Some corrections: - Tile(1, 1) is not an aperiodic monotile. It's an exception because a brown edge of one tile can be placed next to a green edge of another, which is impossible with all the other tiles. (Thanks Scigatt)

It should be noted that Tile(1,1) has a periodic tiling.

@mostly_mental

Жыл бұрын

Good catch! Can't believe I missed that.

@bryanbischof4351

Жыл бұрын

Why doesn’t this contradict the second proof?

@mostly_mental

Жыл бұрын

@@bryanbischof4351 There's an implicit assumption that we put green edges next to green edges and brown edges next to brown. But if the lengths are the same, you can put a green edge and a brown one together. So Tile(1,1) can be arranged in ways none of the other tiles can.

@Scigatt

Жыл бұрын

@@bryanbischof4351 'Long' and 'short' edges equal --> 'long' and 'short' edges can meet edge-to-edge --> more arrangements possible, including a periodic tiling.

@AmberSZ

Жыл бұрын

@@mostly_mental Does that mean if one length is a multiple of the other there could be a similar problem? Like Tile(1,2)

Imagine how cool it would be a random map generator for games using this! Incredible!

@mostly_mental

Жыл бұрын

That could be interesting. I know Wang tiles are used for random generation, so I imagine you could probably do something similar with hats.

I found your channel earlier this week - you are now by far my favourite math creator on the site. Keep up the amazing work!!

@mostly_mental

Жыл бұрын

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

I would love to see your presentation on more of the fine details about the paper -- yes, I read it when it came out, but your presentation of the data is always very visual and clearly sectioned. Great work dude, as always!! It makes my day to see you post :D

@mostly_mental

Жыл бұрын

Thanks, I really appreciate it!

This quality is so high for such a small channel, amazing video!

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

Thank you, this is one of the best explanations right now 😊

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

0:24 and i’m already subbed because your hand motions are so graceful

@mostly_mental

10 ай бұрын

Glad to see all those hours practicing magic have paid off. Thanks for watching!

This is beautiful, and a very clear explanation. Thanks!

@mostly_mental

Жыл бұрын

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

oh i am THRILLED seeing all my favorite math youtubers bring up the topic, but i think you may be the first so far!! fantastic fantastic fantastic!!!

@mostly_mental

Жыл бұрын

I'm glad you like it. And yeah, I'm honestly shocked none of the big names beat me to it (even if Numberphile did sneak a few hats into a graphic on their last video).

This is a wonderful presentation of the paper! I was struggling to understand the Tile(a,b) method for proving aperiodicity when reading the paper, but your explanation was very clear. Easy subscribe.

@mostly_mental

Жыл бұрын

Glad you liked it. Thanks for watching!

Awesome👏🏼👏🏽💯

Good explanation.

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

This is awesome! Just what I needed. I wanted to be able to look at a dot pattern and discern the location of it. Armed with an aperiodic monotile I can do this. Thanks!

@mostly_mental

Жыл бұрын

Glad you like it. Thanks for watching!

yippeee a video about the hat !! and explaining the details of the proof ! ill be honest. i couldnt fully understand it by just reading the paper. so this is a really nice video :) 816 !!

@mostly_mental

Жыл бұрын

Glad you liked it. Thanks for watching!

Love it! Kinda reminds of the 4 primary iterations of cellular automata Professor Wolfram discovered in NKS. ^.^

@mostly_mental

Жыл бұрын

Thanks for watching! And now that you mention it, I kind of see the resemblance.

This was an interesting programme on tiling the surface. A passing mention of yours recalled my mother, Mrs M.A. Ploum-Coopmans (1922-2020), who drew geometric designs from the early 1970s onwards until the late 1990s, or maybe early into the new millennium, . After she had once been asked to draw macramé knots, for talks she was asked to give, she became interested in drawing knots in their own right, then knot shapes before moving onto geometric shapes generally, using graph papers of different geometries as backing. As some point she also investigated pentagons and discovered different ways to tile the surface, ie without leaving a gap, using sheets of A1 paper sellotaped together as verification of any cyclical repeats (or not). These designs were complex patterns, well beyond the four pentagon divisions of an hexagon, a design that is symmetrical along the "x" and "y" axis cutting through the middle of two opposite sides on , say "x", and the apexes on two other opposites on "y". I have kept all the designs she drew on small sheets as on large, as I plan to transfer them from paper to computer files, since some of the sheets have become brittle. The drawings were always fascinating as she never knew where those lines might lead her. She organised a couple of exhibitions near Grasse in France and also used some of her designs for Christmas cards...

@mostly_mental

Жыл бұрын

Those sound really cool. I'd be interested to see them whenever you upload them.

