Genuinely brilliant interview, by both both sides. Brady asks all the right questions, and Craig gives real answers to them. It's rare for it to bring the human element into the research without going too far one way or the the other.
To me it seems like he's repeating a bit. I haven't paid 100% attention, but it feels like he asked him "How does this make you feel?" like four times, each time worded slightly differently. It was still a fun interview nonetheless.
@@QuantumHistorian Should be? According to whom? I believe that such a long interview would benefit from more varied questions (or from being shorter).
Loved seeing the emails between the researchers, as they added more people onto the team. You can imagine what it must be like for a mathematician to get a mesage from your peers saying "We have a promising lead on the biggest open question in our field, and we think you're the ideal person to work on it." (In more cautious language of course, but they know exactly what it means.)
I wish people would stop with those comments. There's nothing 'random' about a person just because he is not some professor in a university. Ande also, it makes it sound like he just found it by accident. And, no: he is an artist and he found the shape deep in his field of interest in which he was working.
Craig wasn’t my professor, but we had common office hours in first year and I went to visit him every week. He was a great teacher, and I never expected him to show up on numberphile before computerphile!
@@thesenamesaretaken if the tile is made thicker and tiled into a plane and then layers of these planes are stacked, does that not count as a aperiodic polygonal tiling? just asking... thanks
Exactly what I was thinking. He basically found the solution and then prof. Kaplan verified and made rigorous. I'd love to hear more about David's process.
I've been waiting for a Numberphile video about this monotile! I've been interested in this subject since I read Martin Gardner's columns on Penrose tiles years ago. Thanks for sharing information about this interesting and important discovery.
I don't know much at all about tilings but it's so much fun seeing how important and exciting this is. I love how you talk to guests, who are often academic (and frankly, typically stifled by strictness and disallowedness), but you as well as they are shown to just be normal people.
Someone on Reddit mentioned that an aperiodic monotile would make tiling in video games look more realistic. Like how water from above is just squares repeating and breaks immersion, a monotile could break it up more naturally.
The shape they called a hat was a t shirt to me lol. It's crazy to see these shapes and so clearly see how they could tile a plane. I wish I could see their reaction when they realized that they found it.
I want a t-shirt that just is the tile shape. WIth the point at the bottom and the asymmetrical wonky sleeves, and offset v-neck, but it would be amazing.
It would be cool if these tilings could be used as texture assets in videogames. Then somewhat simple mathematic formulae could be used to make complex graphics.
Aperiodic tiles have already been used as a texturing trick for quite a while - not using weird-shaped *mono*tiles, but several square Wang tiles with rules for what can connect on what side.
Not sure there would be much appetite for aperiodic tiling in computer graphics. It would be more complicated than triangle or square tiling, which is what everyone uses now, and as long as you keep your textures subtle you don't really have to worry much about the periodicity being obvious.
@@ZekeRaiden In old games there would be visible artifacts is large fields of similar texture, like in for example grass. But it would be overkill to apply this tiling for that issue. It would be interesting to see if some one makes a board game like Carcassonne with this tiling.
why to use more complex part instead of smaller generic ones? nobody cared to even find this answer you got here just simply checking all possible diamond built tiles it means this discovery might be just art for art and all you people hyped about it just pumping empty balloon
When I read of the einstein I'd been waiting for the numberphile about it to come out! Exciting to hear that it was delayed because there's a new and better one.
When I first saw the 'one stone' tile and heard what it could do, it felt 'broken' to me. Couldn't explain it, so a made a bunch and played with them. In very short time, I was making periodic structures in 30 degree increments. 'Specter' tiling fixed the problem; I can look at piles of tiles without getting a sick headache anymore.
18:11 - the careful distinctions between things like calculating vs computing, polyomimos and polyforms, is when you know you're listening to a passionate expert in a very specific field
I really want to tile my new bathroom with the hat. This is a must. I need those tiles if only to bug my friends as they try to find a repeat, and fail.
Interesting to hear about the timeline of discovery and how fast it moved, especially from the hat to the spectre. In terms of does using flips count as a true monotile I feel like it depends. In a purely 2d space I'd say without flips is best, but physically in a 3d space I'd say with flips counts so long as the material you're using doesn't look different depending on whether the tile is flipped or not. So generally I'd say physically in a 3d space the hat is a monotile as is the spectre, but in terms of a purely 2d space I'd say probably just the spectre although it's up to interpretation.
