Gro-TsenHere's a mathematical fact which I find amazingly counter-intuitive:
There exists a polytope (=the convex hull of a finite set) in ℝ⁴ which is not combinatorially equivalent to one with rational vertex coordinates (=the convex hull of a finite subset of ℚ⁴)!
🤯
•1/4
jcreed@gro-tsen.bsky.social I found this very strange, too... https://en.wikipedia.org/wiki/Perles_configuration "In polyhedral combinatorics" seems to be the referent in case you want to dig into it
domotorp@jcreed @gro-tsen.bsky.social Nice! I find this believable, and after this I can also believe the result about the polytope.
@domotorp @gro-tsen.bsky.social can you say more about why you find it intuitive? I guess given polytope faces that *aren't* simplices I can't move vertices independently to the nearest rational point without necessarily "breaking" a face... but the rationals are dense! Surely there's so many rationals around to choose from! :)