Rao17] showed through a computer search that the list of 15 known families of convex pentagons that tile the plane is complete, thereby eliminating any remaining possibility that a convex polygon could be an einstein

Also the proof is like 90 pages, with a very large number of diagrams, and looks like a lot of work. You can’t tell just by looking if it’s aperiodic! :)

Even when a single tile admits periodic tilings, that periodicity may be more or less abstruse, in a way that offers tantalizing hints about aperiodicity

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oh, wow! thanks for this :D I should give that proof a read, it sounds really fun

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Yeah I realized that I don’t think the new monotiles are “one better” or “two better” than the classic Penrose tiles, mainly because of the convex/concave issue.

Subjectively, the convex pair of Penrose tiles still look nicer, and I wonder if there are “fewer” of them. It seems like there are probably many of the concave aperiodic monotiles, but the problem is actually proving that they’re aperiodic. Subjectively, they don’t look that special to me.

I would make an analogy to regular polytopes (if only because I like them)

• in 2D you have infinitely many: pentagon, hexagon, etc.
• in 3D, you have 5
• in 4D, you have 6
• in 5 and more dimensions, you have 3

https://en.wikipedia.org/wiki/Regular_polytope#Higher-dimensional_polytopes

So the 2 extra ones in 3D (dodecahderon and icosahedron), the 1 extra one in 4D (24-cell) are the most special and interesting to me! :)

I was interested enough to spend several weeks making this animation: https://www.oilshell.org/recurse/120-cell-bathroom.original.html

Also there are a small finite number of concave regular polytopes – those are also interesting to me, but I haven’t explored them much.

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The first one is equilateral too! I appreciated this; the reflections bugged me.

Still can’t get simpler than kites & darts if you wanted to manufacture something you could easily lay on a floor though.

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Be warned about kites and darts: the rhombuses themselves aren’t aperiodic unless they have control lines printed on them, where you have to line up the control lines. Here they are on the tiles outside some building in Oxford (where Penrose is): https://i.pinimg.com/originals/dc/5a/0a/dc5a0a7e51c51db73ec330005f4e7d01.jpg

So you can transcribe a penrose tiling using kites and darts, but they’re no good for trying to explore tilings by hand (I discovered this the hard way, after obtaining some rhombi). This is in contest to Hats and Spectres, which can really only be laid out one aperiodically.

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Q: is it possible to make periodic tiling with this? Is it hard or would it be likely if you’re just laying them randomly? Is it possible to get stuck when tiling, forcing you to backtrack to avoid overlaps?

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Q: is it possible to make periodic tiling with this?

No! Hats (the previous paper) and Specters (this paper) only admit aperiodic tilings. This is in contrast to Penrose tilings (either rhombuses, or kits&darts) which can be tiled periodically unless you add additional matching rules (lines drawn on top of the tiles that must match up across tiles). A quote from the original paper about this: “In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions.”

Is it possible to get stuck when tiling, forcing you to backtrack to avoid overlaps?

I suspect so? But very curious if anyone knows for sure.

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The base shape (with straight edges) is “weakly aperiodic”, which (in the terminology of the paper) means that it can tile periodically if you allow both the tile and its reflection, but must be aperiodic if you disallow reflections (but allow translation and rotation). The spectre variant has curved edges which prevent tiles from making a tiling with their reflections, so it is aperiodic whatever planar isomorphisms you allow.