How to Exhibit Toroidal Maps in Space

Dan Archdeacon, C. Paul Bonnington, Jo Ellis-Monaghan


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 p
oints 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
that every polyhedral toroidal map can be exhibited. As a corollary,
every toroidal triangulation has a geometric realization.

toroidal maps, triangulations, geometric realizations

Math Review Classification
Primary 05C10

Last Updated
18 July 2004


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