Perhaps a simpler way to put it: up to chirality, there is at least one polygonal (straight-edged) aperiodic 2D monotile. If chirality is enforced, there is no known polygonal aperiodic monotile, but you can construct an infinite family of monotiles where the vertices are connected by congruent curves rather than straight edges. The "hat" is nice because it is polygonal, but it requires you to ignore chirality (or be in a space where 2D chirality is irrelevant, e.g. 3D space or higher.) The "spectres" are nice because they are genuinely monotiles (fully achiral), but you have to give up the straight edges. Now, the next question is: is there a polygonal aperiodic chiral monotile?
I love that the octo-kite is actually a symmetrical pentagonal bi-kite to which all three possible mirror image bi-kites are attached by each side type. And since the pentagon is symmetrical, two of these mirror image bi-kites have two sides to which they can be attached, while the third has only one. So there are four possible octo-kites that you could construct by this approach. I wonder if all four would be aperiodic monotiles, or just the one.
its just the fact that this simple shape that seemingly comes out of nowhere has a VERY unique property. these two (families of) monotiles have been out there in the space of possible shapes and its just never been found until now. why do they exist? what makes this combination of kites special?
are there tilings that go a long ways out and seem to be periodic or aperiodic but then change from seemingly periodic to aperiodic or the reverse? are there tilings that go a long ways out before they break and stop being able to tile at all? is there a maximum finite tiling that knowingly breaks? is it possible to construct a tiling that's unknowably periodic? i.e. it's impossible to prove if it's periodic or aperiodic?
Its more than one tile. PROOF: if someone did it with zero flipped tiles you would agree that this is stronger than doing it with flips. If it is stronger then you cant agree that they're both aperiodic monotiles and so its two tiles thank you and goodnight 😂
Год назад+1
30:13 What a wonderful thing to say and such a high note to end the video. Amazing interview, excellent questions and honest answers.
Von Neumann probes would build their circuitry and sensory through these shapes, rather than straight edges. The curved areas would allow for more ports/slots on the edges to connect these pieces for whatever data is needed. Theoretically speaking, of course.
There is a fundamental connection between the flipped-tile interval in the aperiodic tiling of "the hat" and the 3x+1 Collatz conjecture. I have discovered a truly marvelous demonstration of this proposition that this comment section is too small to contain.
What's so weird about this is how obvious a potential solution it is. There are not that many combinations of kites from hexagons, and yet nobody tried them!
because they didn't try by brute force it's weird nobody else cared to use computing power to get this low hanging fruit that's why chatgpt will make us even more lazy cleaning up all low hanging fruits leaving only hard problems lol
What is the performance of these for a game board? Square tilings distort distance on the diagonal by root 2 to the centre of the square. Hexes are better but still distort at 2 away from the origin. What is the best periodic tiling where the number of shapes you have to traverse is closest to the distance between the centres of the shapes?
Hexagons are the best regular polygon tiling for that. I don't know if there's a better irregular shape - my intuition is not, but it is just an intuition.
I couldn't have come up with it, but the hat shape is really just two congruent inverted pentagons under two congruent overlapping rectangles with their opposite corners aligned.
I wonder if there is a higher dimensional periodic tile that results in this aperiodic monotile via the cut-and-project method that is used to describe quasi crystals?
It seems that there are 12 orientations used in the chiral aperiodic tiling. Is it possible to use only 1 orientation to make an aperiodic tiling? Or what is the minimum orientation needed to do that? :P
@@bitfloggerI wouldn't be surprised if it is soon proven there are infinitely many. It seems that once a breakthrough like this is made, it can unlock further discoveries very quickly
Should be possible to make a segment of 3D printer filament with preexisiting bracing that can interleave to create adamantine stability without resorting to custom Chiral space filling.
I'm wondering, do they necessarily tile asymmetrically or is it simply that they can? I mean, if you were building a tiling, do you have to put them together particularly to maintain asymmetry, or is there no other way to put them together? Are there missteps that you could make, where you cannot tile further or completely? Are there similarly asymetric blocks?
With the kite-and-dart Penrose tiling, there are rules that you have to follow to ensure that the pattern can keep going. So I'm guessing the same logic applies here.
An aperiodic tile set means it CAN'T tile periodically no matter how hard you try. It's easy to make an aperiodic tiling, but finding a tile which only tiles aperiodically is much harder.
