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### Truncating Platonic Solids

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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.
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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.

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.
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/
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Associated Topics:
High School Geometry
High School Polyhedra
Middle School Geometry
Middle School Polyhedra

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