More about tiles

May 29, 2023 6:59 PM

A chiral aperiodic monotile

Further results from the world of aperiodic monotiling. - previously.
"The 'hat' aperiodic monotile resolves the question of whether a single shape can force aperiodicity in the plane. However, all tilings by the hat require reflections; that is, they must incorporate both left- and right-handed hats. Mathematically, this leaves open the question of whether a single shape can force aperiodicity using only translations and rotations."

Behold, The Spectre!

posted by thatwhichfalls (16 comments total)

I’ve been waiting for this… I found the “hat” flip issue troublesome, and it seemed to have been mostly ignored by the press.
posted by Stu-Pendous at 8:20 PM on May 29, 2023

I can't believe it turned out to be so obvious a derivative of the 'hat' ... okay well that halved the cost of my bathroom tiles then :-)
posted by nickzoic at 8:27 PM on May 29, 2023 [1 favorite]

Perfect update to the hat, I didn't expect it to be so simple either. I'll be playing with theses shape this week!
posted by crossswords at 8:38 PM on May 29, 2023

yessssss
posted by cortex at 9:22 PM on May 29, 2023 [2 favorites]

And I love Joseph Meyers' perfectly coherent description of it as a "vampire einstein". Do not worry about your difficulties with mortality; I can assure you that mine are still greater.
posted by cortex at 10:10 PM on May 29, 2023 [2 favorites]

What? Where does the term "vampire einstein" come from?
posted by straight at 10:19 PM on May 29, 2023

Vampires have no reflection.
posted by pmdboi at 10:57 PM on May 29, 2023 [3 favorites]

also: ein-stein, a single tile that can cover the plane without periodic repetitions.
posted by k3ninho at 11:10 PM on May 29, 2023 [1 favorite]

Now I have to figure what size mold to draw/make using 1/2 a bag of cement, 10 cm thick.

It's late, good project for tomorrow.
posted by Marky at 12:14 AM on May 30, 2023 [1 favorite]

Huh, one of the authors is Craig S. Kaplan, creator of Slide to Unlock (previously).
posted by wjt at 1:01 AM on May 30, 2023 [2 favorites]

Feels like we’re a step closer to discovering ice-nine here somehow.
posted by Ishbadiddle at 2:44 AM on May 30, 2023 [3 favorites]

Huh, one of the authors is Craig S. Kaplan, creator of Slide to Unlock (previously).

Craig Kaplan does a lot of really cool work in the mathematical art/recreational mathematics spaces. He was on the original monotile design team (which is the exact same group of people, unsurprisingly since the spectre is a development of the same process which made the hat), has worked on finite tiling processes (previously on Metafilter), aperiodic bobbin lace, reconfigurable mazes, generative methods for traditional mosiac patterns, and a lot more.

He also made the design I purloined (with his permission) for my wedding ring, and is higher up than I am in the administration of pretty much every mathematical-art organized space I've ever had the honor of being involved in.
posted by jackbishop at 5:58 AM on May 30, 2023 [4 favorites]

Every tiling by Spectres is closely related to a tiling with a sparse distribution of hats lying within a dense field of turtles, and one with a sparse distribution of turtles lying within a dense field of hats.

I know hats and turtles and spectres are actually just tiles shape names, but this mental imagery is delightful.
posted by ananci at 6:21 AM on May 30, 2023 [1 favorite]

Now I'm curious about how far you can take "sparseness". Can you design a pair of tiles so that one of the pair is still required but arbitrarily sparse, like only one hat for a million or a trillion turtles, but you still have to have those hats?
posted by moonmilk at 7:11 AM on May 30, 2023 [1 favorite]

It's hats, spectres and turtles all the way down.
posted by rouftop at 12:50 PM on May 30, 2023

> Now I'm curious about how far you can take "sparseness". Can you design a pair of tiles so that one of the pair is still required but arbitrarily sparse, like only one hat for a million or a trillion turtles, but you still have to have those hats?

My intuition is that the packing constraints of a single "sparse" seed can only propagate through about 2 layers of "dense" tiles before they either force another sparse one or allow unlimited dense ones. So it would be very hard to reach a ratio of more than 20:1.
posted by Phssthpok at 11:39 AM on May 31, 2023