A $\textit{regular polygon surface}$ $M$ is a surface graph $(��, ��)$ together with a continuous map $��$ from $��$ into Euclidean 3-space which maps faces to regular Euclidean polygons. When $��$ is homeomorphic to the sphere and the degree of every face of $��$ 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 $��$ 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.

25 pages, 9 figures