Polyhedron Problem

Date: 10/29/96 at 23:54:2
From: Lottie English
Subject: Geometry

As a homework problem in geometry, we have to answer the following 
question:  

Let P be a polyhedron with v vertices, e edges, and f faces. 
Show that e>=(3/2)f  and e>=(3/2)v.  

I know Euler's formula but the professor said that this formula is not 
necessary when solving this problem. I only know the faces, edges, and 
vertices of the platonic solids, and I know there are more polyhedra 
than those. How do I go about solving this problem?

Date: 12/06/96 at 16:12:21
From: Doctor Lorenzo
Subject: Re: Geometry

Ask yourself how many edges each face has.  How many faces share each 
edge?  If ownership of each edge were divided equally between the 
faces that share the edge, each triangle would own 3/2 edges, each 
quadrilateral would own 4/2 edges, and so on.  

So what can you say about how many edges the *average* face would own?  
This average is another way of saying e/f.  

A similar argument with counting edges at each vertex and vertices at 
each edge gives you a lower bound on e/v.

-Doctor Lorenzo,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/