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Homework due February 12.

This assignment concerns applications of the Euler characteristc to Platonic polyhedra. We define a Platonic polyhedron (or a Platonic triangulation of the underlying topological space) to be one in which each face has the say number of vertices, say p, and each vertex has the same number of faces meeting at it, say q.  Thus if it has E edges then it has 2E/q vertices and 2E/p faces.  The 1-skeleton of a polyhedron is the graph consisting of its vertices and edges.  We assume here that the polyhedron is a surface, i.e., each edge belongs to exactly two faces.

1. Prove that if a Platonic polyhedron is homeomorphic to a 2-sphere (so its Euler characteristic is 2), then the only possible values of (p,q) are the five familiar ones.

 
2. Determine the possible values of (p,q) if the polyhedron is homeomorphic to the real projective plane, so its Euler characteristic is 1.  Describe the resulting polyhedra.  In particular find one with the complete graph on 6 points as its 1-skeleton.  Its dual has six faces with sharing an edge with every other one.  This shows that 6 colors may be needed to color a map of the projective plane.

 
3. Determine the possible values of (p,q) if the polyhedron is homeomorphic to a torus, so its Euler characteristic is 0.  Describe the resulting polyhedra; there are infinitely many of them.  In particular find ones having the complete graph on 7 points and the houses and utilities graph as their 1-skeleta.

 
4. Do the same if the polyhedron is homeomorphic to a torus with two holes, so its Euler characteristic is -2.  Using Euler's formula will give a list of over 20 possible values of (p, q), so feel free to use a computer program to help list them all.  You then need to check to see which of them are actually asociated with polyhedra.  You may work as a team to do this.

This page was last revised on January 25, 2002, by Doug Ravenel .