Some other links:

* Twitter thread: https://twitter.com/cs_kaplan/status/1637996326129393666

* Page on the subject: https://cs.uwaterloo.ca/~csk/hat/

* Interactive application: https://cs.uwaterloo.ca/~csk/hat/app.html

Don’t miss the interactive editor! https://cs.uwaterloo.ca/~csk/hat/app.html

It looks like flipping the tile over is allowed. Is that right?

https://www.youtube.com/watch?v=W-ECvtIA-5A

From the paper:

> A single hat is asymmetric, and so in any patch we can distinguish between “unreflected” and “reflected” orientations of tiles.

Tweet showing what the actual monotile looks like: https://twitter.com/cs_kaplan/status/1637996332475359232?s=2...

Wow! I remember reading the Penrose tiles article (1977 Scientific American, Martin Gardner's Mathematical Games column https://www.scientificamerican.com/article/mathematical-game...) in the mid-80s, where if I remember right they were very excited to get the number of tiles required for a nonperiodic tiling down to 2.

Roger Penrose is still alive, I hope he is pleased to see this.

I was recently reading a fun paper by one of the authors on this paper, Chaim Goodman-Strauss. Very exciting to see him here!

For those who might be interested, it's a whirlwind of various problems in computer science around decidability. "Can't Decide? Undecide!" https://www.ams.org/notices/201003/rtx100300343p.pdf

Interesting, thanks for sharing! Also it's the first time I ever find graphical advertising in a paper, at the very end though.

Monumental achievement, but what's really extraordinary here is that this monograph is completely written/illustrated/typeset using LaTeX. Can't even begin to imagine how much effort this took.

The images aren’t necessarily generated with LaTeX! That being said, TikZ [https://pgf-tikz.github.io/pgf/pgfmanual.pdf] is really nice for making pictures — just look at the front page of their manual!

Less effort than MS Word, I assure you.

Almost all math papers are written in LaTeX. There is nothing unusual about this.

Probably not that much. The figures are almost certainly just included vector images.

I started to think if this hat tile shape could be used to make some sort of tile-laying game. In figure 1.1 you can see sort of hexagonal overlay grid, so maybe using that somehow

It's hard to make aperiodicity matter in a board game: as players place tiles, they either extend an aperiodic layout to a bigger one (unexciting) or they are unable to place a tile because someone, possibly much earlier, deviated from the "correct" construction.

Moreover, with a single einstein there is no tile unpredictability or choice whatsoever, eliminating two of the main strategic dimensions of tile-laying games.

One idea could be to have the tiles have different colorings, then your game goal could be to make some patterns or continuous areas. Example tile coloring scheme https://ibb.co/n0JJXF3

Could make for more interesting "terrain" of tiles used to form hex-based playfields.

I wonder what would happen with a molecule with this shape, what would happen to a crystal with this configuration?

These articles should come with a warning "Giant rabbit hole ahead" on them...

And you can't talk about quasicrystals without mentioning this mesmerizing animation https://www.jasondavies.com/animated-quasicrystals/

For extra added fun, try blinking while looking at the animation

i need this in my bathroom

This shape seems exceptionally suitable for matching continuous decoration patterns (e.g. knotwork) across tile edges and half edges.