Existence of regular 3-polytopes of order 2𝑛

In this paper, we prove that for any positive integers {n,s,t} such that {n\geq 10}, {s,t\geq 2} and {n-1\geq s+t}, there exists a regular polytope with Schläfli type {\{2^{s},2^{t}\}} and its automorphism group is of order {2^{n}}. Furthermore, we classify regular polytopes with automorphism groups of order {2^{n}} and Schläfli types {\{4,2^{n-3}\},\{4,2^{n-4}\}} and {\{4,2^{n-5}\}}, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hungar. 53 2006, 1–2, 231–255].

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.

Monodromy Groups and Self-Invariance

For every polytope 𝒫 there is the universal regular polytope of the same rank as 𝒫 corresponding to the Coxeter group 𝒞 = [∞, … ,∞]. For a given automorphism d of 𝒞, using monodromy groups, we construct a combinatorial structure 𝒫d. When 𝒫d is a polytope isomorphic to 𝒫 we say that 𝒫 is self-invariant with respect to d, or d-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a d-automorphismof a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Critical intensities of Boolean models with different underlying convex shapes

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Critical intensities of Boolean models with different underlying convex shapes

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes

This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature. For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.