An extension of the classification of high rank regular polytopes
Abstract
Up to isomorphism and duality, there are exactly two nondegenerate abstract regular polytopes of rank greater than $n3$, one of rank $n1$ and one of rank $n2$, with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n3$ and $n4$. The result is, up to a isomorphism and duality, seven abstract regular polytopes of rank $n3$ for each $n\geq 9$, and nine abstract regular polytopes of rank $n4$ for each $n \geq 11$. Moreover we show that if a transitive permutation group $\Gamma$ of degree $n \geq 11$ is the automorphism group of an abstract regular polytope of rank at least $n4$, then $\Gamma\cong S_n$.
 Publication:

arXiv eprints
 Pub Date:
 September 2015
 arXiv:
 arXiv:1509.01032
 Bibcode:
 2015arXiv150901032F
 Keywords:

 Mathematics  Combinatorics