Thanks for watching! Check the description for some notes and some links to bonus content I've released recently. And consider checking out my Patreon here: www.patreon.com/comboclass
I love how this channel feels like a channel from 2012, it feels pretty nostalgic and at the same time this video is 20 hours old and talking about really new maths discoveries. Also you sometimes give Big Joel vibes
I had heard about the hat, and assumed that this video was just about that. I was one of the people left slightly unsatisfied with the need for reflection, and so I was quite please when you mentioned that an even better one had been found. Now i just need to tile my house with it.
If anyone is interested in learning more about this kind of stuff, the branch of mathematics which studies tilings (and much more) is called geometric group theory
you have the BEST set i’ve ever seen!!! i love your presentation, on par with some of the best Numberphile contributors!!! Super cool to see there’ve been even new discoveries in shapes, i can’t believe the time i’m living in
The pupils at our school in the U.K. have had SO much fun with these tiles. I ran off 600 of them and the pupils have noticed it is increasingly difficult to tile them as you spread outwards. If you have a postal address I’ll mail you some Domotro. 😊
4:50 I think you're wrong in your explanation of aperiodicity. Even for many if not all aperiodic tilings, if you pick arbitrarily large but finite chunk of it, you will find another place in the tiling that matches that chunk exactly. The matter is, you cannot shift and/or rotate the whole tiling in a way that would make it match itself everywhere.
With aperiodic patterns (according to the sources I read) they need to avoid lining up with “translations”, not necessarily “rotations”, and “aperiodic” (different than “non-periodic” which is another term some authors use) was defined as needing to avoid “arbitrarily large” periodic sections. Do you have other sources you are referencing? (If so I’m happy to read more about it. I’m not really an expert on this topic but did study a variety of different sources on it while researching this episode)
@@ComboClass Where I'm coming from is Penrose's tiling which is aperiodic but if you pick any finite part, you can find infinite copies of it. Given how the new tile's aperiodic tiling is constructed according to my very fast look at the paper, the same is true for that tiling as well.
Thanks for the reply. I still have seen aperiodicness defined as avoiding arbitrarily large periodic stretches, but I may have defined “periodic” incompletely. I’ll look into that further and consider adding a correction/clarification of some of the definitions to the description
@@ComboClass It does indeed avoid arbitrarily large periodic subregions, but that is not the same as sufficiently large subregions having no copies anywhere.
Love your style. Outside. Clocks falling down. Fantastic. Found out about Chiral Aperiodic Monotiles last night and today made a mushroom 🍄 shaped spectre Chiral Aperiodic Monotile in Inkscape. 🤓
This is very exciting. I myself have done quite a bit meddling around, but the only monotile I've found is the golden rhombus, which in 3d space can project the Penrose rhombic tiling on a 2d plane. There's also a counterpart for the kites and darts set. I'm not really sure I like this particular aperiodic monotile, but once they make a breakthrough, new discoveries tend to follow, so very exciting indeed.
Do aperiodic monotiles exist in 3D, 4D, or higher dimensions? I was thinking about it and, as a novice, I couldn't figure out if it would be more or less difficult
Hmmm, it might be really difficult. In 2d, you can make any polygon regular just by moving the vertices and inflating it. In 3d, regular polyhedra are way more finnicky. These 2d monotiles are irregular, but regular polygons seem to be an important part of their construction.
Im more and more convinced that the "theory of everything" will eventually be discovered by a hobbyist mathemetician just messing around for fun... And the fundamental will end up being a shape or a group or a symmetry that is transcendent in some way .. spinning/flipping/tiling in such a way to explain the weak force mirror breaking thing, antimatter/matter ratio, etc... I know im sounding like a Platonist, which im not.. because i think we'll never get there without experiment giving us clues aaaand that a TOE isnt really that useful since emergent properties are so complex that the whole universe has taken 14 billion years just to get to here with its calculation... So the only way to get ahead of the universe in any meanignful way is to recognize the qualitative difference created by complexity... Buuut regardless, a TOE would be so amazing and could help us steamline our higher level discoveries
The example of an aperiodic tiling at 5:02 is actually periodic because the tiling could be translated vertically and create identical copies of itself.
