How to exhibit toroidal maps in space

Dan Archdeacon, Craig Paul Bonnington, Joanna A Ellis-Monaghan

Research output: Contribution to journalArticleResearchpeer-review

13 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)573 - 594
Number of pages22
JournalDiscrete & Computational Geometry
Volume38
Issue number3
DOIs
Publication statusPublished - 2007
Externally publishedYes

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