Next: Problem 73: Congruent Partitions of Polygons
Previous: Problem 71: Stretch-Factor for Points in Convex Position
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?
Richard Kenyon, first posed in 2006.
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).
Re-posed at Oberwolfach Workshop, Jan. 2009.
J. O’Rourke, 23 Jan. 2009.