Barry Monson

Regular 4-polytopes related to general orthogonal groups

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p }. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

Modular reduction in abstract polytopes

The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in Z[ ] (with the golden ratio), to construct new regular 4-polytopes of hyperbolic types{3,5,3} and{5,3,5} with automorphism groups given by finite orthogonal

Realizations of Regular Toroidal Maps

We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types{3,6} and{6,3}.

Eisentein Integers and Related C-groups

Using modular quotients of linear groups defined over the Eisenstein ring Z[ω], we construct infinite families of finite regular or chiral polytopes of types {3,3,6}, {3,6,3} and {6,3,6}.

Medial layer graphs of equivelar 4-polytopes

In any abstract 4-polytope P, the faces of ranks 1 and 2 constitute, in a natural way, the vertices of a medial layer graph G. We prove that when P is finite, self-dual and regular (or chiral) of type {3, q, 3}, then the graph G is finite, trivalent, connected and 3-transitive (or 2-transitive). Given such a graph, a reverse construction yields a poset with some structure (a polystroma); and from a few well-known symmetric graphs we actually construct new 4-polytopes. As a by-product, any such 2- or 3-transitive graph yields at least a regular map (i.e. 3-polytope) of type {3, q}.

Polytopes related to the Picard group

Abstract In a previous paper we constructed a finite regular (abstract) 4-polytope of type {3,3, p } for each odd prime p . Here we extend this construction, allowing p to be any positive integer. Distinct polytopes of the same type can then arise, some of which may be chiral; but in each instance, facets and vertex figures are regular.

Realizations of Regular Toroidal Maps of Type {4,4}

Abstract. We determine and completely describe all pure realizations of the finite toroidal maps of types {4,4}(b,0) and {4,4}(b,b) , b \geq; 2 . For large values of b , most such realizations are eight-dimensional.

A family of uniform polytopes with symmetric shadows

A peculiar manipulation of the Coxeter diagrams used in Wythoffs construction provides a family of orthogonal projections of one uniform polytope onto another.

Locally toroidal polytopes and modular linear groups

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, here we 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.