@thierryploum5923

Жыл бұрын

It may take a while, as I improve certain aspects of my own life; and yet I do think it important for her ideas to be better known since some of her designs were very inventive.

I appreciate this version of the argument about lattice lengths in the chevron and comet tilings! It was easier for me to follow than the version in the paper. To my understanding, the paper gives that argument as a proof that the mapping between the two tilings cannot be a similarity, but doesn't rule out other possibilities. It then gives a second argument using some additional facts about the supposed periodic tilings to show that the map would have to be a similarity, which you don't cover in the video (fair enough, it's pretty long and technical). Is the fact that the mapping would have to be a similarity implicitly assumed in the version of the proof you give here? Or is the proof somehow refined to not require it?

@mostly_mental

Жыл бұрын

Yeah, I left that implicit. I was focusing on the intuition, so I glossed over a lot of details needed to make these arguments rigorous.

I always wondered what the first set of 20426 tiles would look like drawn out.

Thanks for asking the questions at the end. Can´t wait to hear more about this topic. This can´t be the end. Or will there be a alrge time spane where nothing happens again, who knows

@mostly_mental

9 ай бұрын

Well, there was a followup paper a few months later. The same team found an aperiodic monotile that doesn't need to be flipped over to tile the plane. Video here: kzread.info/dash/bejne/ZXesqI-Am87Qmbg.htmlsi=PtLXR4nAujCJeCwS Now we have the new techniques from these papers, and there's been a huge surge of interest in the problem. I'd be shocked if there aren't more exciting results in short order.

@heralddobre8114

9 ай бұрын

@@mostly_mental So, one question still open would be a shape with less "sides" or more sides or "zero" sides. Seems to me with no knowlege whatsoever you´d have to "fuse" two or even more repeating repetative tilling shapes together. And the questions you have mentionend in the follow up video.

minerals do this, there is a constrain on the growth of the edges as they compete for resources at the same temperatures and pressures until they meet each other. The numbers could be considered proportional to the forces place on the growth edge relative to the distribution of the elements in the solution. Its quite a problem to work out why certain patterns start. Minerals are the crystals are the tiles, rocks are the tiled surface, the edges are kind of like vectors perpendicular to the resource density, so like the edge with the most elements/resources available in the correct ratios needed to grow the grain/crystal with get bigger faster. depending on the shape of the initial crystals/tiles the subsequent grains have less degrees of freeedom and make these meta patterns which we call textures. Some grains can eat others, but first they eat the liquid. you get the same sort of arrangement, of these touching ones that are long and the same and then smaller one that fit in between, if you have resorbtion then the longer one will be more if you have rapid cooling or many nucleation points the theory goes you get more of the single type tiles/grain/cystals (e.g. deep granophyric granites with poikiolitic textures vs near surface dolerites which are mostly fine grained). Anyways if you considere the edges as a surface then maybe is has some applications to understand mineral growth.

@mostly_mental

Жыл бұрын

I wouldn't be surprised if there are physical applications. I know aperiodic tilings are closely connected to quasicrystals, so this result will probably lead to some interesting research there, too.

Hi Mostly Mental, can you please explain why this t-shirt (not a hat) is considered one shape, and not two? If you have an asymmetrical shape and you turn it around it is a different shape. Imagine you buy these new tiles and the manual tells you to lay one tile face down, would you be happy to fill your infinite floor?

@mostly_mental

Жыл бұрын

That's a good observation. It comes down to what we mean when we say tiles are "the same". In this case, we mean the tiles are congruent, and the t-shirt qualifies. Now, you might reasonably say that's the wrong definition of "same", and you want a tile that uses only rotations with no reflections, in which case, you've stumbled across an open problem. Find me a tile like that and I'll be happy to feature it.

Are the hats ordered in space except for rotations? I guess I am thinking of crystal symmetry in physics. There can be a difference between long range translational order and rotational order of the elements.

It’s a T-shirt, I can’t believe they said hat over T-shirt

another question, do you know where can i find something more specific about the "ein stein" problem that the hat broke? an article talking about this problem or something? thanks :)

@mostly_mental

7 ай бұрын

"Einstein" is just another name for "aperiodic monotile". "Ein stein" literally translates to "one stone", so it's bit of a play on words.

hey, i would really appreciate if you could share your references for the aperiodic tiling with trapezoide tile demonstration, starting at 1:34. thank you!

@mostly_mental

7 ай бұрын

strauss.hosted.uark.edu/papers/AHT.pdf describes the whole process in a bit more depth.

