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!
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!
@@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.
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!
Great explanation. In our remodeling we were considering using Penrose tiles. Then the hat and soon after spectre were discovered, now we are looking for a tile maker who'd make this for us.
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 :)
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?
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.
Who is the guy on the Mostly Mental videos on RUclips? I learned great gobs of stuff from his video on the Spectre tile. I would love to be able to email him, and bounce some ideas around with him. He introduces himself on his videos as Foxy or Doxy or something like that.
Hello, Foxy here. I'm just some guy on the internet who likes talking about math. My contact info is on my channel page, though I can't promise I'm very responsive.
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?
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.
The hat and turtle were both discovered on a grid of kites, which are really just two 30-60-90 right triangles glued together along the hypotenuse. So the ratio of sides is sin(60), or sqrt(3).
If monotiles cannot exist in 3d (which we do not know yet), I don't think the same would apply to 4d or any other nD where n=x^2, but I have no reason to believe this other than intuition
It's certainly possible, but I'd be a bit surprised. My guess is that either there are monotiles in every dimension (with possible exceptions of 4D, 8D, or 24D, since those are always weird), or there's some cutoff where it's possible for nD or lower but not (n+1)D or higher.
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!
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!
@@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.
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!
Great explanation. In our remodeling we were considering using Penrose tiles. Then the hat and soon after spectre were discovered, now we are looking for a tile maker who'd make this for us.
Time to redecorate the bathroom!
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 :)
@@ThePian0Man88 by fiat? You mean by only buying one of the two!
this channel is a hidden gem! thank you for this alpha
Glad you like it. Thanks for watching!
Great explanation of the proof!
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!!!!
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?
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.
Such high quality and so few views. Keep Going.
I'm glad you like it. Thanks for watching!
Who is the guy on the Mostly Mental videos on RUclips? I learned great gobs of stuff from his video on the Spectre tile. I would love to be able to email him, and bounce some ideas around with him. He introduces himself on his videos as Foxy or Doxy or something like that.
Hello, Foxy here. I'm just some guy on the internet who likes talking about math. My contact info is on my channel page, though I can't promise I'm very responsive.
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?
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.
@@mostly_mental Thanks, I get it!
how did you define the length rad3?
The hat and turtle were both discovered on a grid of kites, which are really just two 30-60-90 right triangles glued together along the hypotenuse. So the ratio of sides is sin(60), or sqrt(3).
🤯♾
Why is this shape named a Spectre?
I guess it looks kind of like a ghost when you add the curvy edges? Naming things is hard.
If monotiles cannot exist in 3d (which we do not know yet), I don't think the same would apply to 4d or any other nD where n=x^2, but I have no reason to believe this other than intuition
It's certainly possible, but I'd be a bit surprised. My guess is that either there are monotiles in every dimension (with possible exceptions of 4D, 8D, or 24D, since those are always weird), or there's some cutoff where it's possible for nD or lower but not (n+1)D or higher.
That's a t shirt not a hat