There is no way to put them together in a periodic way. That is what it means for a tile, or set of tiles, to be aperiodic. They force a non-periodic pattern. By the way, the fact that the Penrose tiles have an edge-colour-matching rule is not important because you can add bumps/dents to enforce such rules to make it purely about the shape. If you tile the hat by hand, there are lots of ways to get stuck, often only after unknowingly placing many tiles incorrectly.
Does anyone know if with a given spectre tile there is only infinite aperiodic tiling; more than one; or an infinite number? I.e. if there was only one, no matter how you start, eventually you would generate the same tiling as from any other starting position. Is this an open question, or do we have the answer?
Since there are multiple ways of connecting two tiles, my guess is that is is possible to make more than one different tiling, but I'm not 100% sure. You can, for example, connect them in a straight line, all having the same orientation, going off to infinity. (With the monotile on this video, specifically.)
Genuinely brilliant interview, by both both sides. Brady asks all the right questions, and Craig gives real answers to them. It's rare for it to bring the human element into the research without going too far one way or the the other.
I absolutely agree. Great format/style for a numberphile video - thrilling and captivating!
To me it seems like he's repeating a bit. I haven't paid 100% attention, but it feels like he asked him "How does this make you feel?" like four times, each time worded slightly differently.
It was still a fun interview nonetheless.
@@excelelmira Yeah, asked him how he felt _about different aspects_ of it. Which is... exactly how a half hour interview should be conducted lol
@@QuantumHistorian Should be? According to whom? I believe that such a long interview would benefit from more varied questions (or from being shorter).
@@excelelmira And I believe it's best to pay 100% attention to something before making recommendations on it.
Brady is such a great interviewer. I miss Hello Internet.
Also more Numberphile Podcast pls
Yeah, hello internet was the GOAT…
many many moons now
Definitely needs to make a comeback.
I miss it too😢
Loved seeing the emails between the researchers, as they added more people onto the team. You can imagine what it must be like for a mathematician to get a mesage from your peers saying "We have a promising lead on the biggest open question in our field, and we think you're the ideal person to work on it." (In more cautious language of course, but they know exactly what it means.)
"I'm putting together a team"
" you son of a bitch...I'm in"
For anyone trying to find the Japanese artist Prof. Kaplan is mentioning on several occasions, the proper spelling is "Yoshiaki Araki".
+
I love how it was found by a random shape enthusiast. Just so cool that this guy could find it with awesome intuition
A recreational mathematician, just like Fermat.
Not random. Dave. That man is a true mathematician
I wish people would stop with those comments. There's nothing 'random' about a person just because he is not some professor in a university. Ande also, it makes it sound like he just found it by accident. And, no: he is an artist and he found the shape deep in his field of interest in which he was working.
Craig wasn’t my professor, but we had common office hours in first year and I went to visit him every week.
He was a great teacher, and I never expected him to show up on numberphile before computerphile!
What a wonderful interview. The guest was very generous to all involved, from his coauthors to the listeners.
I’m glad that David Smith got top billing on the article.
If Craig is looking for a new quest... well, he can always go one dimension higher and look for an aperiodic monosolid.
Nice, but maybe it only works in even dimensions.
stolz
Or maybe a chiral aperiodic polygonal tile
@@fburton8 well now I want to see a 3d projection of an aperiodic 4d hypertile
@@thesenamesaretaken if the tile is made thicker and tiled into a plane and then layers of these planes are stacked, does that not count as a aperiodic polygonal tiling? just asking... thanks
Craig, really enjoyed the talk. Great to relive the moment. Fantastic journey. Many thanks.
+
Should interview this David guy too. Interesting to see a non-mathematician get real work done in math.
Exactly what I was thinking. He basically found the solution and then prof. Kaplan verified and made rigorous. I'd love to hear more about David's process.
Brady, you are amazing at interviewing. The window you open to the world's incredible nature is mind-blowing. Thank you for sharing with us.
I like how it looks like a Tshirt
Was thinking the same thing. I'd call it a t-shirt tile.
A torn-up t-shirt?
@@asheep7797you might call it "high fashion"
The other one looks like a pancho
@@asheep7797 Or a shirt where one side's tucked in and the other side isn't.
I've been waiting for a Numberphile video about this monotile! I've been interested in this subject since I read Martin Gardner's columns on Penrose tiles years ago. Thanks for sharing information about this interesting and important discovery.
Me too! I have a copy of that issue of SciAm where he talks about the Penrose tiles.
Great attitude to see the criticism of flipping the shape as another solution to solve
Wonderful closing words and beautiful interview. Thank you both very much!
