Polyhedron with Regular Pentagon Faces

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

$S^2$ is the two-dimensional sphere. Let $M$ be a closed polyhedral surface homeomorphic to $S^2$, with every face a regular pentagon of the same size. Consider an immersion of this surface into Euclidean three-space; self-intersections are permitted. A solid regular dodecahedron has twelve such regular-pentagonal facets.

Statement recorded in the supplied source.

Must every such immersed $M$ be the boundary of a union of solid dodecahedra glued at common facets? The source also asks the same question when the surface is embedded in three-space. [S2]

Short English statement

Must every such immersed $M$ be the boundary of a union of solid dodecahedra glued at common facets? The source also asks the same question when the surface is embedded in three-space.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5500072 (AMR-054-0072). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke (editors), The Open Problems Project, maintained original source, Problem 72, Statement and complete Partial/Related Results. https://github.com/edemaine/topp/blob/main/Problems/P.000072.