Refurio AnachroRefurioAnachro@mathstodon.xyz
Nov 15, 2026, 4:45 PMOoh, there are planar zonotopes, too! Gorgeous!
Thank you, @MWinter !
Replies
Martin WinterMWinter@mathstodon.xyzNov 15, 2026, 5:22 PMTotally! The planar counterparts are tilings with polygons that are both inscribed and centrally-symmetric. It was a surprise to me how non-tivial it is to construct such. Already playing with just hexagons is interesting. The difficulty probably has the same source as for zonotopes.
Such tilings are dual to (inscribable) simplicial line arrangements. You get some interesting ones from restricting higher-dimensional reflection arrangements.
Still, the planar setting seems not quite as restricted as the spherical one:
- Refurio AnachroRefurioAnachro@mathstodon.xyzNov 15, 2026, 6:15 PM
Thanks! I have no intuition about the obstructions, does it get even easier for hyperbolic arrangements? I tend to guess it doesn't, but I can't tell why.
- Martin WinterMWinter@mathstodon.xyzNov 15, 2026, 6:32 PM
@RefurioAnachro It does, actually. The reason is that there is an infinite zoo of reflection groups in hyperbolic space. Each gives a simplicial line arrangement, which in turn is dual to such a tiling with inscribed centrally-symmetric polygons. The sphere and plane are the hard case.