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When David Smith, Joseph Myers (@jsm28), Chaim Goodman-Strauss and I posted our paper "An aperiodic monotile" (https://arxiv.org/abs/2303.10798) back in March, we answered the open problem of whether a single shape could tile the plane aperiodically.
Some people observed that tilings by the "hat" used both unreflected and reflected tiles. Although the einstein problem was answered, in some contexts (e.g., tile floors), you'd likely have to manufacture two separate tiles. Our paper left open the question of whether a shape could tile aperiodically using translations and rotations only, with no reflections.
I hate to sound like a broken record, but then, on March 26th, Dave noticed something interesting. (1/n)
Recall that the hat and the turtle belong to a continuum of shapes denoted Tile(a,b) for edge lengths a and b. The hat is Tile(1,sqrt(3)), and the "turtle" is Tile(sqrt(3),1). All members of that continuum are aperiodic except for the equilateral polygon Tile(1,1). Dave became curious about Tile(1,1) after seeing Yoshi Araki playing around visually with its close relative Tile(1,1.01). (2/n)
Because Tile(1,1) is equilateral, it permits a wider range of adjacencies, which then allow it to tile isohedrally using equal numbers of unreflected and reflected tiles. (3/n)
But what if we counterbalanced that extra freedom by, say, just forbidding reflections outright? Dave discovered that if he tried to place copies of Tile(1,1) by translation and rotation only, well, he didn't get stuck but he couldn't find a block of tiles that repeated by translation. Needless to say, the four of us began studying this shape more intensively. (4/n)
After a couple of months of work, we cracked it: If you only allow yourself to tile by translations and rotations, then Tile(1,1) admits only non-periodic tilings! We call this a "weakly chiral aperiodic monotile" -- it's aperiodic in a reflection-free universe, but tiles periodically if you're allowed to use reflections.
The tiling is reminiscent of, but not identical to, hat tilings -- it contains a sparse population of "odd" tiles, which are rotated by odd multiples of 30 degrees relative to all other tiles. (5/n)
Does that matter? Hang on, there's one more step! Because Tile(1,1) is equilateral, and because we're not using reflections, it's easy to modify its edges to *force* it to tile without reflections.
These shapes, which we call "Spectres", are "strict chiral aperiodic monotiles": shapes that are forced to tile aperiodically, and can't use reflections! If you objected to the hat because of its reflections, this is the shape for you. (6/n)
Yuck, my thread broke apart. For completeness, the remaining posts are here:
(7/n): https://mathstodon.xyz/@csk/110454755267183546
(8/n): https://mathstodon.xyz/@csk/110454766587723863
(9/n): https://mathstodon.xyz/@csk/110454792622698781
Also, I neglected to include @Chaimgoodmanstrauss in my original post. Sorry!
@csk @Chaimgoodmanstrauss I decided that it looked like a fiddler crab. https://sigmoid.social/@moultano/110495545831485317
@csk @Chaimgoodmanstrauss threads don’t federate anyway, I had to go to your instance to read it, and I hate that because Mastodon breaks using the cursor ↑/↓ keys to scroll 
Better do one large post, really.