Truncating Platonic Solids

Date: 08/04/98 at 07:53:00
From: Daisuke Nomura
Subject: Truncation of Platonic solids

Dear Dr Math, 

I was wondering if you could give me some information on Platonic 
solids. I am doing an essay and I need to know these things:

1. Investigate the effects of truncation on each of the Platonic 
   solids.

2. Historical and current practical applications of Platonic solids 
   and their truncated forms.

Date: 08/04/98 at 18:22:04
From: Doctor Rick
Subject: Re: Truncation of Platonic solids

Hi, Daisuke. 

Platonic solids, as you probably know, are the five polyhedra whose 
faces are all identical regular polygons. They are named for Plato,
the Greek philosopher, who theorized that the elements (there were 
believed to be four of them) were made up of four of these shapes. 
If you're interested, you can read about this at:

   http://weber.u.washington.edu/~smcohen/timaeus.htm   

The Platonic solids aren't the only interesting polyhedra. Here is a 
page that shows you a lot more polyhedra than you need (80, in fact):

   http://www.mathconsult.ch/showroom/unipoly/list.html   

The Platonic solids are there - see numbers 1, 5, 6, 22, and 23. If you 
can get or make 3-D models of these five shapes, it will help you a lot 
to see what truncation is and what it does.

Truncation is slicing off the corners (vertices) of a polyhedron. It 
adds a face at each corner - the cut surface. If three faces meet at a 
vertex, as in a cube, then the new face is a triangle, with an edge 
meeting each of the three original faces. What happens to those 
original faces - how many edges do they have now? How many vertices 
does the polyhedron have now?

You can truncate just a little, or a lot. You can truncate so much that 
the new faces meet. This will change the number of vertices and the 
number of edges on the original faces. You can truncate even more. 
What happens then?

Some of the polyhedra that you make by truncation are sort of regular. 
Not as regular as the Platonic solids, but they are interesting enough 
that they are named after another Greek philosopher, Archimedes. The 
Archimedean solids have regular polygons for faces, but they are not 
all the same. Can you figure out how much to truncate each Platonic 
solid so its faces are all regular polygons? For that matter, can you 
truncate a Platonic solid and end up with another Platonic solid?

Some of the polyhedra on the Web page I mentioned above are 
Archimedean solids. Some of these that you can make by truncating the 
Platonic solids are numbers 2, 7, 8, 9, 24, 25, and 26. See if you can 
figure out how to make them.

You can see that there are a lot of good questions to ask and answer 
as you explore truncation. Try to make tables of the different solids 
you make, and how you got them. Some can be made in more than one way. 
There are amazing connections among them. Have fun exploring!

As for history and applications, I mentioned Plato and Archimedes. 
Kepler, the astronomer, also had ideas about these solids. You will 
recognize one truncated form that is associated with a very popular 
sport! In chemistry, there are polyhedral molecules known as 
"buckyballs" that have gotten a lot of attention lately. The man for 
whom they are named, Buckminster Fuller, also designed a globe or map 
shaped like a dodecahedron. Those are some things that come to my mind 
right away.

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