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- Title
How to Exhibit Toroidal Maps in Space.
- Authors
Dan Archdeacon; C. Paul Bonnington; Joanna A. Ellis-Monaghan
- Abstract
Abstract Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.
- Subjects
NEWTON diagrams; POLYGONS; GENERALIZED polygons; TOROIDAL magnetic circuits
- Publication
Discrete & Computational Geometry, 2007, Vol 38, Issue 3, p573
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-007-1354-3