alex @alexthecamel@schelling.pt

(2)for any T we can construct an S so that T->S

this time whenever we make a decision we have finitely many options

if on the other hand we're always making infinitely many decisions, we have continuum many tilings
(we can imagine making each decision based on the next unused bit of an infinite bitstring, say)
so lemma (1) done.

if we can build our tiling so we're only making finitely many decisions, after we've made all those decisions we're left with some patch P.
for every occurrence of P in T it extends to S, and as it occurs infinitely many times T is periodic

we also want T->S
we say a "decision" happens when there is more than one way to extend our tiling so that it appears in T (and thus appears in T infinitely many times)

lets imagine building up a tiling S step by step (for concreteness in say, a square grid we can enumerate the squares in a spiral around the origin and say at each stage we're adding a tile to cover the uncovered square with the smallest number, so we reach everywhere eventually)

for any tilings of the plane T,S we write T->S if every finite patch of tiles in S occurs infinitely often in T.

(1) we first show that if T->T either T is periodic or there are uncountably many tilings

ok here's a more concrete + human understandable proof of uncountability
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RT @pawnofcthulhu
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? https://twitter.com/cs_kaplan/status/1637996332475359232
https://twitter.com/pawnofcthulhu/status/1638004102352498688

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

this is decidedly less awkward if you don't sell the indigenous people guns and horses and let them fight it out for a century or two before you move in

ok the fact that land acknowlegements were a thing in Australia well before spreading to the US explains ... a fair bit

be a shame if labour couldn't reach majority government 🤔

apropos of nothing
(snp + the lib dems aren't going to support a tory government)

gotta say 'the Liberal Party are out of touch! how can they win back the woke metropolitan elites?' is a very entertaining conventional wisdom and I'm going to enjoy it while it lasts.

we should not have exposed a bunch of nerds who were bullied in high school to the concept of 'weirdness points'

it does tend to get annoying when contextualising norms are so strong they don't let you say anything more complicated than 'red team good blue team bad' tho

of course for a viewpoint like this there are going to be people you can't persuade and who are going to be angry at you either way

contextualising norms are for when you have influence and the people you are talking to aren't sure whether to trust you (esp to make policy decisions)

(in case you couldn't guess the half dozen dots off the main sequence are very remote/outback areas with high indigenous population)

nitpick but if you're citing the 5 million number melbourne has 29 metropolitan seats

sorry i had to scale the dots by population, can't have you thinking we're *too* poor; resolution is not great but they range from like 40k to 800k