Polyhedra: Classification, Theorem

Date: 02/12/98 at 21:52:36
From: Andrew Barnett
Subject: Geometry, polyhedrons

I would like to know how polyhedrons are classified and which figures 
can be used for the faces. I would also like to see the theorem 
relating the faces, edges, and vertices. Thank you for any available 
information.

Date: 02/13/98 at 13:20:10
From: Doctor Martin
Subject: Re: Geometry, polyhedrons

The classification of polyhedra is a fantastic classical result. The 
simplest classification is that of the "regular" polyhedra. A regular 
polyhedron is one which has the following properties:

(1) Every face is a regular polygon.
(2) Every face is congruent to every other face.
(3) Every vertex has the same number of faces around it.  

There are precisely 5 regular polyhedra:

(1) The Tetrahedron, a triangular pyramid which possesses 4 vertices, 
    six edges, and 4 faces.
(2) The cube, or hexahedron, which you probably know well.
(3) The octahedron, which has 8 triangular faces, and looks like two 
    square-based pyramids connected at their bases.
(4) The dodecahedron, which has 12 pentagonal faces, and 20 vertices.
(5) The icosahedron, which has 20 triangular faces, and 12 vertices.

These are called the Platonic solids, though they were certainly known 
before Plato's time. Plato connected them with the "five elements":  
fire, air, earth, water, and spirit. Water, for instance was connected 
with the icosahedron, perhaps because the icosahedron looks most like 
a spherical drop of water. The cube was connected with the earth, 
perhaps because it seems so stable when it sits down. 

There is a great deal of information on the Platonic solids, which you 
can find on the web or in history of math texts. If you're interested 
in a proof that there are only these five regular polyhedra, write 
back, and I'll give it to you - it's not too hard, only requiring a 
little geometry and algebra.  

Non-regular polyhedra are also categorized and classified - in 
particular, I think that there are 13 "Archimedean" or semi-regular 
polyhedra. Check out the Geometry Center's Web site for more about 
those, and other types of polyhedra.

The formula that I think you're looking for is:  

   V - E + F = 2

where V, E, and F represent the number of vertices, edges, and faces 
in (almost any) polyhedron. You might test it out on the cube, or try 
to figure out how many edges there are in the dodecahedron and 
icosahedron from the data I gave above.  

I hope that this helps -- this is very interesting stuff!

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