Nope, while the representation may have been just a handful of squares they were representing infinitely long columns. It’s then aperiodic since every column is offset by a different transcendental number.
I love your unique approach to presenting math ideas. With sloppy "accidents" happening around you, you're still very careful and precise with your words. So, it is with a touch of humility and respect that I point out a small grammatical error on your part: The singular form of "vertices" is "vertex", not "vertice." 😁
Incredible video as always! I was wondering if there was a known application for these monotiles. I wonder if they could be used in an encryption method
One possible application (of many possibilities) is that certain crystal-like structures in nature can form periodic or aperiodic patterns, and by knowing which sorts of “tiles” offer which opportunities, we can learn which structures might emerge from which types of molecules
Kitty reaping the benefits of a friendly human hand, controlled by a subconscious while the conscious is distracted talking or some other human-ish thing
Thank you for explaining how you would disregard a "glitch" in checking periodicity. I had never heard it explained that way. Also for petting the big black cat Num Num Num. You didn't mention any specific people until David Smith as an amateur mathematician, only presented the ideas and the unanswered questions.
Can’t wait to see the Hat and mirrored T-shirt on a Futurama character, but I can’t help but feel new discoveries in this field aren’t going to get the same press even if they’re somehow more impressive. Please cover them if they do!
About the colors/matching rules. Are those really imperfections of the wang tiles and penrose tiles. Or are they just there because it makes the tiles visually cleaner and because it is trivial to make two sides only fit together like puzzle pieces anyway?
EDIT: Sorry. I could tile it with this compound left-right tile, without flipping it over. I was misremembering. sorry for being dumb. I laser cut out some of the Hat/Shirt as both the left and right handed versions and made them magnets to mess around with on my fridge. I found that if i pair a left with a right always, i could get a periodic tiling of them... which i haven't heard anyone talk about... i guess it's not interesting. ;)
I think you must have cut them out imprecisely or something. I wouldn't trust a physical version of one to properly represent whether there are gaps/overlaps/etc. There shouldn't be a way to combine the hat and its mirror image into a periodic tiling (they don't just fit side-by-side as pairs on both directions without gaps, a more complex pattern is required to tile with them, which is an aperiodic pattern).
Sometimes if you only have a small number of tiles, the pattern can look periodic, but if you continued the pattern, it would break down and reveal its aperiodic nature. For example, see the pattern at 08:10 in this video. There are areas which match other areas, but that doesn't mean that the pattern is periodic as if you tried to translate the larger pattern you'd find other areas don't match. BTW, I'm glad you called the shape Hat/Shirt. I was beginning to think I was the only person who sees that shape and thinks 'shirt' (not hat).
Are you sure you copied the hat (10:11) and not the unadorned spectre (12:23)? The hat has 6 sides of length 1, 1 of length 2 and 6 of length sqrt{3} ~~ 1.732. The spectre has 12 sides of length 1 and 1 of length 2. The spectre can indeed make a periodic tiling in the way you describe; in fact Smith et al's first paper showed this tiling, and that's why they specifically excluded that shape from their set of aperiodic monotiles. To tile with hats you need (7+3\sqrt5)/2 ~~ 6.8541 times as many of one handedness as of the other.
@@landsgevaer I think I was tiling them aperiodically... I was misremembering the situation. i was able to tile it all without flipping over any pairs of tiles though. nevermind.
I might be misunderstanding your definition of periodic. My understanding was that for a tiling to be periodic you need to be able to translate the whole tiling. There are many aperiodic tilings where you can find a translation of any arbitrarily large subset. But you cannot translate the entire tiling.
