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/
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