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.

I've been posting some "shorts" on this channel (6 so far) without sending them to subscriptions/notification feeds, so check those out here: https://www.youtube.com/@ComboClass/s...
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!
One of those livestreams was right after the most recent monotile discovery came out, and contains another description of aperiodic tilings and a look through the papers I'm referencing: https://www.youtube.com/live/u8WifFDQ...

Here are the two papers published this year with the new discoveries:
https://arxiv.org/abs/2303.10798
https://arxiv.org/abs/2305.17743
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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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!

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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.