What you’re describing could be called “non-periodic” by some sources, but not “aperiodic”. An “aperiodic tiling” means it does not contain arbitrarily large periodic patches, and if you can translate an arbitrarily large subset of it then the tiling wouldn’t be called aperiodic. It sometimes might be called non-periodic though (there’s not a 100% clear convention in how authors use the terms)
@@ComboClass well then by your definition the penrose tiling is periodic... Any penrose tiling can be reduced and expanded in 100% determinable way, choose an arbitrarily large patch P of any infinite penrose tiling, reduce the tiling repeatedly until the entirity of P is on one tile T1 of a new infinite penrose tiling, find a tile T2 of the same type, and with the same orientation as T1 (every orientation that can occur will occur infinitely many times within a tiling) , repeatedly expand until you get back to the original tiling and then translate the patch that corresponds to T1 onto the patch that corresponds to T2. There you have translated the whole of P onto a matching part of the tessellation. Similar constuctions are possible in most infinite aperiodic tilings with uniquely determinable expansion and reduction rules.
@@ComboClass I have reread your response and I think I see where we are getting tripped up. You are using translation as a shorthand for periodically translational, but that's just what periodic means. Aperiodic tilings have segments (arbitrarily large segments) that can be translated onto identical segments (see my proof post) but these will not occur periodically. The existance of a translation of part of a tiling is not an indication of periodicity.
Like you said, it is easier to find a tileing where you can translate arbitrarily large subsets vs one where you can translate the whole thing. But that means that it is harder for a tile to not allow the former, i.e. be aperiodic: With the latter definition you would just need to have a finite size starting configuration that cannot appear anywhere else.
@@someknave Do you have any particular sources about penrose tilings apparently being able to match arbitrarily large sections of themselves with just translation? I'm happy to read more about it (as I'm not an expert in this particular subject and did most of my research on it in the past couple months). I may have used "periodic" at times when "not aperiodic" would be clearer, but all of the sources I read DID use "arbitrarily large" within their definition of "aperiodic" tilings, so I'd like to see what source(s) you're referring to.
None of the tilings with a square are 'completely' aperiodic, i.e. in every direction. Theres always one direction along which you can move (an arbitrary large portion of) the tiling along such that the tiles line up (namely along the rows/columns)
5:43 Hmm, are there any (I donʼt know if thereʼs already a word for this) strictly periodic tilings? Ones where you can make periodic tilings but *not* aperiodic ones? I can think of examples I think qualify but I donʼt know how to prove it.
i imagine you could just use a "traditional" periodic tiling like squares, triangles, or hexagons, but then add notches to the sides (like puzzle pieces) to ensure they could only be placed in the traditional fashion, and not slid around like the squares can.
Hexagons can only tile periodically, as they only fit together one way without leaving gaps. Squares can be either aperiodic or periodic as you can slide the rows out of alignment by any amount. (Triangles can be slid similarly to squares)
@@retrogiftsuk4812 Squares and triangles can be slid, to break the symmetry in one direction, but it would remain periodic in the other direction afaik.
Was I interested in another 'einstein tile discovery' video? Not necessarily. Was I excited to see what an excited Domotro would have to say about it? HECK YES
Yes I made a correction in the description. I I tried to use squares as a more unique example but I should have used rectangles which would have demonstrated the point (that some periodic tilings can be adjusted to make a form that’s not periodic) more accurately
Fun fact: The builders of ancient Mosques in Iran and probably other countries in the region used aperiodic tesselation already in the 15th century. Here is a lengthy lecture on the topic: th-cam.com/video/rldnu9rNpH8/w-d-xo.html&ab_channel=PeterLu
I suppose we could call a vertex a vertice to make it pluralize simply to vertices, but then anyone joining that church is going to wonder, "What's a vertex?" (For approximately as long as someone might wonder, "Why's the plural vertices and not vertexes?" in the other church) whenever encountering legacy literature on vertex (or vertice) related topics, dating from some unknown date after 2023 has long passed. Until that old 2020's stuff has become fully redundant, this problem will replace the old, "Why not vertexes?" problem of the olden days. (The olden days we're living in, and will continue to live in for a few years hence.) I wonder if that's worth the bit of consistency gained this way? And what about the people who start insisting on pronouncing the "ice" in vertice the same as you pronounce the "ice" in "ice"? (For consistency, they'd argue.) Now you gain the new problem of the vertices that rhyme with "ices" (with the ice in ice and lice) making old fashioned recordings, where the ancestors talk of something called a "vertiss-sea" and "vertiss-seas" when clearly what they're talking about is vert-Ices almost impossible to understand (or something milder than that). Weird old fossils rambling on this strange way of theirs. Why can't they just call a vert-Ice a vert-Ice like everyone else does? I suppose there's the answer: Fix the spelling as well as the pronunciation and the plurals. Do away with "vertices", and write it as "vertiss-seas". Phew! Problem solved. :D
Video could've used a 'recap' section along the way. I had to stop and go back for definitions at the start and then go back, sorry..I'm just old and slow.
