The chiral polytopes of full rank are very interesting. It was once erroneously proven by McMullen that there were none, however it has since been revealed to not be the case. I am aware of two chiral polytopes of full rank: (I'm also aware of chiral polytopes of full rank in projective space, but I'm not as interested in that. I'm just asking about real Euclidean space.)
* [Roli's cube](https://polytope.miraheze.org/wiki/Roli%27s_cube), a finite 4-polytope described by Bracho, Hubard, and Pellicer in 2014
* [$\mathcal P(3,8,3,3)$](https://polytope.miraheze.org/wiki/%F0%9D%93%9F(3,8,3,3)), an infinite 5-polytope described by Pellicer in 2021
Pellicer gives a proof that every chiral polytope of full rank is either rank 4 or 5. Pellicer states:
> Up to now, chiral $n$-polytopes of full rank were known only for $n\in \{4,5\}$.
and gives the two examples I have given.
Each of these two examples is the first of its kind. It would not be surprising if there were other discoveries after which are harder to locate because they were not the first. Are there any more known cases or is it just this two?
WheatWizard
·
2025-04-10T15:46:18Z (9 days ago)