Unravelled abstract regular polytopes

regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.

regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grunbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS3, connected sums of handlesS 1 × S 2 , euclidean and spherical space forms, and other examples with non-trivial fundamental group.

Regular polytopes (RPs) are an extension of 2D (two-dimensional) regular polygons and 3D regular polyhedra in n-dimensional ( n ≥ 4 ) space. The high abstraction and perfect symmetry are their most prominent features. The traditional projections only show vertex and edge information. Although such projections can preserve the highest degree of symmetry of the RPs, they can not transmit their metric or topological information. Based on the generalized stereographic projection, this paper establishes visualization methods for 5D RPs, which can preserve symmetries and convey general metric and topological data. It is a general strategy that can be extended to visualize n-dimensional RPs ( n > 5 ).

This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schlafli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.

This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schläfli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.

Almost simple groups with socle [formula omitted] acting on abstract regular polytopes

Regular and Chiral Polytopes

Regular complex polytopes in unitary space have been studied since the 1950s. The advent of computer graphics has enabled us to represent these on the real plane in a meaningful way with very little effort. Here we are chiefly concerned with a particular sequence of complex polytopes in two, three and four dimensions and with some associated real polytopes.

Regular Polytopes

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders globally and locally. They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled.

Here, for $W$ the Coxeter group $\mathrm{D}_n$ where $n > 4$, it is proved that the maximal rank of an abstract regular polytope for $W$ is $n - 1$ if $n$ is even and $n$ if $n$ is odd. Further it is shown that $W$ has abstract regular polytopes of rank $r$ for all $r$ such that $3 \leq r \leq n - 1$, if $n$ is even, and $3 \leq r \leq n$, if $n$ is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.

regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schlafli types {4,4,3} and {4,4,4}.

Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that each of these subcones is isomorphic to a set of positive semi-definite hermitian matrices of dimension $m$ over either the real numbers, the complex numbers or the quaternions. In particular, we correct an erroneous computation of the dimension of these subcones by McMullen and Monson. We show that the automorphism group of an abstract regular polytope can have an irreducible character $\chi$ with $\chi\neq \overline{\chi}$ and with arbitrarily large essential Wythoff dimension. This gives counterexamples to a result of Herman and Monson, which was derived from the erroneous computation mentioned before. We also discuss a relation between cosine vectors of certain pure realizations and the spherical functions appearing in the theory of Gelfand pairs.

On the number of abstract regular polytopes whose automorphism group is a Suzuki simple group [formula omitted

For each almost simple group G such that S ≤ G ≤ Aut(S) and S is a simple group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract regular polytopes on which G acts regularly. The results have been obtained using a series of Magma programs. All these polytopes are made available on the first author's website, at http://cso.ulb.ac.be/~dleemans/polytopes/. © Birkhauser Verlag, Basel 2007.