I said that a hobbyist mathematician discovered them and then worked with other mathematicians. All of that is true. Sorry if you don’t consider hobbyist mathematicians “true” mathematicians, or don’t consider mathematics a science
It seems to me that a square NEVER could make an aperiodic tiling. I seriously doubt it. At least, the example you show is not (if you multiply the translations in one axis by n*pi, remains perfectly periodic in the axis) I challenge you to show even one totally aperiodic tiling example with squares ;-)
Thanks for watching! Check the description for some notes and some links to bonus content I've released recently. And consider checking out my Patreon here: www.patreon.com/comboclass
Take a shot each time the word tile is said
1:00 for the busy: the previous aperiodic monotiles had the drawback of being composed of disconnected parts.
I love how this channel feels like a channel from 2012, it feels pretty nostalgic and at the same time this video is 20 hours old and talking about really new maths discoveries. Also you sometimes give Big Joel vibes
weed
I had heard about the hat, and assumed that this video was just about that. I was one of the people left slightly unsatisfied with the need for reflection, and so I was quite please when you mentioned that an even better one had been found. Now i just need to tile my house with it.
The obly acceptable kitchen tiling
Love it! I've been meaning to get into tiling patterns for textile design, super excited to try some escher-esque animals out of this spectre 👻
Only quite pleased? Dave S.
You sound fun…
This could be used for game developers who want to hide pattern repetitions on large textures.
If anyone is interested in learning more about this kind of stuff, the branch of mathematics which studies tilings (and much more) is called geometric group theory
Bro this channel must be popular some day
you have the BEST set i’ve ever seen!!! i love your presentation, on par with some of the best Numberphile contributors!!! Super cool to see there’ve been even new discoveries in shapes, i can’t believe the time i’m living in
also you remind me a bit of Vsauce3, who rocks
I can imagine puzzles made from such tiles. Taking frustration to a whole new level...
The pupils at our school in the U.K. have had SO much fun with these tiles. I ran off 600 of them and the pupils have noticed it is increasingly difficult to tile them as you spread outwards. If you have a postal address I’ll mail you some Domotro. 😊
i never thought of doing an aperiodic tiling of something, but this sounds amazing, the infinite family of them, thats pure mathematics!
I imagine there could be interesting implications should we find any natural molecules (or engineer some) which arrange like this monotile.
There already are crystals with pentagonal symmetry. Those cannot tile space regularly either.
en.m.wikipedia.org/wiki/Quasicrystal
The thought that one single tile can be aperiodic kind of breaks my mind... These are by far some of the coolest shapes ever.
“Artists” saying there’s no such thing as novelty, and mathematicians still discovering new shapes.
Popular art has to reference or react to the past to be accepted. There are plenty of unpopular artists still innovating.
4:50 I think you're wrong in your explanation of aperiodicity. Even for many if not all aperiodic tilings, if you pick arbitrarily large but finite chunk of it, you will find another place in the tiling that matches that chunk exactly. The matter is, you cannot shift and/or rotate the whole tiling in a way that would make it match itself everywhere.
With aperiodic patterns (according to the sources I read) they need to avoid lining up with “translations”, not necessarily “rotations”, and “aperiodic” (different than “non-periodic” which is another term some authors use) was defined as needing to avoid “arbitrarily large” periodic sections. Do you have other sources you are referencing? (If so I’m happy to read more about it. I’m not really an expert on this topic but did study a variety of different sources on it while researching this episode)
@@ComboClass Where I'm coming from is Penrose's tiling which is aperiodic but if you pick any finite part, you can find infinite copies of it.