I don't know much at all about tilings but it's so much fun seeing how important and exciting this is.
I love how you talk to guests, who are often academic (and frankly, typically stifled by strictness and disallowedness), but you as well as they are shown to just be normal people.
As a gamedev/artist/vfx geek, this is super interesting. Love this stuff 🖤
Someone on Reddit mentioned that an aperiodic monotile would make tiling in video games look more realistic. Like how water from above is just squares repeating and breaks immersion, a monotile could break it up more naturally.
Nice that you mention David Smith. You know, the guy that discovered this thing.
the man who discovered the shape twice. a true mathematician!
The shape they called a hat was a t shirt to me lol. It's crazy to see these shapes and so clearly see how they could tile a plane. I wish I could see their reaction when they realized that they found it.
Agreed! Looks like a v-neck.
Craig seems like a nice bloke. Happy for him.
why would it matter?
Respect for david
Didn't expect the double upload
Me neither
Me neither
Me neither
I did
Me neither
Now: What is the smallest number of edges that a polygonal aperiodic monotile can have?
I love that this hippie shape enjoyer created some shape and was like hey man I made this shape but it’s not working properly 😂
He knew perfectly well what he found
UWaterloo content! Love it when I get to see someone local.
WATER WATER WATER! LOO LOO LOO!
thank mr goose
@@raytonlin1 nonsense, Waterloo STEM students don't do the chear.
Dave Smith is a genius.
I've been called many things before but never 'a genius'. You are too kind.
I want a t-shirt that just is the tile shape. WIth the point at the bottom and the asymmetrical wonky sleeves, and offset v-neck, but it would be amazing.
0:54 Did anyone else appreciate how Craig's background perfectly defined one of the kites that makes up the hat tile?
Yeah it looked great tbh
As brilliant as this story is, as incredible as this interview is, the editing is pure joy. :D
Clever and amusing presentation design putting the videos shaped windows!
It would be cool if these tilings could be used as texture assets in videogames. Then somewhat simple mathematic formulae could be used to make complex graphics.
Aperiodic tiles have already been used as a texturing trick for quite a while - not using weird-shaped *mono*tiles, but several square Wang tiles with rules for what can connect on what side.
Not sure there would be much appetite for aperiodic tiling in computer graphics. It would be more complicated than triangle or square tiling, which is what everyone uses now, and as long as you keep your textures subtle you don't really have to worry much about the periodicity being obvious.
@@ZekeRaiden In old games there would be visible artifacts is large fields of similar texture, like in for example grass.
But it would be overkill to apply this tiling for that issue.
It would be interesting to see if some one makes a board game like Carcassonne with this tiling.
why to use more complex part instead of smaller generic ones?
nobody cared to even find this answer you got here just simply checking all possible diamond built tiles
it means this discovery might be just art for art and all you people hyped about it just pumping empty balloon
seemed like an impossible problem, turns out to be the exact opposite,
as a huge geometry fan, this discovery is HUGE for me
i love this
Those arrangements of cardboard cutouts are really wonderful.
When I read of the einstein I'd been waiting for the numberphile about it to come out! Exciting to hear that it was delayed because there's a new and better one.
When I first saw the 'one stone' tile and heard what it could do, it felt 'broken' to me.
Couldn't explain it, so a made a bunch and played with them.
In very short time, I was making periodic structures in 30 degree increments.
'Specter' tiling fixed the problem; I can look at piles of tiles without getting a sick headache anymore.
I too am a shape hobbyist. I have not experienced this level of success
18:11 - the careful distinctions between things like calculating vs computing, polyomimos and polyforms, is when you know you're listening to a passionate expert in a very specific field
An interesting thing about the distribution of the reflective hats is that they seem to be 2 connected hats apart from each other?
I'm really amazed about how easy the discovered monotile is no generate
I really want to tile my new bathroom with the hat. This is a must. I need those tiles if only to bug my friends as they try to find a repeat, and fail.
great story, what a time to be alive.
Interesting to hear about the timeline of discovery and how fast it moved, especially from the hat to the spectre.
In terms of does using flips count as a true monotile I feel like it depends. In a purely 2d space I'd say without flips is best, but physically in a 3d space I'd say with flips counts so long as the material you're using doesn't look different depending on whether the tile is flipped or not. So generally I'd say physically in a 3d space the hat is a monotile as is the spectre, but in terms of a purely 2d space I'd say probably just the spectre although it's up to interpretation.
