Polytopes Of Rank Research Articles

Chiral polytopes of full rank exist only in ranks 4 and 5

Skeletal chiral polytopes of rank n in a Euclidean space must have as ambient space $${\mathbb {E}}^d$$ for some $$d \ge n$$ if they are finite, or some $$d \ge n-1$$ if they are infinite. If the dimension attains the lower bound just mentioned, we say that the polytope is of full rank. In this article it is proven that a chiral polytope of full rank can only have rank 4 or 5.

Polytopes of large rank for [formula omitted

This paper examines abstract regular polytopes whose automorphism group is the projective special linear group PSL(4,Fq). For q odd we show that polytopes of rank 4 exist by explicitly constructing PSL(4,Fq) as a string C-group of that rank. On the other hand, we show that no abstract regular polytope exists whose group of automorphisms is PSL(4,F2k).

ABSTRACT REGULAR POLYTOPES FOR THE O'NAN GROUP

In this paper, we consider how the O'Nan sporadic simple group acts as the automorphism group of an abstract regular polytope. In particular, we prove that there is no regular polytope of rank at least five with automorphism group isomorphic to O′N. Moreover, we classify all rank four regular polytopes having O′N as their automorphism group.

Mixing chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A "mixing" construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.

Constructions of Chiral Polytopes of Small Rank

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.

Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group S n of degree 5 ⩽ n ⩽ 9 are available. Two observations arise when we look at the results: (1) for n ⩾ 5 , the ( n − 1 ) -simplex is, up to isomorphism, the unique regular ( n − 1 ) -polytope having S n as automorphism group and, (2) for n ⩾ 7 , there exists, up to isomorphism and duality, a unique regular ( n − 2 ) -polytope whose automorphism group is S n . We prove that (1) is true for n ≠ 4 and (2) is true for n ⩾ 7 . Finally, we also prove that S n acts regularly on at least one abstract polytope of rank r for every 3 ⩽ r ⩽ n − 1 .

Eulerian abstract polytopes

The classical theory of regular convex polytopes has inspired many combinatorial analogues. In this article, we examine two of them, the eulerian posets and the abstract regular polytopes, and see what the overlap between the concepts is. It is shown that a section regular polytope is eulerian if and only if it is spherical, or it has even rank and is locally spherical. Equivelar polytopes of rank less than 4 are eulerian, and some progress is made towards a characterisation of equivelar eulerian posets in higher rank. In particular, necessary conditions are given for an equivelar quotient of a cube or a torus to be eulerian.

Generalized CPR-graphs and applications

We give conditions for oriented labeled graphs that must be satisfied in order that such are permutations representations for groups of automorphisms of chiral or orientably regular polytopes. We develop a technique for construction of highly symmetric polytopes using such graphs. In particular, we construct chiral polytopes with the automorphism group $S_n$ for each $n>5$, an infinite family of finite chiral polytopes of rank $4$, a polytope of rank $5$, as well as several infinite chiral polyhedra.

Constructions for chiral polytopes

AbstractAn abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on flags, with adjacent flags lying in different orbits. In this paper, we describe a method for constructing finite chiral n-polytopes, by seeking particular normal subgroups of the orientation-preserving subgroup of an n-generator Coxeter group (having the property that the subgroup is not normalized by any reflection and is therefore not normal in the full Coxeter group). This technique is used to identify the smallest examples of chiral 3- and 4-polytopes, in both the self-dual and non-self-dual cases, and then to give the first known examples of finite chiral 5-polytopes, again in both the self-dual and non-self-dual cases.