Given how the new tile's aperiodic tiling is constructed according to my very fast look at the paper, the same is true for that tiling as well.
@@kasuha Agreed. +1
Thanks for the reply. I still have seen aperiodicness defined as avoiding arbitrarily large periodic stretches, but I may have defined “periodic” incompletely. I’ll look into that further and consider adding a correction/clarification of some of the definitions to the description
@@ComboClass It does indeed avoid arbitrarily large periodic subregions, but that is not the same as sufficiently large subregions having no copies anywhere.
Love your style. Outside. Clocks falling down. Fantastic. Found out about Chiral Aperiodic Monotiles last night and today made a mushroom 🍄 shaped spectre Chiral Aperiodic Monotile in Inkscape. 🤓
this channel is a gem
This channel is the definition of chaotic good. 😂
Thanks, Domotro! I thought this will be an episode about pentatiles, didn't know about those discoveries, good to know!
Great vid man ive been interested in these shapes forever!
One of your best, the editing is getting fancy - I like it!!
This is very exciting. I myself have done quite a bit meddling around, but the only monotile I've found is the golden rhombus, which in 3d space can project the Penrose rhombic tiling on a 2d plane. There's also a counterpart for the kites and darts set.
I'm not really sure I like this particular aperiodic monotile, but once they make a breakthrough, new discoveries tend to follow, so very exciting indeed.
Great episode. I love the energy!
Cool tiles.
I also like your clumsy-backyard-professor-persona and I get the feeling that it's not just a role you're playing.
Fantastic reporting Domotro
The singular of vertices is vertex.
Do aperiodic monotiles exist in 3D, 4D, or higher dimensions? I was thinking about it and, as a novice, I couldn't figure out if it would be more or less difficult
Hmmm, it might be really difficult. In 2d, you can make any polygon regular just by moving the vertices and inflating it. In 3d, regular polyhedra are way more finnicky.
These 2d monotiles are irregular, but regular polygons seem to be an important part of their construction.
You have lots of vertices, but if you only had one, it would be a vertex.
Im more and more convinced that the "theory of everything" will eventually be discovered by a hobbyist mathemetician just messing around for fun... And the fundamental will end up being a shape or a group or a symmetry that is transcendent in some way .. spinning/flipping/tiling in such a way to explain the weak force mirror breaking thing, antimatter/matter ratio, etc... I know im sounding like a Platonist, which im not.. because i think we'll never get there without experiment giving us clues aaaand that a TOE isnt really that useful since emergent properties are so complex that the whole universe has taken 14 billion years just to get to here with its calculation... So the only way to get ahead of the universe in any meanignful way is to recognize the qualitative difference created by complexity... Buuut regardless, a TOE would be so amazing and could help us steamline our higher level discoveries
The example of an aperiodic tiling at 5:02 is actually periodic because the tiling could be translated vertically and create identical copies of itself.
Nope, while the representation may have been just a handful of squares they were representing infinitely long columns. It’s then aperiodic since every column is offset by a different transcendental number.
@@ladyravendale1and those columns can be shifted all together to match with themselves
this stuff is really interesting, and the way this video presents it does justice to that quite well imo!
I love your unique approach to presenting math ideas. With sloppy "accidents" happening around you, you're still very careful and precise with your words. So, it is with a touch of humility and respect that I point out a small grammatical error on your part: The singular form of "vertices" is "vertex", not "vertice." 😁
I have been looking for a video on this!
Nice video! Make one on fractals!
Incredible video as always! I was wondering if there was a known application for these monotiles. I wonder if they could be used in an encryption method
One possible application (of many possibilities) is that certain crystal-like structures in nature can form periodic or aperiodic patterns, and by knowing which sorts of “tiles” offer which opportunities, we can learn which structures might emerge from which types of molecules
Suggestion for video: 2’s compliment, in other bases than 2, and related patterns that emerge. 😊
Kitty reaping the benefits of a friendly human hand, controlled by a subconscious while the conscious is distracted talking or some other human-ish thing
Thank you for explaining how you would disregard a "glitch" in checking periodicity. I had never heard it explained that way.