Perhaps a simpler way to put it: up to chirality, there is at least one polygonal (straight-edged) aperiodic 2D monotile. If chirality is enforced, there is no known polygonal aperiodic monotile, but you can construct an infinite family of monotiles where the vertices are connected by congruent curves rather than straight edges.
The "hat" is nice because it is polygonal, but it requires you to ignore chirality (or be in a space where 2D chirality is irrelevant, e.g. 3D space or higher.) The "spectres" are nice because they are genuinely monotiles (fully achiral), but you have to give up the straight edges.
Now, the next question is: is there a polygonal aperiodic chiral monotile?
@@chalichaligha3234 shh, don't tell them, i learn that it is disrespectful to backseat experts
@@starrmayhem if he didn't recognize the problem he posed was solved before he posed it by the very video he watched then he's not an expert
@@SilverLining1 hi~
Mathematically, the hat isn't a monotile.
I love that the octo-kite is actually a symmetrical pentagonal bi-kite to which all three possible mirror image bi-kites are attached by each side type. And since the pentagon is symmetrical, two of these mirror image bi-kites have two sides to which they can be attached, while the third has only one. So there are four possible octo-kites that you could construct by this approach. I wonder if all four would be aperiodic monotiles, or just the one.
🔺 Bravo David Smith! 🔻
☀☀☀☀☀☀☀☀☀
Well, hats off to you all! That's great!
0:10 These look like the kiki and boba of aperiodic monotiles.
Way beyond cool that these tiles are being discovered (and I'm around to see it happen!)
Superb interview
Amazing interview!
finally early to a numberphie video, and it's about tiling, honestly i see this as an absolute win
its like finding a prime number in shapes or something. what a weird problem space. i aint never heard of this before
its just the fact that this simple shape that seemingly comes out of nowhere has a VERY unique property. these two (families of) monotiles have been out there in the space of possible shapes and its just never been found until now. why do they exist? what makes this combination of kites special?
Cute and creative video editing! Now time to work on an even simpler specter shape!
Finally you did this video
are there tilings that go a long ways out and seem to be periodic or aperiodic but then change from seemingly periodic to aperiodic or the reverse?
are there tilings that go a long ways out before they break and stop being able to tile at all?
is there a maximum finite tiling that knowingly breaks?
is it possible to construct a tiling that's unknowably periodic? i.e. it's impossible to prove if it's periodic or aperiodic?
To be fair, if "einstein" means "one stone" then even if you have to flip it, you only need one shape of that stone
Its more than one tile. PROOF: if someone did it with zero flipped tiles you would agree that this is stronger than doing it with flips. If it is stronger then you cant agree that they're both aperiodic monotiles and so its two tiles thank you and goodnight 😂
30:13 What a wonderful thing to say and such a high note to end the video. Amazing interview, excellent questions and honest answers.
Fantastic!
Von Neumann probes would build their circuitry and sensory through these shapes, rather than straight edges. The curved areas would allow for more ports/slots on the edges to connect these pieces for whatever data is needed.
Theoretically speaking, of course.
There is a fundamental connection between the flipped-tile interval in the aperiodic tiling of "the hat" and the 3x+1 Collatz conjecture. I have discovered a truly marvelous demonstration of this proposition that this comment section is too small to contain.
Cool story bro
I must go milk the cow, but tomorrow I will prove this theory.
What's so weird about this is how obvious a potential solution it is. There are not that many combinations of kites from hexagons, and yet nobody tried them!
because they didn't try by brute force
it's weird nobody else cared to use computing power to get this low hanging fruit
that's why chatgpt will make us even more lazy cleaning up all low hanging fruits leaving only hard problems lol
What is the performance of these for a game board? Square tilings distort distance on the diagonal by root 2 to the centre of the square. Hexes are better but still distort at 2 away from the origin. What is the best periodic tiling where the number of shapes you have to traverse is closest to the distance between the centres of the shapes?
Hexagons are the best regular polygon tiling for that. I don't know if there's a better irregular shape - my intuition is not, but it is just an intuition.
is there a 3-D analog? what about in n-D?
How did you do the irregular curvy shape as a mask in the video?
Brady, please do a video on young Daniel Larsen and his amazing paper on Carmichael numbers.
What's going on with the arcs drawn on the Penrose Tiles and the Trilobite and Crab? Is it a guideline for how to place them to successfully tile?
Yes, they're basically rule enforcements.
