Wythoffian Skeletal Polyhedra in Ordinary Space, I

TL;DR

This paper explores Wythoff's construction applied to regular skeletal polyhedra, generating new highly symmetric structures with diverse face configurations, and discusses conditions for these to be uniform in ordinary space.

Contribution

It introduces a systematic approach to constructing Wythoffians of regular skeletal polyhedra and analyzes their symmetry and uniformity properties.

Findings

Wythoffians are vertex-transitive with diverse face shapes.

Construction produces new symmetric skeletal polyhedra.

Conditions for uniformity of these polyhedra are examined.

Abstract

Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction is applied to the forty-eight regular skeletal polyhedra (Grunbaum-Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as "truncations" of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the blueprint for the construction and treats the Wythoffians for distinguished classes of regular polyhedra. The Wythoffians for the remaining classes of regular polyhedra will be discussed in Part II, by the second author. We also examine when the construction produces uniform skeletal polyhedra.