@mateussteffler

7 ай бұрын

thank you so much man! :)

The example presented from 1:33 to 3:44 may be considered periodic if you don't consider the trapezoidal shape, but the hexagonal shape it forms instead... or am I missing something? Is there is reason to only consider the trapezoidal shape of the "super shapes" and not their hexagonal organisation?

@mostly_mental

Жыл бұрын

You're right, I completely screwed up. I modified another proof to make it easier to explain, and I broke it in the process. Thanks for catching that.

@andrewkepert923

Жыл бұрын

No I don’t think you screwed up. Yes the tiling you have sits inside a periodic tiling (regular hexagons) but the actual tiling of traps is aperiodic. There are two ways to understand this (more probably). The discussion of the labelling via the hierarchy of larger + larger traps gives labels each tile with a sequence based on its relationship to the rest of the tiling. This is unique, or has at most finitely many tiles with the same labels (eg you can tile inside a 60 degree wedge based on your 4-fold replication rule which sort of breaks things, but not too badly - you just need 6 copies of this). So there is a slight oversight in the claim that you can always find a common ‘ancestor’ tile of any two tiles, but this is not a big deal - it can be fixed. The second way is just to focus on the hexagonal grid and the pattern of ‘splits’. It should not be surprising that the orientation of these splits can accommodate aperiodic tilings. I could use them to encode digits of π if I wanted. So the hexagonal grid is not a reason to discard this example.

@mostly_mental

Жыл бұрын

@@andrewkepert923 Yeah, I think I've been staring at this diagram too long, and I'm starting to see patterns that aren't there. You can definitely recover the hierarchy, so it can't be periodic. It just looks it because I didn't draw the grid out far enough.

@andrewkepert923

Жыл бұрын

@@mostly_mental we've all been there. :-)

This is a good explanation for why the hat’s tiling is aperiodic, but what I don’t get now is why all attempts at a periodic tiling using just the hat will fail?

@mostly_mental

Жыл бұрын

Say you have some tiling with hats. Then you can find two corresponding tilings, one with chevrons and one with comets. If the original tiling is periodic, then both of the new ones will be two, and they have to repeat at the same distance. But by looking at the grids that the chevron and comet fit into, we see there is no distance where they can both repeat. So the tiling with hats also can't repeat. Which means that tiling isn't periodic. And since we can do this for *every* tiling with hats, we see that every tiling will be aperiodic.

will we get part 2? where Tile(1,1) is suddenly aperiodic when its reflection isn't taken, so it's an actual "true" (chiral) monotile? (with slight wiggle in edges removing periodic tiling with mirror copy)

@mostly_mental

Жыл бұрын

Yes, that video's currently in production. It should be out soon.

When the hat first appeared, I thought it was just a variation on hexagonal (and therefore triangular) tiling. Are there more aperiodic variations of hexagonal/triangular tiling?

@mostly_mental

Жыл бұрын

The hat is a polykite made of 8 kites, and Tile(sqrt(3), 1) is also a polykite, made of 10 kites. The authors ran a computer search and found those are the only two aperiodic polykites with up to 21 kites, but there may be some with more.

My biggest question is why they would call it a hat 🎩 when it's clearly a t-shirt 👕

@4:21 you say, "these both use more than on tile..." but the middle section only shows 11 squares with coloured bits, are these all the same size and shape square tiles with patterns painted on them? meaning that again, the orientation to neighbour property comes in to play? thanks, a confused viewer.

@mostly_mental

Жыл бұрын

Yeah, I see how that could be confusing. For those tiles (known as Wang tiles), neighboring tiles need to match colors. They also can't be rotated or reflected. It's a different model of tiling than everything else I was showing, so I didn't want to draw too much attention to it.

interesting that the side lengths are all like the proportions of the sides of a 30-60-90 degree triangle...

@mostly_mental

Жыл бұрын

That's not a coincidence. The kites I mentioned at 5:10 can be seen as two 30-60-90 triangles glued together along the hypotenuse.

Is there any 3 dimensional analogue to these tiles?

@mostly_mental

Жыл бұрын

There are a few known tiles that work, with some caveats. The disconnected Socolar-Taylor tile I showed actually has a connected 3D counterpart (en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile ), but that's only weakly aperiodic. And there's a Schmitt-Conway-Danzer tile (en.wikipedia.org/wiki/Gyrobifastigium#Schmitt%E2%80%93Conway%E2%80%93Danzer_biprism ), but that can have screw symmetry. As far as I know, no one's discovered a strongly aperiodic monotile in 3D.

3:10 Wouldn't the same applied with four scare making a bigger one? Yet, a tilling of scare is periodique, but it will have a way to put a coordonate different in each one, wouldn't it?

