Skeletal Snub Polyhedra in Ordinary Space, I

Abstract

Skeletal polyhedra are discrete connected structures consisting of finite (planar or skew) or infinite (linear, planar, or spatial) polygons as faces, with two faces on each edge and a circular vertex figure at each vertex. The present paper describes the blueprint for the snub construction and shows that it can be applied to both regular and chiral skeletal polyhedra in ordinary space. The resulting skeletal snub polyhedra are vertex-transitive and highly locally symmetric. Their properties - from a combinatorial, topological, and geometric perspective - are described and illustrated on some particularly interesting examples. We examine when the construction yields uniform skeletal polyhedra and discuss the completeness of our list of generated structures.

Figures (5)

• Figure 1: Vertices, edges, and faces of $S_P(v)$ adjacent to the initial vertex $v$
• Figure 2: Adjacent vertices and faces at vertex $v$
• Figure 3: Faces and vertices around $v$
• Figure 4: Crossed quadrilateral (bow tie)
• Figure 5: Crossed quadrilateral (butterfly)

Theorems & Definitions (24)

• Theorem 4.1
• Lemma 4.2
• proof
• Theorem 5.1
• proof
• Remark 1
• Theorem 5.2
• proof
• Lemma 5.3
• proof