Also for petting the big black cat Num Num Num.
You didn't mention any specific people until David Smith as an amateur mathematician, only presented the ideas and the unanswered questions.
Can’t wait to see the Hat and mirrored T-shirt on a Futurama character, but I can’t help but feel new discoveries in this field aren’t going to get the same press even if they’re somehow more impressive. Please cover them if they do!
I love this dude, it's like maths in a fairy tale setting
About the colors/matching rules. Are those really imperfections of the wang tiles and penrose tiles. Or are they just there because it makes the tiles visually cleaner and because it is trivial to make two sides only fit together like puzzle pieces anyway?
At least some can be modified to look like puzzle pieces. I’m not sure if all can
I find this extremely interesting! My background is art and graphic design. Thank you for explaining this in an easy way!
Two ways I can think of to press on:
Can we make one that needs less rotations?
Can we make one with less edges?
EDIT: Sorry. I could tile it with this compound left-right tile, without flipping it over. I was misremembering. sorry for being dumb.
I laser cut out some of the Hat/Shirt as both the left and right handed versions and made them magnets to mess around with on my fridge. I found that if i pair a left with a right always, i could get a periodic tiling of them... which i haven't heard anyone talk about... i guess it's not interesting. ;)
I think you must have cut them out imprecisely or something. I wouldn't trust a physical version of one to properly represent whether there are gaps/overlaps/etc. There shouldn't be a way to combine the hat and its mirror image into a periodic tiling (they don't just fit side-by-side as pairs on both directions without gaps, a more complex pattern is required to tile with them, which is an aperiodic pattern).
Sometimes if you only have a small number of tiles, the pattern can look periodic, but if you continued the pattern, it would break down and reveal its aperiodic nature. For example, see the pattern at 08:10 in this video. There are areas which match other areas, but that doesn't mean that the pattern is periodic as if you tried to translate the larger pattern you'd find other areas don't match.
BTW, I'm glad you called the shape Hat/Shirt. I was beginning to think I was the only person who sees that shape and thinks 'shirt' (not hat).
Are you sure you copied the hat (10:11) and not the unadorned spectre (12:23)? The hat has 6 sides of length 1, 1 of length 2 and 6 of length sqrt{3} ~~ 1.732. The spectre has 12 sides of length 1 and 1 of length 2. The spectre can indeed make a periodic tiling in the way you describe; in fact Smith et al's first paper showed this tiling, and that's why they specifically excluded that shape from their set of aperiodic monotiles. To tile with hats you need (7+3\sqrt5)/2 ~~ 6.8541 times as many of one handedness as of the other.
It would be interesting, since that was proven impossible.
The surface of that fridge of yours might be worth studying...
@@landsgevaer I think I was tiling them aperiodically... I was misremembering the situation. i was able to tile it all without flipping over any pairs of tiles though. nevermind.
12:19. i love how the top part of the spectre almost exactly overlays your bird feeder. :D
Haha!!! Are those tiles from Blokus? I loved that board game.
They are actually from a Tetris board game (yes, there is a Tetris board game haha)
i keep finding lost blokus tiles they haunt me continually
I might be misunderstanding your definition of periodic. My understanding was that for a tiling to be periodic you need to be able to translate the whole tiling.
There are many aperiodic tilings where you can find a translation of any arbitrarily large subset. But you cannot translate the entire tiling.
What you’re describing could be called “non-periodic” by some sources, but not “aperiodic”. An “aperiodic tiling” means it does not contain arbitrarily large periodic patches, and if you can translate an arbitrarily large subset of it then the tiling wouldn’t be called aperiodic. It sometimes might be called non-periodic though (there’s not a 100% clear convention in how authors use the terms)
@@ComboClass well then by your definition the penrose tiling is periodic...
