Mathematicians Just Discovered These Shapes!

A few months ago, mathematicians discovered the first "aperiodic monotile", and in the weeks since then, they've discovered new and improved families of these shapes. Let me tell you about the history of aperiodic tilings, from “Penrose tiles” to the “hat" monotile to the newest “spectre" monotiles that were just discovered!

Some clarifications/corrections:
- In the example I gave at 5 minutes, I used squares (which I thought was more unique than the more common examples that use rectangles) but I should have used rectangles, as I realized afterwards that shifting columns of a square grid wouldn't be quite enough to make it not periodic in all directions. The overall point there, that many periodic tilings can be altered to create non-periodic tilings, is true.
- When I describe how some tilings have a "glitch" but then have an arbitrarily large section that can be translated onto a matching section, some sources would call that tiling non-periodic due to the glitch. I should have called that type "not aperiodic" as opposed to "periodic" to be clearer. It's true that it still wouldn't be "aperiodic" (a different term than when sources say "non-periodic") if it had an arbitrarily large section that can do that.

Also make sure you're tuned in to my @Domotro channel where I post a bunch of bonus math videos, and also do livestreams a couple days a week!

Here are the two papers published this year with the new discoveries:
In this video, I used some images from those two papers, as well as some images from Wolfram “Mathworld” (images created by Eric W. Weisstein) and Wikipedia (images contributed by the users Maksim, Inductiveload, Parcly Taxel, and Geometry Guy).

This was filmed by Carlo Trappenberg.
Special thanks to Evan Clark and to all of my Patreon supporters:
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Domotro
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Berkeley, CA 94709

If you want to try to help with Combo Class in some way, or collaborate in some form, reach out at combouniversity(at)gmail(dot)com

In case people search any of these terms, some of the topics discussed in this episode are: 2D tilings/tesselations, periodic tilings, aperiodic tilings, squares / triangles / pentagons / hexagons, Wang tiles / Wang dominoes, Penrose tiles (including the "kite" and "dart" shapes), the new "hat" aperiodic monotile discovered by David Smith, the even newer "spectre" tiles that are chiral aperiodic monotiles he discovered, and more!

If you're reading this, you must be interested in Combo Class. Make sure to leave a comment on this video so the algorithm shows it to more people :)

DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.

[zuzaaa]

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

[benjamingrant]

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.

[autismuskaefer]

This could be used for game developers who want to hide pattern repetitions on large textures.

[keremkelleboz]

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

[frankmansour]

I can imagine puzzles made from such tiles. Taking frustration to a whole new level...

[roflanix]

Bro this channel must be popular some day

[PotatoVariety]

The thought that one single tile can be aperiodic kind of breaks my mind... These are by far some of the coolest shapes ever.

[peppermann]

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. 😊

[fireballninja]

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

[kasuha]

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.

[adamsouza]

“Artists” saying there’s no such thing as novelty, and mathematicians still discovering new shapes.

[soninhodev]

i never thought of doing an aperiodic tiling of something, but this sounds amazing, the infinite family of them, thats pure mathematics!

[pensiv]

I imagine there could be interesting implications should we find any natural molecules (or engineer some) which arrange like this monotile.

[kimjamuru]

This channel is the definition of chaotic good. 😂

[rmdodsonbills]

You have lots of vertices, but if you only had one, it would be a vertex.

[publiconions]

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

[theinternetis]

Great vid man ive been interested in these shapes forever!

[stooshie]

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.

[eugenezenzen]

One of your best, the editing is getting fancy - I like it!!

[bdhanes]

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. 🤓