A Polyhedral Model in Euclidean 3-Space of the Six-Pentagon Map of the Projective Plane

In a private communication, Branko Grünbaum asked: “I wonder whether you know anything about the possibility of realizing as a polyhedron in Euclidean 3-space the family of six pentagons, that is a model of the projective plane arising by identifying antipodal points of the regular dodecahedron. Naturally, any realization must have some self-intersections—but is there any realization that is not completely contained in a plane?”We show that it is possible to realize this polyhedron; in our realization five of the six faces are simple polygons. In this model there are sets of three faces, which form a realization of the Möbius strip without self-intersections. There are four variants of the model. We conjecture that in any model of this polyhedron there must be at least one self-intersecting face.