Abstract Regular Polytopes

Abstract

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.

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

“…There are a number of standard operations on maps which create new regular maps from old (see [6,22]). The duality operation δ replaces a map P by its dual P δ (often denoted P * ); algebraically this corresponds to reversing the order of the generators ρ 0 , ρ 1 , ρ 2 of the group.…”

“…Throughout we insist that any two adjacent faces, which share a common edge, do not lie in the same plane. We usually identify P with the underlying map (cell complex) on the surface, or with the abstract polyhedron consisting of the vertices, edges, and faces (as well as ∅ and P as improper elements), partially ordered by inclusion (see Coxeter and Moser [5] and McMullen and Schulte [6]). In any case, since P is embedded in E 3 , the underlying surface is free of self-intersections and is necessarily orientable; and since the faces of P are convex, the underlying abstract polyhedron is necessarily a lattice, meaning here that any two distinct faces meet, if at all, in a common vertex or a common edge.…”

“…They can be studied from a combinatorial and topological viewpoint as cell-complexes (tessellations) on surfaces, essentially as abstract polyhedra (see [6]), or be viewed as complex algebraic curves over algebraic number fields (see Jones and Singerman [7]). …”

Regular 4-polytopes from the Livingstone graph of Janko’s first group

“…In this paper, we decide to take the incidence geometry point of view since our aim is to construct the two polytopes mentioned in the introduction as incidence geometries. The interested reader can find the basic theory of regular abstract polytopes in [14].…”

Aesthetic Patterns with Symmetries of the Regular Polyhedron

“…Due to the perfect symmetry of regular polyhedra, they have been the subject of wide attention [1][2][3][4][5][6]. Dutch artist Escher et al [7] designed several amazing woodcarvings of polyhedral symmetries.…”