@mostly_mental

Жыл бұрын

The difference is that in a square tiling, you don't know which of the neighbors get grouped together into the larger square. Which means there's not a unique way to assign coordinates. With the trapezoids, the grouping is forced by the directions of the neighbors. So the coordinates can only be chosen in one way.

@edgardeshayes4177

Жыл бұрын

@@mostly_mental Right; thanks you!😊

@mostly_mental

Жыл бұрын

@@edgardeshayes4177 Turns out the trapezoids are actually periodic. The argument works much better with the chair tiling (en.wikipedia.org/wiki/Chair_tiling ). Sorry for the confusion.

@edgardeshayes4177

Жыл бұрын

@@mostly_mental No need for excuses, everything is fine...😉

Technically the tile at the exact midpoint of the series does tile periodically

@mostly_mental

Жыл бұрын

Yeah, don't know how I missed that one. I even drew it because I wanted to point it out, and then I completely forgot to mention it.

I'm struggling with the definition of aperiodic. For me seems like the trapezoid @1:50 repeats all over the place. If I had a plane tiled with that I could find so many repeats. Literally the bottom of the hexagon has a solid repeat of the same trapezoid we were talking about.

@mostly_mental

Жыл бұрын

It's okay if sections repeat, so long as you don't repeat the whole plane all at once. In this case, if you took two copies of the plane and moved them so that the top trapezoid on one lines up with the bottom trapezoid on the other, then somewhere else you would find trapezoids that don't line up.

whoever figured out this monotile first knew very little about clothing.... it looks like a shirt, not a hat, dammit!

🕉

Looks more like a shirt than a hat to me.

looks more like a t-shirt to me

I tried to implement this inflation/deflation, but found that the supermetatiles are not congruent to the "1st-generation" metatiles. 😞 The parallellogram is wider and the propellorblade has a different shape. So you need to inflate repeatedly, then a final different inflation, and then fill in the hats.

@mostly_mental

Жыл бұрын

Yeah, you don't quite get similarity. There's similarity in the limiting behavior, but no level of metatile ever quite reaches it.

@landsgevaer

Жыл бұрын

@@mostly_mental None of the supertiles are congruent across levels of inflation? I just noticed the first level being different, but that may well be true. 😢 I did implement Penrose inflation on my computer some time ago, for funzies. Definitely doable. But this I wouldn't be able to get right, I think. Not without simply porting the app of the authors.

I'm still kind of confused about why this is considered a valid monotile, seeing as all tilings I've seen so far include both the hat and its mirror image. How are those the same tile? Unless it has been proven that no tiling could possibly exist *without* mirroring, I don't see why this counts.

@mostly_mental

Жыл бұрын

That's a good observation. It comes down to what you mean by "same". The hat works if you mean "congruent". But as you noticed, the hat requires reflections. The existence of a monotile with only rotations and no reflections is an open problem. If you find one, let me know, and I'll make a follow-up video.

@stanleydodds9

Жыл бұрын

The allowed transformations are the orthogonal transformations, i.e. the rotations and reflections in 2D. These are all the linear transformations which map an orthonormal basis to an orthonormal basis, and are also exactly the transformations which preserve lengths, so it's a fairly natural and simple way to restrict to specific transformations. What you are talking about is the special orthogonal group, which is an index 2 (normal) subgroup of the orthogonal group. These are the orthogonal transformations which have a determinant of +1, or alternatively, these are the transformations that preserve both length and orientation, or preserve both length and signed area/volume. It takes a little more description, but it's still quite a natural group of transformations. I guess my question is, is there a particular reason that you feel it's fine to allow rotations, but not fine to allow reflections? Either choice makes sense to me as a restriction. It's sort of the same as the difference between the symmetric groups, and the alternating groups. The symmetric groups feel more "complete" (they contain all permutations), while the alternating groups feel more "simple" (they are literally simple, but also in the sense that it's a natural way to split the symmetric group in half).

@mostly_mental

Жыл бұрын

@@stanleydodds9 If you bought hat-shaped tiles which were red on top and blue on bottom, you could tile a floor with them. And the tiles would all be the same shape, so you could say it's good enough. But also, the floor would be two colors, so it's not uniform. Depending on your sense of aesthetics, that might not be acceptable.

Googling the aperiodic trapezoid example, how come I found nobody mentions it? BTW, it would help if you kept the example further out in order to see how it doesn't repeat

@mostly_mental

Жыл бұрын

I see this proof a lot more often with the chair tiling (en.wikipedia.org/wiki/Chair_tiling ). But I found that a bit too visually busy (and a pain to draw) so I went with trapezoids instead. Here's a paper (by one of the same authors) that shows them both, plus a few others: strauss.hosted.uark.edu/papers/AHT.pdf

@netzerk

Жыл бұрын

@@mostly_mental Thanks! for sure this should have been mentioned more often in all of the countless videos on the subject

Yah I pulled that paper from arxive as soon as I learned it exists. ((:

What about a 3d aperiodic monotile?

