Gro-TsenJan 13
Here'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
Show threadMoritz Firsching

@gro-tsen.bsky.social
All 4-polytopes with up to 9 vertices can be realised with rational coordinates, see
https://firsching.ch/papers/j006-enumeration-of-4-polytopes.html

I think the smallest explicitly known example of a 4-polytope that cannot be realised with rational coordinates has 33 vertices (unless I missed some more recent development). So it is an open question what the smallest number of vertices of a non-realisable 4-polytope is!

The complete enumeration of 4-polytopes and 3-spheres with nine vertices

We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 are non-polytopal.

Moritz Firsching
Feb 3 at 4:45pmWeb