Diagonals and Symmetry in Polyhedra

Date: 09/15/2002 at 21:39:55
From: David
Subject: Platonic Solids

I would like to know a formula for finding the number of diagonals in 
a Platonic solid. 

I already know that there are 0 diagonals in a tetrahedron, 4 in a 
cube, and 3 in an octahedron. I still don't know how many diagonals 
are in an dodecahedron or a icosahedron, but I believe that they are 
48 and 160. If you could send me the formula for the number of 
diagonals in a Platonic solid, it would be very helpful. 

Thank you for your time.
David

Date: 10/26/2002 at 23:04:05
From: Doctor Nitrogen
Subject: Re: Platonic Solids

Hi, David:

There is a much better and more insightful way you can determine the 
number of diagonals of a Platonic solid than using a mere formula, 
provided that by "diagonal" you mean diagonals that pass through 
opposite vertices. You can study their symmetry properties.

First of all, note that below I use a symbol like {p, q}. This symbol 
just stands for a Platonic solid and it means the Platonic solid at 
issue has polygonal faces that are {p}, meaning if p = 3, 4, or 5 
the faces are equilateral triangles, squares, or regular pentagons, 
respectively, and that exactly q of these faces, where q might be 3, 
4, or 5, meet a common vertex. 

You said you noted that the cube, or {4, 3} had 4 diagonals through 
opposite vertices, and that the tetrahedron ( {3, 3} ) had no 
diagonals going through opposite vertices. Well, the reason is plain 
if you look at the symmetry of both of these solids. The square {4, 3} 
has opposite vertices; the tetrahedron (or,  {3, 3} ) does not. So the 
total number of diagonals running through two opposite diagonals of a 
square is 4, one for each two opposite vertices. Can you visualize 
that? 

For the dodecahedron, which has 20 vertices, there are no opposite 
diagonals because there are no opposite vertices, although you can 
pass a line through the centers of two opposite faces. 

Now for the octahedron, which has 6 vertices, there are three sets of 
two opposite vertices, so you can pass exactly 3 diagonals through 
opposite vertices. 

Lastly, for the icosahedron, which has 12 vertices, there are 6 sets 
of sets of two opposite vertices, so you can pass exactly 6 diagonals 
through opposite vertices.

Summarizing, you can pass 4 diagonals through the opposite poles of a 
cube {4, 3}, you can pass exactly 3 diagonals through the opposite 
vertices of an octahedron ( {3, 4} ), and you can pass exactly 6 
diagonals through the opposite vertices of an icosahedron. 

Mathematicians call the sets that describe how you can rotate or 
reflect these five Platonic solids in 3-dimensional space "symmetry 
groups," because certain rotations in 3-space keep the solid looking 
exactly the way it did before you rotated it, and because these 
rotations leave the solid and distances between different parts the 
same.

The specific symmetry groups involved are also called the 
"tetrahedral," "octahedral," and "icosahedral" groups. 

You can read more about the symbol {p, q} for the Platonic solids at:
    
   4D Platonic Solids - Polytopes (geometry-research)
   http://westview.tdsb.on.ca/Mathematics/4DPlatonicSolids.html 

Did this help answer your question? You are welcome to return to The 
Math Forum and Doctor Math whenever you have any math-related 
questions.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/