@mostly_mental

Жыл бұрын

There are a few known tiles that work, with some caveats. The disconnected Socolar-Taylor tile I showed actually has a connected 3D counterpart (en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile ), but that's only weakly aperiodic. And there's a Schmitt-Conway-Danzer tile (en.wikipedia.org/wiki/Gyrobifastigium#Schmitt%E2%80%93Conway%E2%80%93Danzer_biprism ), but that can have screw symmetry. As far as I know, no one's discovered a strongly aperiodic monotile in 3D.

I can cover an area with tiles to makes a hex grid, or triangles. The area covered is two tiles wide; 'Chains' that are 'all perimeter'. Right angles, multiples of 15 degrees are easy/peasy. If one lays the tiles out alternating 'up' and 'down', two different chains can be formed. Keeping the tiles 'right-side up', a third type of chain is formed. 6 tiles can form a hexagon, and from each 'face' chains of all three types can be formed. The 'hole' is not a 'hole'; it is enclosed perimeter. kzread.info/dash/bejne/mnWhxamxdpW4Yqw.html "The tiles are Software" They make p-n junction transistors, logic gates and amplifiers.

the same team got a monotile that doesn't need reflection!!

@mostly_mental

Жыл бұрын

Very exciting! I'm in the process of recording the follow-up video.

@NoNameAtAll2

Жыл бұрын

they didn't "get" a monotile it's the same Tile(1,1) from this video! \o/

yoour tile make some interesting 3d objects when you fold them. then they make a repeating pattern.

The section at 3:28 doesn't make sense. Any periodic tiling can have an arbitrary encoding method to produce a unique identifier for any tile, how does this show that this makes a shape suddenly non-periodic? If you take a square with side length 1, you can arbitrarily define a midpoint, and each square you add to the plane has a line segment connecting that midpoint to the midpoint of the first square you started with and the data from that line segment is a unique identifier. How is a perfectly bland infinite tiling of squares suddenly "non-periodic"? This makes no sense to me.

@mostly_mental

Жыл бұрын

The trick is that there's a *unique* encoding for the trapezoids. If you look at any given tile, there's only one way to group it with its neighbors into a larger trapezoid, only one way to group that larger trapezoid into an even larger one, and so on. That means there's only one coordinate system of this type. With the squares, there are four ways to group any given square into a larger square, four ways to group those, and so on, which gives you infinitely many indistinguishable coordinate systems.

@GodOfReality

Жыл бұрын

@@mostly_mental Doesn't this assume that you can't just rotate the plane? If the plane is fixed then yeah I see your point, though assuming the plane is fixed in place seems rather arbitrary to me.

@mostly_mental

Жыл бұрын

@@GodOfReality Even if you do rotate the plane, you don't change the groupings. And those determines the coordinates, so the coordinates don't change either.

@GodOfReality

Жыл бұрын

@@mostly_mental I get it, thanks for replying.

@1:38, Should the first and single trapezoid unit have a number? is it the number 4 in the next composite trapezoid, in the middle? and is it, 41, 42, 43 and 44 in the last and large composite tile? would using colours made it all clearer? or would naming the first single tile 1 and then numbering it 1 in the next middle composite... be less confusing? thanks

@mostly_mental

Жыл бұрын

Yeah, I agree that whole section is a bit confusing. Here's a better explanation from one of the authors: strauss.hosted.uark.edu/papers/AHT.pdf The numbers tell you where a trapezoid is relative to its neighbors, and the first trapezoid doesn't have any. It could be number 4, but you could also rotate it into any of the other three positions, so any other number is just as valid. You don't know without additional context. The coordinates are written from the smallest trapezoid to the largest. You could just as well do them from largest to smallest. I did use color, but apparently my camera doesn't pick up the difference between dark brown and hot pink. I probably need better lighting.

@rickyardo2944

Жыл бұрын

@@mostly_mental Look your stuff is great and I like it and your style, I did not mean to be critical in a negative way as I realise how difficult it is to present anything clearly as it does take a lot of hard work and you clearly do that, I also notice that the fact that a lot of people show surprise that this is something new and this is due to the failure to clarify that there are some that do both periodic and aperiodic and some that only do aperiodic and I don't know what is the name for tiles that only tile periodically and not aperiodically, the name for tiles that only do it aperiodically and the name for tiles that do both periodically and aperiodically? do you see what i mean? and if you know can you help?

