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
(in actuality S_α will become constant while α is still countable
because our basis is countable
and the set of things B_α in our basis which are a subset of the open set S\S_ α always gets bigger as α goes up
and as S\S_α is the union of all things in B_α...)
https://twitter.com/CihanPostsThms/status/1644700418583273473
here's a result you get with very similar argument lol
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RT @CihanPostsThms
The following is a theorem of ZFC (in particular CH is not assumed):
Let G be a group of cardinality ≤ |ℕ|. Then the cardinality of the set of subgroups of G is
• either ≤ |ℕ|,
• or equal to |ℝ|.
https://twitter.com/CihanPostsThms/status/1644700418583273473