Problem 72: Polyhedron with Regular Pentagon Faces

Statement

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?

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 S² (V - E + F = - 6).

Appearances

Re-posed at Oberwolfach Workshop, Jan. 2009.

Categories

polyhedra

Entry Revision History

J. O'Rourke, 23 Jan. 2009.