@mostly_mental

Жыл бұрын

@@rickyardo2944 A tile that only tiles aperiodically is called an "aperiodic monotile". I don't think there's a standard name for either a tile that only tiles periodically or a tile that can do both. I might call them "purely periodic tile" and "semi-periodic tile", but I don't think anyone would know what I mean.

@rickyardo2944

Жыл бұрын

@@mostly_mental' Thanks for that, you see what the problem is, naming is so important as I am sure you agree, periodic, "byperiodic" and aperiodic do come to mind...

I'm not sure I get the "different coordinates" proof. It seems that it is possible to make a very similar construction with squares, where 4 smaller squares make up a larger square ad infinitum and each of the smallest squares has unique coordinates (i. e. an infinite sequence in the set {1,2,3,4}, as in your proof), yet the tiling is periodic. Did I maybe miss a key point in your proof? After re-watching the proof: So, in my construction, you can't tell which larger square a smaller square belongs to. (Unless you are allowed to look at the coordinate sequence, in which case the larger squares are exactly the ones made up of small squares with sequences that start arbitrarily, but continue in a prescribed way.) Another difference in my construction: It allows translations that are smaller than a larger square. I think that is in fact the key to your proof, too: If there was a translation symmetry, the symmetry map would move the pattern by a finite distance, but that means there is a trapezoid that is larger than that distance, so the symmetry cannot map it to another trapezoid, hence there is no such symmetry.

@mostly_mental

Жыл бұрын

Yes, that's exactly it. You need to be able to uniquely reconstruct the hierarchy. Having translational symmetry would give you multiple ways to reconstruct it.

@cannot-handle-handles

Жыл бұрын

@@mostly_mental Fair enough. I still think a distance argument (similar to the one at the end of the video or the one I provided above) would make for a more intuitive proof, because it does not hinge on the introduction of coordinates.

@mostly_mental

Жыл бұрын

@@cannot-handle-handles You're probably right. It would also help if the trapezoids weren't actually periodic. The argument works much better with the chair tiling (en.wikipedia.org/wiki/Chair_tiling ). Sorry for the confusion.

@cannot-handle-handles

Жыл бұрын

@@mostly_mental That's totally okay, confusion is normal and can even be a useful motivator to start thinking more deeply about something. And thank you for sharing the chair tiling as another example!

@mostly_mental

Жыл бұрын

@@cannot-handle-handles On further reflection, the trapezoids really do seem to be aperiodic. I've just been staring at the diagram so long I've started to see patterns that aren't there. I still find the chairs easier to think about.

How about 3D Aperiodic Monotile ?

@mostly_mental

10 ай бұрын

There are a few known almost aperiodic monotiles in 3D. The disconnected Socolar-Taylor tile has a connected 3D analog that can tile layers aperiodically, but the layers might repeat (en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile ). And there are some shapes like the Schmitt-Conway-Danzer tile (en.wikipedia.org/wiki/Gyrobifastigium#Schmitt%E2%80%93Conway%E2%80%93Danzer_biprism ) that aren't periodic, but can still have screw symmetry. So far, no one's found a true 3D aperiodic monotile.

so... im failing to see how your first example of Aperiodic tiles (the trapezoids) are actually aperiodic. because to tile trapezoids like that. you would need to have them be able to form Hexagons as is evident in your example and hexagons are a Periodic tile.

@mostly_mental

Жыл бұрын

You're right that if we only consider the hexagons, we get something periodic. But each hexagon can be divided into trapezoids in three different directions. And if you list out the directions along any row of hexagons (or with any constant displacement), you'll find there's no repeating pattern.

I totally don't get from "you can write recursive coordinates" to "that means it's aperiodic." You could do recursive coordinates for packed squares. There's a step missing.

@mostly_mental

Жыл бұрын

The key idea is that there are *unique* recursive coordinates, determined by the orientation relative to the neighbors. For the squares, there are lots of ways you could group them together, so there are lots of coordinates you could use for any given tile. For the trapezoids, there's only one possible set of coordinates for each.

if the trapezoid is an aperiodic monotile and tessellates the plane aperiodically then what is new?

@svenreichard8726

Жыл бұрын

The trapezoid also has periodic tilings. The hat doesn't, so it is is purely aperiodic.

@cannot-handle-handles

Жыл бұрын

@@svenreichard8726 That's something that I found was missing from many articles I skimmed: They often failed to point out that only aperiodic tilings are possible with the discovered family of tiles, so they skipped one of the most important parts, which disappointed me a bit as a math communicator.

@rickyardo2944

Жыл бұрын

@@cannot-handle-handles what is "the most important parts" ?

