It is possible to represent a convex 3-dimensional polytope by
a graph in the plane. Such graphs can be plane graphs (i.e. have
crossings only at vertices) or show crossings. It may be desirable
for a drawing to have crossings if by doing so it displays some
aspect of the original polytope, for example, more of its symmetries.
Problem
1. Examine the relation between the isometries of 3-dimensional
polytopes and the symmetries that are displayed by different graphs
that might be used to represent them, with or without crossings,
in the plane. As specific examples one might consider drawings
of the tetrahedron, cube, and octahedron in the the plane.
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