How to Exhibit Toroidal Maps in Space |
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Authors: | Dan Archdeacon C Paul Bonnington Joanna A Ellis-Monaghan |
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Institution: | (1) Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA;(2) Department of Mathematics, University of Auckland, Auckland, New Zealand;(3) Department of Mathematics, Saint Michael's College, Colchester, VT 05439, USA |
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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. |
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Keywords: | |
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