Oh what a great time to be alive!
at Queen Mary's university in London, one of the walls has Penrose tile design
Someone please make a “hat” shaped cookie cutter so that professor Kaplan can safely eat (a bunch of) his hat!
That's interesting. Really cool in fact.
Can these shapes also tile non-planar surfaces eg. a cylinder, sphere or moebius strip?
I couldn't have come up with it, but the hat shape is really just two congruent inverted pentagons under two congruent overlapping rectangles with their opposite corners aligned.
or, as David said, a bunch of kites.
its so amazing that it was discovered by a hobbyist!!!!!
I love the conversation but Dave smith always feels like an after thought to me, where the interview with him!??!
Yes: it would be quite challenging to prove a negative, but finding a single aperiodic monotile is demonstrating the positive.❤
No visual tiling of the new shape? :(
CLIFFHANGER
It does take time to make all those fancy animations
I'm curious about the use of the hat as polygonal masonry. Earthquake proof walls??
Canada on Numberphile! Hurray!
It's a shirt that's not tucked into the pants on one side obviously.
Great vid Brady.
Unfortunately tiles are usually only glazed on one side. An unglazed tile stamped both sides would work.
I want to see photos of some of these things made with the tiles!
Thank you
Is there a relationship between the aperiodic monotiles and the transcendental numbers?
very interesting
what about non-euclidean tiling? such as the surface of a sphere
Q: When can I buy these in ceramic for redoing my kitchen tiles?
I wonder if there is a higher dimensional periodic tile that results in this aperiodic monotile via the cut-and-project method that is used to describe quasi crystals?
Imagine tiling the entire maths building with one tile❤
where to find the works Prof Kaplan talks about? i wonder and wander. Thanks in advance to anyone
well done you guys
It seems that there are 12 orientations used in the chiral aperiodic tiling. Is it possible to use only 1 orientation to make an aperiodic tiling? Or what is the minimum orientation needed to do that? :P
watching this video made me miss the Numberphile podcast
how many tiles are needed to build a minimum bigger tile of the same shape?
Do you think there are other aperiodic monotiles out there waiting to be discovered?
Or a proof that there "can be only one" (reference to Highlander).
@@bitfloggerI wouldn't be surprised if it is soon proven there are infinitely many. It seems that once a breakthrough like this is made, it can unlock further discoveries very quickly
Should be possible to make a segment of 3D printer filament with preexisiting bracing that can interleave to create adamantine stability without resorting to custom
Chiral space filling.
I'm wondering, do they necessarily tile asymmetrically or is it simply that they can? I mean, if you were building a tiling, do you have to put them together particularly to maintain asymmetry, or is there no other way to put them together? Are there missteps that you could make, where you cannot tile further or completely? Are there similarly asymetric blocks?
With the kite-and-dart Penrose tiling, there are rules that you have to follow to ensure that the pattern can keep going. So I'm guessing the same logic applies here.
An aperiodic tile set means it CAN'T tile periodically no matter how hard you try. It's easy to make an aperiodic tiling, but finding a tile which only tiles aperiodically is much harder.
There is no way to put them together in a periodic way. That is what it means for a tile, or set of tiles, to be aperiodic. They force a non-periodic pattern. By the way, the fact that the Penrose tiles have an edge-colour-matching rule is not important because you can add bumps/dents to enforce such rules to make it purely about the shape. If you tile the hat by hand, there are lots of ways to get stuck, often only after unknowingly placing many tiles incorrectly.
@@jonahbranch5625 Why is it easier to make a tiling set that can be aperiodic but is not necessarily aperiodic? I would think that much harder.
@@willcool713 I'm not sure why. But you can tile 2x1 rectangles aperiodically just by rotating two of them so it doesn't seem that hard
Is the hat they talk about related to the shirt tile they show?
Does the 'hat' now provide a blueprint for creating other such shapes?
The hat is made of 8 kites. Is that the minimum number? Or could we make an aperiodic monotile with fewer kites or darts?
Does anyone know if with a given spectre tile there is only infinite aperiodic tiling; more than one; or an infinite number? I.e. if there was only one, no matter how you start, eventually you would generate the same tiling as from any other starting position. Is this an open question, or do we have the answer?
Since there are multiple ways of connecting two tiles, my guess is that is is possible to make more than one different tiling, but I'm not 100% sure.
You can, for example, connect them in a straight line, all having the same orientation, going off to infinity. (With the monotile on this video, specifically.)
Pity some of the tiles are mirror images ----- thats like two tiles.