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Problem 72: Polyhedron with Regular Pentagon Faces

Let M be a closed polyhedral surface homeomorphic to S2 which is entirely composed of equal regular pentagons. If M is immersed in 3-space, is it necessarily the boundary of a union of solid dodecahedra that are glued together at common facets?
Richard Kenyon, first posed in 2006.
Partial and Related Results
The corresponding question for equal squares has a positive answer. The question for surfaces embedded in 3-space is also interesting and open. The Kepler-Poinsot great dodecahedron has regular pentagon faces, and is immersed, but is not homeomorphic to S2 (V - E + F = - 6).
Re-posed at Oberwolfach Workshop, Jan. 2009.
Entry Revision History
J. O'Rourke, 23 Jan. 2009.

The Open Problems Project - September 19, 2017