Regular Polygon 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.