How to Build Minimal Polyhedral Models of the Boy Surface

Abstract

In the middle of the last century A. Möbius gave a combinatorial description of a closed, one-sided, polyhedral surface, which was soon recognized as a topological model of the real projective plane. It also turned out to be the surface that J. Steiner had defined geometrically. Soon, algebraic definitions followed which were used to construct plaster models of the cross-cap and the Roman surfaces. Until the Klein bottle, none of these non-orientable closed surfaces, whether smooth or polyhedral, was known to be “immersible” in R ³. A topological immersion i : M → R ³ is a locally injective continuous mapping. An immersion i : M → R ³ of a compact 2-manifold (without boundary) is called polyhedral if the image of i is contained in the union of finitely many planes.