Regular Polygon Surfaces

TL;DR

This paper characterizes how regular polygon surfaces with specific face degrees can be realized as boundaries of polyhedral unions, focusing on spheres and providing counterexamples for higher genus surfaces.

Contribution

It proves realizability results for regular polygon surfaces with face degrees five, four, or eight, and shows limitations for higher genus cases.

Findings

Surfaces with degree five faces are realizable as dodecahedral unions.

Surfaces with degree four or eight faces are realizable as cube and octagonal prism unions.

Counterexamples demonstrate the failure of these realizability results for higher genus surfaces.

Abstract

A $\textit{regular polygon surface}$ $M$ is a surface graph $(\Sigma, \Gamma)$ together with a continuous map $\psi$ from $\Sigma$ into Euclidean 3-space which maps faces to regular Euclidean polygons. When $\Sigma$ is homeomorphic to the sphere and the degree of every face of $\Gamma$ is five, we prove that $M$ can be realized as the boundary of a union of dodecahedra glued together along common facets. Under the same assumptions but when the faces of $\Gamma$ have degree four or eight, we prove that $M$ can be realized as the boundary of a union of cubes and octagonal prisms glued together along common facets. We exhibit counterexamples showing the failure of both theorems for higher genus surfaces.