@cannot-handle-handles

Жыл бұрын

@@rickyardo2944 I don't know what all the most important parts are, explicitly, but ONE OF the most important parts probably is the fact that being aperiodic not only means allowing non-periodic tilings but enforcing them (i. e. not allowing periodic tilings).

@rickyardo2944

Жыл бұрын

@@cannot-handle-handles Thanks

Your videos are great! Have you considered participating on the math video competitions hosted by 3b1b? It'd bring more attention to your channel.

@mostly_mental

Жыл бұрын

I'm glad you like them. And yes, I've participated the past two years, and I have a plan for this year, too.

Excuse me but still boggled what's not periodic about the initialish trapezoidal tiling.

@mostly_mental

Жыл бұрын

Pick any tile. If you look at its neighbors, you can see which larger trapezoid it belongs to. Then if you look at those trapezoids (ignoring individual tiles), you can see how those group together to make even larger trapezoids. And so on, until you can reconstruct the tile's position within every level of the hierarchy. You can assign coordinates along the way, to tell you which position a tile is in within each group. Any two tiles will be in different positions within in the first trapezoid that includes both of them, so they'll differ in at least some coordinate. So the coordinates are unique. If the pattern repeated, there would be two tiles with identical neighbors, neighbors of neighbors, and so on. Which would mean this process gives identical coordinates. Since the coordinates are unique, we can't have that. So there isn't a repeat, and this tiling is aperiodic.

@davidrogers8030

Жыл бұрын

@@mostly_mental Thanks. I mostly got that from the vid. The problem I have is that every trapezoid (rotated) of any particular size order is identical, and the pattern repeats in a regular (and periodic) way, so at the appropriate scale the internal coordinate numbers must repeat at regularly spaced intervals (without any variation) for that design. Admittedly I'm a bit hazy on the exact difference twixt nonp. and ap. Does a tiling of hexagons (or equilateral triangles or squares) halved at pseudo-random (to trapezoids, rite trangles or oblong, as) count?

@davidrogers8030

Жыл бұрын

@@mostly_mental Thanks. I mostly got that from the vid. The problem I have is that any trapezoid of a given order is identical to regularly spaced repeats, arranged in an inviolably fixed repeating pattern. Admittedly I'm hazy twixt nonp. and ap. Would hexagons (or equilateral triangles or squares) split into two at (appropriate) pseudo-random angles count?

@mostly_mental

Жыл бұрын

@@davidrogers8030 The trapezoids aren't going to be regularly spaced. It's a little hard to see in my diagram, but whichever line of repetitions you follow (excluding the central line of symmetry), you'll always run into a trapezoid facing a different direction eventually. And I'm fairly sure a randomly split tiling will be aperiodic (with probability 1). There are only a countably infinite set of tilings that repeat in two independent directions, and uncountably many tilings, so that has probability 0. I don't have a good argument for tilings that only repeat in one direction, but that seems very likely to also be probability 0.

got the tile... but not the time

HATS and COMETS!!??. Those are clearly VEE-NECK TEE SHIRTS and WHISTLES. Chevrons are gonna chevron.

That the new method of proving aperiodicity is more important or exciting than the discovery of the tile itself is a judgement call that I don't quite agree with.

@mostly_mental

Жыл бұрын

Fair enough. For me, the value of a problem is that it pushes you to think differently to solve it. So the hat itself is a cute curiosity. But a new proof technique opens the door to answering all sorts of interesting questions. So I always find the new tool more exciting.

@rosiefay7283

Жыл бұрын

tldr: Perhaps think of it this way. What's exciting is not the proof technique itself, but the discovery of a tile family so flexible that it contains two shapes which enable such a proof technique to work. In a tiling, call a point where three or more tiles meet a "junction". In every other previously discovered family of aperiodic tile-sets, the only difference among the tile-sets in the family is the rather trivial one of changing the edge-decoration that enforces the matching-rules. Changing the edge-decoration doesn't change where the junctions are. If you started, for example, with a set of Wang tiles based on a lattice of squares, you could change it to a different set, but it would still be based on that same lattice. By contrast, with any tile in Smith et al's family, if we change the edge lengths (while at all times keeping equal edges equal), the tile shape changes, and the arrangement of junctions also changes, but the tiles still tile (i.e. they still cover the whole plane and their interiors don't overlap). That doesn't happen with Wang or Penrose tiles. And, as Mostly Mental demonstrated, two results of doing this, while keeping the tile area constant, are a 4-iamond on a lattice of triangles of edge length sqrt 2, and an 8-iamond on a lattice of triangles of edge length 1.

petrography go wild

Your aperiodic proof doesn't make sense. You could do the exact same thing with square tilngs. (Each square is a quadrant in a larger square- no other square is *that* square but the tiling is still periodic) aperiodic means that there is never repetition along any N-1 dimensional surface ever - if you were to draw a line out to infinity over the plane it would never cross a tile in the exact same way twice.

