Twisted Groups and Locally Toroidal Regular Polytopes (original) (raw)
Related papers
2002
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.
Locally toroidal regular polytopes of rank 4
Commentarii Mathematici Helvetici, 1992
The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3, p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schl~ifli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (sections of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.
Higher Toroidal Regular Polytopes
Advances in Mathematics, 1996
A regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids. The locally toroidal polytopes of rank 4 have been extensively discussed, and their classification is now nearly complete. In this paper, the locally toroidal polytopes of higher rank are investigated. Again, an almost complete classification of these regular polytopes is obtained; as well as sporadic examples, there are several infinite families.
Symmetry Properties of Generalized Regular Polytopes
arXiv (Cornell University), 2005
1)-dimensional elements is not necessarily integer, though all the combinatorial and metric properties meet those of regular polytopes in a classic sense. New relationships between Schlafli symbol { f 1 ,..., f n-1 } of the regular polytope and its metric parameters have been established. Using the generalized regular polytopes concept, group and metric properties of arbitrary metric space tessellations into regular honeycombs were investigated. It has been shown that sequential tessellations of space into regular honeycombs determine an infinite discrete group, having finite cyclic, dihedral, symmetric, and other subgroups. Set of generators and generating relations of the group are identified. Eigenvectors of regular honeycombs have been studied, and some of them shown to correspond to Schlafli symbols of known integer regular polytopes in 3 and 4 dimensions. It was discovered that group of all regular honeycombs comprises subsets having eigenvectors inducing a metric of the (p, q) signature, and in particular, (+-). These eigenvectors can be interpreted as self-reproducing generalized regular polytopes (eigentopes).
On locally spherical polytopes of type {5, 3, 5
Discrete Mathematics, 2009
There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is J 1 × J 1 × L 2 (19), where J 1 is the first Janko group, of order 175560, and L 2 (19) is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one.
Locally toroidal polytopes and modular linear groups
Discrete Mathematics, 2010
When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d ≥ 2, one obtains a finite group G d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.