Explore Quasi-Periodic Tiling

December 1, 2022 9:39 AM

Pattern Collider is an interactive tool to explore Penrose (and other "quasi-periodic") Tilings. (here's another webtoy) minutephysics on YT: Why Penrose Tiles Never Repeat.

Numberphile: 5 and Penrose Tiling (7mins)
Veritasium: The Infinite Pattern That Never Repeats (21mins)
The trouble with five (article)

Previously: The Tiling Database, Penrose Tiling Quilt

posted by gwint (12 comments total)

Very nice!
posted by brambleboy at 9:55 AM on December 1, 2022

Fun fact, the question of whether there is an aperiodic set of size 1 in the Euclidean plane is known as the Einstein problem, for ein stein = "one shape", not to be confused with Albert.
posted by allegedly at 9:56 AM on December 1, 2022 [2 favorites]

I'm unclear on whether global fivefold rotational symmetry or reflection symmetry are avoidable in a Penrose tiling. Can anybody point me to a construction method that guarantees nonexistence of either or (preferably) both?

All the example tilings I've seen have got both these symmetries, which those presenting them usually describe as beautiful, but I see them as really annoying flaws. If I'm ever going to Penrose tile my bathroom, I'd prefer that the pattern I'm making would never have either, regardless of how far it was extended.
posted by flabdablet at 10:16 AM on December 1, 2022 [1 favorite]

So, are there 3D Penrose shapes, or quasicrystals?
posted by newdaddy at 10:16 AM on December 1, 2022

Oh, and if I can break all the local fivefold rotational symmetries as well, so much the better.
posted by flabdablet at 10:20 AM on December 1, 2022 [1 favorite]

The ? link leads to an explanation of how it works. The magic is de Bruijn's 1981 algorithm for generating quasiperiodic tilings via grid intersections. I had no idea such a thing was possible! What a nice interactive demonstration of the method.
posted by Nelson at 10:23 AM on December 1, 2022 [2 favorites]

I've dreamed of making something like this for years and am only frustrated by how snazzy the site it. Penrose tilings have real weird properties and I remember really liking this 2006 thesis on how a small patch of tiles can force the positions of distant tiles.

Thanks for the link!
posted by crossswords at 12:48 PM on December 1, 2022

I did a science fair project on quasicrystals in junior high because of a Scientific American article about them (sometime in the late '80s) which featured a gorgeous tile floor of Penrose tiles from (iirc) a church somewhere.
posted by joannemerriam at 2:46 PM on December 1, 2022

Oh, this is very neat. I've been familiar with Penrose tilings and quasicrystals for a while now, but I had never caught on to the underlying grid intersection stuff from de Bruijn, which is just eye-opening. The notion that all these Penrose-like variants are in fact strictly determined from the underlying grid's intersections and angles thereof is just fuckin' magical.

I'm sure I've read about the proof-by-irrationality detail for why Penrose tilings are inherently non-repeating over translation, but it apparently hadn't stuck before because that was another a ha moment going through the links.

Still makes my eye twitch a little that patterns with rotational symmetry are described as not being periodic, but I know it's a definitional thing about translation symmetry and definitions matter. Quasiperiodic as a nod is enough to make me shut up about it but *shakes fist*
posted by cortex at 7:30 AM on December 2, 2022 [1 favorite]

The rotational symmetry annoys me too. If you fiddle the rotation and pan sliders a bit you can move the center of symmetry off-screen.

One thing I'm curious about is how big a subset of the space of all non-periodic tilings the De Bruijn construction gives. There's plenty of ways to lay down Penrose-shaped tiles that are not quasi-periodic, part of the Penrose construction is selecting a subset of all tilings that ensures non-periodicity. The De Bruijn method seems to result in an even more restricted subset of all possible tilings. How big of a subset is it? (Infinitessimally small is my guess but doesn't capture the intuition of what I really want which is "does this method make things that look too same-y?")
posted by Nelson at 8:15 AM on December 2, 2022

That music: I wonder about its generation.
posted by the Real Dan at 9:20 AM on December 2, 2022

Wish I had found this before I posted, but going to add for historical purposes:

Tilings Encyclopedia!
posted by gwint at 7:19 PM on December 2, 2022