@mostly_mental

Жыл бұрын

The difference is that with the trapezoids, there's a *unique* way to group the tiles into larger trapezoids. That gives you a unique coordinate system, which is impossible in a periodic tiling. With a square tiling, you could group the tiles in many different ways. So two different tiles can end up with the same coordinates in different coordinate systems. And the lack of uniqueness allows for a periodic tiling.

@siraaron4462

Жыл бұрын

@@mostly_mental what's unique about it? Assuming the trapazoids have uniform properties (aka there's a finite number of different trapazoids you can draw) any region on this plane (bounded or unbounded) repeats itself just a few tiles over - periodically. I'n other words If I were to zoom out really far and zoom in somewhere else how could I even find those coordinates again? I could pass any number of coordinates indistinguishable from where I stared.Even if I did, by some anomaly of chance, pass those coordinates again I wouldn't even have any way of knowing. Alternatively you can mirror the trapazoids and you have a symmetric hexagon. If you know anything about tilings you should understand the significance of any arrangement of shapes that can be simplified as a hexagon if the trapazoids have no uniformity (as was the case in your freehand drawing) then it's only aperiodic because there's no consistency - no pattern. the variety is as infinite as your ability to draw inconsistantly The other flaw with your logic (and arguably the largest flaw) is that your thought experiment can be done with a single uniform trapezoid (all trapezoids on the plane are the exact same only rotated and translated) AKA a monotile. therefore if it were aperiodic - we would already have an aperiodic monotile and this new discovery becomes meaningless.

@mostly_mental

Жыл бұрын

​@@siraaron4462 You can absolutely reconstruct the coordinates. Zoom out and zoom in on some arbitrary tile. Just by looking at the neighbors, you can tell which group of four it belongs to, and which one of those four it is (which you can't do with the squares or the periodic trapezoids). That gives you the first coordinate. Now look at the larger trapezoids you've just created, ignoring the individual tiles within them. Once again, you can tell which group of four it's in, and which of the four it is. That gives you a second coordinate. Repeating this process, we get an infinite sequence of coordinates. And there's no choice at any step here, so these coordinates are unique. And notice that for any two tiles, there will be some smallest level of trapezoid that contains both. That means those two tiles will have to differ in the corresponding coordinate. So no two tiles will have the same coordinates. Yes, if you pair the trapezoids together into hexagons and ignore the boundary between them, you do get a periodic tiling. But that's only because you've thrown out an important part of the structure: the orientations of the divisions. You could easily encode the digits of pi in binary in two different orientations of hexagons, and that would never repeat. All the trapezoids on the plane *are* uniform. And it has an aperiodic tiling, but it isn't an aperiodic monotile. That's because it doesn't *only* give aperiodic tilings. As you say, there are many periodic tilings with the same trapezoid. The hat is interesting because you *only* get aperiodic tilings.

@siraaron4462

Жыл бұрын

@@mostly_mental ok wow... the Dunning-Kreuger effect is real my friend Yes, each trapezoid inside of the larger trapezoid has a unique orientation. within that finite bounded region the trapazoids do not have not translational symmetry with each-other but the aperiodic/periodic descriptor refers to the repetition. A pattern that doesn't repeat itself in some way can't be either.

@mostly_mental

Жыл бұрын

@@siraaron4462 Here's a paper (conveniently by one of the same authors) with a more rigorous proof: strauss.hosted.uark.edu/papers/AHT.pdf . You'll get a lot more out of it if you put aside your antagonism and read through with the intent to learn, not to criticize.

Oh I hate this, why? This is so much pain. Why would they do this to us all?

We’ve been using this in 3D printing for about 5 yrs now. Started on Marlin machines and was adopted across all machine and controllers. It’s not new and they didn’t come up with it.

@mostly_mental

Жыл бұрын

I wouldn't be surprised if someone's come up with the hat before. It's not that complicated, and there are a lot of people who play around with shapes. But in math, the proof is the important bit, and that's certainly new.

its not a hat its a t-shirt

Is this an aperiodic monotile? I guess it's just a variation of the hat with more surfaces? pasteboard.co/7BtGHj9PnTqk.png

@mostly_mental

Жыл бұрын

That's an interesting shape, but I think it's periodic. You've got clusters of 6 tiles meeting to form jagged hexagons, and those hexagons are all evenly spaced. So you could slide the center of one hexagon onto another and the tiling would line up.

@isaackeyet2938

Жыл бұрын

That's very helpful actually, I'm a novice at all this and this explains how tiles are periodic for me :)