Generic Orthotopes

TL;DR

This paper introduces the concept of generic orthotopes, a broad class of orthogonal polytopes with desirable topological and combinatorial properties, useful for representing Coxeter complexes.

Contribution

It defines generic orthotopes, explores their properties, and provides formulas for volume and Euler characteristic based on floral arrangements.

Findings

Generic orthotopes have a homogeneity property similar to smoothly bounded sets.

Faces and cross-sections of these polytopes are also described by floral arrangements.

Formulas for volume and Euler characteristic are derived using natural statistics of floral arrangements.

Abstract

This article studies a large, general class of orthogonal polytopes which we may call "generic orthotopes". These objects emerged from a desire to represent a Coxeter complex by an orthogonal polytope that is particularly nice with respect to traditional topological, structural, or combinatorial considerations. Generic orthotopes have a pleasant "homogeneity" property, somewhat like a smoothly bounded compact subset of Euclidean space. Thus, as soon as we demand that every vertex of an orthogonal polytope be a floral arrangement, as defined here, many derivative structures such as faces and cross-sections are also described by floral arrangements. We also give formulas for the volume and Euler characteristic of a generic orthotope using a couple of statistics that are defined naturally for floral arrangements.