Playing with Plato

This post first appeared at kkdegner@blog.com on 7/29/13.

This weekend I came across this article, in the Iowa City Press Citizen (Sunday, July 28, 2013 edition to be exact).

Which made me feel a little nostalgic, because when I used to teach high school and middle school I used to teach my students to do this:

photo[1]

And when I saw the article this weekend I was reminded about how interesting these shapes can be, so interesting in fact it made me find and read these:

photo

And then, it inspired me to write a blog post (lucky you!).

In the article the author, Kim Cook, says that geometric shapes are all the rage for fall decorating.  The shapes can be used as lamp shades, and knick-nacks, and even wall paper!  Cook lists numerous places to find these geometric shapes, and even gives a suggestion about how to make your own.  (This DIY blogger also tells you how to make your own 3-D geometric shapes).

Well, would you believe me if I told you that this isn’t the first time 3-D geometric shapes are all the rage in decorating?  In fact, the use of regular polyhedra, is routinely found in the work of Renaissance artists and craftsman.

leonardo-dodec      pacioli

Taken from http://www.georgehart.com/virtual-polyhedra/

And, the aesthetic  appeal of these shapes is probably one of the main reasons the Greeks became obsessed with studying them.  Its also probably the same reason so many people in the year 2013 are willing to put them on their mantels (even if they don’t really know anything about them!).

So, let’s talk a little more about these 3-D geometric shapes that may soon be coming to an “en vogue” home near you.  The most basic of these are the regular polyhedra.  (Sometimes called the Platonic Solids).  The solids are made by having the same number of the same regular polygons meet at each vertex (the regular polygons are called faces).  For example, the regular polyhedron with the least number of faces is the pyramid.  This solid is made up of four regular (or equilateral) triangles, three meeting at each side.  Following this same pattern, we get another regular polyhedra, the octahedron; which is made up of 8 regular triangles, meeting at each side.  You keep playing the game, until regular triangles don’t make a solid anymore (why would that happen?).  Then, we can start with regular quadrilaterals (commonly known as squares).  How many square faces can meet at each vertex to form a solid? Next, we can check regular pentagons and so on . . .

Go ahead and try this and while you’re trying this think about these few things:

1. Why must there be a minimum of 3 faces meeting at each vertex?

2. How do you know when you can’t make anymore solids out of the given shape?

3. Regular polyhedra can only be made out of a few regular polygons, why?

We could talk for days about this topic, but that’d make for a pretty long blog post, so let’s agree that this is a good starting point and revisit the idea of regular polyhedra (and polyhedra in general) at a later date, ok?

The books in my photo at the top of this post include:

Combinatorics and Graph Theory by John M. Harris, Jeffry L. Hirst, and Michael J. Mossinghoff. Springer. 2000.

The Crest of the Peacock by George Gheverghese Joseph. Princeton University Press. 2000.

The Joy of Mathematics by Theoni Pappas. Wide World Publishing. 2004.

Mathematics and its History (2nd Edition) by John Stillwell. Springer. 2000.

Polyhedra by Peter R. Cromwell. Cambridge University Press. 1997.

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