Any penrose tiling can be reduced and expanded in 100% determinable way, choose an arbitrarily large patch P of any infinite penrose tiling, reduce the tiling repeatedly until the entirity of P is on one tile T1 of a new infinite penrose tiling, find a tile T2 of the same type, and with the same orientation as T1 (every orientation that can occur will occur infinitely many times within a tiling) , repeatedly expand until you get back to the original tiling and then translate the patch that corresponds to T1 onto the patch that corresponds to T2. There you have translated the whole of P onto a matching part of the tessellation.
Similar constuctions are possible in most infinite aperiodic tilings with uniquely determinable expansion and reduction rules.
@@ComboClass I have reread your response and I think I see where we are getting tripped up. You are using translation as a shorthand for periodically translational, but that's just what periodic means. Aperiodic tilings have segments (arbitrarily large segments) that can be translated onto identical segments (see my proof post) but these will not occur periodically. The existance of a translation of part of a tiling is not an indication of periodicity.
Like you said, it is easier to find a tileing where you can translate arbitrarily large subsets vs one where you can translate the whole thing. But that means that it is harder for a tile to not allow the former, i.e. be aperiodic: With the latter definition you would just need to have a finite size starting configuration that cannot appear anywhere else.
@@someknave Do you have any particular sources about penrose tilings apparently being able to match arbitrarily large sections of themselves with just translation? I'm happy to read more about it (as I'm not an expert in this particular subject and did most of my research on it in the past couple months). I may have used "periodic" at times when "not aperiodic" would be clearer, but all of the sources I read DID use "arbitrarily large" within their definition of "aperiodic" tilings, so I'd like to see what source(s) you're referring to.
How about tiling a sphere or a torus?
Could they be quantified as new 2D fractals? Chiral-Fractals?
i have two of the things that dimotro has scattered on his desk at the start are you spying on us?!
One day it will be rare to be one of the first 100 people to see a combo class vid
Nope, I am hiring 100 people just to await a video to be posted, just to keep you out of the first 100. :)
"And I'll see you in the next episode" great cutaway!
None of the tilings with a square are 'completely' aperiodic, i.e. in every direction. Theres always one direction along which you can move (an arbitrary large portion of) the tiling along such that the tiles line up (namely along the rows/columns)
Yeah there is a correction in the description that I tried to use squares as a more unique example but I should have used rectangles
I knew I would find someone in the comments pointing this out, good on you
How do I find a spectre tutorial?
There's a tutorial video called "Chiral Aperiodic Monotiles - Their discovery and their construction using Inkscape"
5:43 Hmm, are there any (I donʼt know if thereʼs already a word for this) strictly periodic tilings? Ones where you can make periodic tilings but *not* aperiodic ones? I can think of examples I think qualify but I donʼt know how to prove it.
i imagine you could just use a "traditional" periodic tiling like squares, triangles, or hexagons, but then add notches to the sides (like puzzle pieces) to ensure they could only be placed in the traditional fashion, and not slid around like the squares can.
Hexagons can only tile periodically, as they only fit together one way without leaving gaps. Squares can be either aperiodic or periodic as you can slide the rows out of alignment by any amount. (Triangles can be slid similarly to squares)
The X pentomino can tile and there is essentially only one way it can (or two ways, but one is just the other one, flipped). This is periodic.
@@retrogiftsuk4812 Squares and triangles can be slid, to break the symmetry in one direction, but it would remain periodic in the other direction afaik.
Was I interested in another 'einstein tile discovery' video?
Not necessarily.
Was I excited to see what an excited Domotro would have to say about it?
HECK YES
You can get rid of the matching rules for the penrose tilings by modifying the pieces so they lock like puzzle pieces, but only in one orientation.
Can't you just add puzzle piece like connectors to the set of two nonperiodic tiles to remove the tiling rules
Do you know abour Scutoids? It's a new geometrical shape (solid) discovered in 2018.
Yeah they are cool, maybe they will show up in an episode someday
Very interesting, my friend.
Great stuff! Now to head to my school’s DT department and set up the laser printer …. Thanks Domotro 😊👍
Albert One Stone sounds like a meme name
mindblowing, thanks a lot for sharing
Love it! Thanks so much 🙏 🙏 🙏 🙏 🙏 🙏 🙏
You scooped Numberphile. Nice.
