These two parts ask about the numbers of faces, edges, and vertices of various p

Question

These two parts ask about the numbers of faces, edges, and vertices of various polyhedral. The edges and vertices of regular polyhedral can be counted efficiently if you yse these facts: (i) each polygonal face has the same number of edges and vertices; (ii) each edge is common to two faces; and (iii) the same number of eges come together at each vertex. Strategies for dealing with semi regular polyhedral are similar.

a. For each of the five Platonic solids, count the number V of Vertices, the number F of faces
and the number E of edges. Check that in each case V-E+F=2

Explanation / Answer

The proof is indeed not simple. This is the Euler's Formula for Platonic solids; the value V-E+F is the Euler's number which equals 2 for these solids and can have other values for other polyhedra. You can find the full proof of the Euler's Formula (actually, nineteen of those proofs) at the link http://www.ics.uci.edu/~eppstein/junkyard/euler/

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