cute
one quote "the tile admits uncountably many tilings": is it even possible to have a set of tiles that aperiodically tile the plane without having uncountably many tilings?
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RT @cs_kaplan
In a new paper, David Smith, Joseph Myers, Chaim Goodman-Strauss and I prove that a polykite that we call "the hat" is an aperiodic monotile, AKA an einstein. We finally got down to 1! https://arxiv.org/abs/2303.10798 4/6
https://twitter.com/cs_kaplan/status/1637996332475359232

ok i guess i'd stipulate "only exists an aperiodic tiling" (i think necessary) and "snaps on to a grid" (i suspect unnecessary) here

like the space S of aperiodic tilings comes with a compact topology where the basis of open sets is "we are given where a finite number of where the tiles are is"
and Z^2 acts freely on this set by translations because the tiling is aperiodic

hrmm. construct for any ordinal α S_α

S_0=S
S_{α+1} is S_α minus all its isolated points

S_λ is the intersection of S_α α<λ

S_α is always closed, nonempty, and infinite

S_α is also eventually constant if for no other reason than that eventually you run out of points. call the resulting set T

pretty sure because T is perfect and this is a nice enough topological space T has cardinality continuum

i think this reasoning still works if we're talking tiles where there's only a finite number of ways to glue 2 tiles together sensibly
you don't quite get the group action but you do still get a compact topology on all the ways to build a tesselation around some base tile

and one point in this space always gives you an infinite number because aperiodic