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### 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/
```
Associated Topics:
College Polyhedra
High School Polyhedra

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