As a casual math viewer, but why? why is energy put into figuring out if you can tile a plane with one tile, and what that shape would be?
This video has some real chaotic energy.
Can hexagons not be aperiodic?
but isn't a square tiling with offset columns still periodic? it's only aperiodic in one direction
Yes I made a correction in the description. I I tried to use squares as a more unique example but I should have used rectangles which would have demonstrated the point (that some periodic tilings can be adjusted to make a form that’s not periodic) more accurately
Now my future house will 100% have these tiles as actual tiles
Fun fact: The builders of ancient Mosques in Iran and probably other countries in the region used aperiodic tesselation already in the 15th century. Here is a lengthy lecture on the topic: th-cam.com/video/rldnu9rNpH8/w-d-xo.html&ab_channel=PeterLu
Domotro, I think you could make a 10 minute video of you petting your cat and it would get 1 million views
FINALLY I can tile an infinite plane without being repetitious. Such a relief.
Mathematicians nicely shrinked the set of tiles from a size of a bunch of pieces to only one. Next step? ZERO tiles!
Yeah you made this video finally!
I cannot wait for aperiodic monotiles to be found to tile a 3d hyperplane
From the start I was thinking “ooo! When is he going to mention Wang tiles?” and then at 6:50 I see that beautiful graphic
This was so cool! Thanks a lot!
I suppose we could call a vertex a vertice to make it pluralize simply to vertices, but then anyone joining that church is going to wonder, "What's a vertex?" (For approximately as long as someone might wonder, "Why's the plural vertices and not vertexes?" in the other church) whenever encountering legacy literature on vertex (or vertice) related topics, dating from some unknown date after 2023 has long passed. Until that old 2020's stuff has become fully redundant, this problem will replace the old, "Why not vertexes?" problem of the olden days. (The olden days we're living in, and will continue to live in for a few years hence.)
I wonder if that's worth the bit of consistency gained this way? And what about the people who start insisting on pronouncing the "ice" in vertice the same as you pronounce the "ice" in "ice"? (For consistency, they'd argue.) Now you gain the new problem of the vertices that rhyme with "ices" (with the ice in ice and lice) making old fashioned recordings, where the ancestors talk of something called a "vertiss-sea" and "vertiss-seas" when clearly what they're talking about is vert-Ices almost impossible to understand (or something milder than that). Weird old fossils rambling on this strange way of theirs. Why can't they just call a vert-Ice a vert-Ice like everyone else does?
I suppose there's the answer: Fix the spelling as well as the pronunciation and the plurals. Do away with "vertices", and write it as "vertiss-seas".
Phew! Problem solved. :D
Отличное видео!
now I want a 3dimensional monotile
Do an episode on "the rabbit sequence!"
Video could've used a 'recap' section along the way. I had to stop and go back for definitions at the start and then go back, sorry..I'm just old and slow.
Cool!
great job with the environment
Can we aperiodically monotile a sphere? Would make a cool football.
Well, time to rebuild the bathroom.
casually sitting there while the hose be leaking sprays some water
Huge board game potential
Going to get this on my kitchen floor
It's impressive to witness someone who speaks at 0.75x speed
Nah, a hobbyist discovered these shapes.
Who then told mathematicians, not scientists.
I said that a hobbyist mathematician discovered them and then worked with other mathematicians. All of that is true. Sorry if you don’t consider hobbyist mathematicians “true” mathematicians, or don’t consider mathematics a science
It seems to me that a square NEVER could make an aperiodic tiling. I seriously doubt it. At least, the example you show is not (if you multiply the translations in one axis by n*pi, remains perfectly periodic in the axis) I challenge you to show even one totally aperiodic tiling example with squares ;-)
Hey man, nice cat...
I like this.
Sick! im so gona lasercut cork specters to "tile" a wall
Now do the same in 3d? Aperiodic 3d monotiles anyone?
1:20 My fellow nerd have a pet rainbow 🌈 in his backyard.
Why does matematician need a labcoat?
Albertiodic Einstein.
albert one stone
I wish you went through the proof of the einstein tile.