Problem 72: Polyhedron with Regular Pentagon Faces

**Statement**- Let
*M*be a closed polyhedral surface homeomorphic to*S*^{2}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? **Origin**- Richard Kenyon, first posed in 2006.
**Status/Conjectures**- Open.
**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
*S*^{2}(*V*-*E*+*F*= - 6). **Appearances**- Re-posed at Oberwolfach Workshop, Jan. 2009.
**Categories**- polyhedra
**Entry Revision History**- J. O'Rourke, 23 Jan. 2009.

The Open Problems Project - September 19, 2017