  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.
