Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes

Gabe Cunningham

doi:10.37236/7070

Non-Flat Regular Polytopes and Restrictions on Chiral Polytopes

The Electronic Journal of Combinatorics ◽

10.37236/7070 ◽

2017 ◽

Vol 24 (3) ◽

Cited By ~ 1

Author(s):

Gabe Cunningham

Keyword(s):

Regular Polytopes ◽

Abstract Polytope ◽

Chiral Polytopes

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

Constructions of Chiral Polytopes of Small Rank

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-033-4 ◽

2011 ◽

Vol 63 (6) ◽

pp. 1254-1283 ◽

Cited By ~ 13

Author(s):

Antonio Breda D’Azevedo ◽

Gareth A. Jones ◽

Egon Schulte

Keyword(s):

Automorphism Group ◽

Small Rank ◽

Abstract Polytope ◽

Chiral Polytopes ◽

General Method

AbstractAn abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4, and 5.

Free Extensions of Chiral Polytopes

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-033-7 ◽

1995 ◽

Vol 47 (3) ◽

pp. 641-654 ◽

Cited By ~ 12

Author(s):

Egon Schulte ◽

Asia Ivić Weiss

Keyword(s):

Classical Theory ◽

Convex Polytope ◽

Regular Polytopes ◽

Geometric Structures ◽

Maximal Symmetry ◽

Abstract Polytopes ◽

Chiral Polytopes ◽

Classical Notion ◽

Extension Result ◽

Amalgamated Product

AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

Quasi-Regular Polytopes of Full Rank

Discrete & Computational Geometry ◽

10.1007/s00454-021-00304-5 ◽

2021 ◽

Author(s):

Peter McMullen

Keyword(s):

Full Rank ◽

Regular Polytopes

The Electronic Journal of Combinatorics ◽

10.37236/2512 ◽

2012 ◽

Vol 19 (3) ◽

Cited By ~ 5

Author(s):

Daniel Pellicer ◽

Gordon Williams

Keyword(s):

Automorphism Group ◽

Regular Polytopes

We discuss representations of non-finite polyhedra as quotients of regular polytopes. We provide some structural results about the minimal regular covers of non-finite polyhedra and about the stabilizer subgroups of their flags under the flag action of the automorphism group of the covering polytope. As motivating examples we discuss the minimal regular covers of the Archimedean tilings, and we construct explicit minimal regular covers for three of them.

Structure of 2-skeletons of higher dimensional regular polytopes

Mathematical journal of Ibaraki University ◽

10.5036/mjiu.50.27 ◽

2018 ◽

Vol 50 (0) ◽

pp. 27-33 ◽

Cited By ~ 1

Author(s):

Fumiko Ohtsuka

Keyword(s):

Regular Polytopes ◽

Higher Dimensional