Honestly for all the people who said that reflections shouldn’t be allowed, the middle shape Tile(1,1) fits the bill perfectly for an aperiodic monotile (disallowing reflections). I can’t wait to see these tilings out in the wild!

One thing that strikes me here if I am understanding it correctly is that you have this kind of spectrum in which the hat side consists of a continuum of hats between chevron and spectre and the turtle side consists of a continuum of turtles between spectre and comet with all these intermediary tiles having the property of tiling the plane only in an aperiodic fashion but involving a certain number of reflected versions (proportion phi to the fourth power) with the spectre in the middle which can be tiled periodically or periodically but which heads a family of spectre mutants having chiral figures taking the place of its non-chiral line segments. This infinite family of spectre mutants has the property of tiling the plane only periodically. Cool!

@carlkuss

6 ай бұрын

Also: that the hat in the strict sense, like the turtle in the strict sense, is constituted by an assemblage of parts that you get from the (periodic!) tiling of the plane into hexagons with their corresponding dual (tiling in triangles). So that the whole aperiodic tiling is there in your face amidst this simple (periodic) tiling into hexagons and triangles. If I am right about that. Weird!

@mostly_mental

6 ай бұрын

@@carlkuss Yeah, you've summed it all up pretty well. It's wild that stitching triangles and hexagons together is all you need to get so much complexity.

Time to redecorate the bathroom!

@ThePian0Man88

Жыл бұрын

You don’t even need to do the squiggly spectre shape for the floor! If you do Tile(1,1) and just as the paper says “by fiat” don’t reflect then you can tile your bathroom aperiodically :)

@josephrhoades8113

Жыл бұрын

​@@ThePian0Man88 by fiat? You mean by only buying one of the two!

this channel is a hidden gem! thank you for this alpha

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

The next big question is who is going to start selling boxes of hats, spectres, and turtles in pleasant colors for the kitchen and bathroom!!!!

Great explanation of the proof!

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

Personally I'm really curious to see if there are any aperiodic monotiles whose tilings have rotational symmetry. One of the cool things about other aperiodic tile sets is they have degrees of rotational symmetry you can't get in periodic tile sets - penrose tilings with fivefold symmetry, or ammann-beenker tilings with eightfold symmetry. As far as I've seen, hat/turtle/specter family tilings don't have any symmetries. I wonder if that's just a quirk of that specific tile family, or if it's a general fact about aperiodic monotiles?

@landsgevaer

Жыл бұрын

If you start from a propellor, with 2pi/3 rotation symmetry, and inflate that arbitrarily often, you must end up with an arbitrarily large (i.e. infinite) tiling that has that rotation symmetry. Analogous to what penrose tiles do for 2pi/5.

There is a crystallographic class callled the triclinic, which is defined as having no symmetry. When I studied mineralogy years ago, there was only one mineral known, called axiinite, that crystallized in this class. Is it possible that the molecular units of this mineral are 3-d aperiofic monoitiles?

@mostly_mental

11 ай бұрын

That's a good thought, and you're on the right track. By definition, true crystals always have translational symmetry (and are thus periodic). Triclinic crystals are periodic in three non-orthogonal directions at three different distances, which is the least symmetry a true crystal can have. But there's a class of minerals called quasicrystals that have aperiodic structure. The whole field of quasicrystals is relatively new, so there are only a handful of known examples. And a quasicrystal only gives us a lattice of centers, so it would still take some work to construct the tiles that fit around them. But for anyone interested in exploring 3d aperiodic tilings, that's a promising place to start.

@user-ld4bt4mf8g

11 ай бұрын

@@mostly_mental Thanks, I get it!

Such high quality and so few views. Keep Going.

@mostly_mental

10 ай бұрын

I'm glad you like it. Thanks for watching!

🤯♾

Why is this shape named a Spectre?

@mostly_mental

Жыл бұрын

I guess it looks kind of like a ghost when you add the curvy edges? Naming things is hard.

That's a t shirt not a hat