Platonic Solids, Symmetry, and the Fourth Dimension

On his blog Azimuth, John Baez has been posting a series called “Symmetry and the Fourth Dimension.” He writes: “The idea is to start with something very familiar and then take it a little further than most people have seen…without getting so technical that only people with PhDs understand what’s going on. I’m more interested in communicating with ordinary folks than in wowing the experts.” He starts with several posts on three-dimensional geometry, focusing on the Platonic solids and their relationship to Coxeter groups, to lay a good foundation before bumping it up a dimension in Part 10.

The icosidodecahedron, a solid "halfway through" the transition from dodecahedron to its dual, the icosahedron. Image: Tomruen, via Wikipedia. Created using Robert Webb's Great Stella software.

The icosidodecahedron, a solid “halfway through” the transition from the dodecahedron to its dual, the icosahedron. Image: Tomruen, via Wikipedia. Created using Robert Webb’s Great Stella software.

Although it’s written for a lay audience, I’ve gotten a lot out of the series, too, and I expect to get even more out of it when he tackles topics such as the Hopf fibration and visualizations of the 120-cell. (I’m just guessing that these will be in future posts based on the Google+ posts he’s using as a jumping-off point for the blog series.)

I love geometry, but my research focuses on two-dimensional surfaces, which has allowed me to get by without spending much time thinking about three- or four-dimensional polytopes. I have a basic understanding, but I don’t have a rich repertoire of examples and visualizations at my fingertips, and I know next to nothing about the history of the subject. Baez has given me a nice big picture overview of some really interesting topics in geometry that hadn’t grabbed me before. His enthusiasm for the Platonic solids is infectious, especially in Posts 6 and 7. I’m keeping his exposition in mind for future use in my classes. I think the first 9 posts would be a good starting point for a section in a geometry course.

Part 11, on the four-dimensional cube, or tesseract, has several really great pictures, including this trippy one. If you cross your eyes just right, you’ll see a 3-d projection of the tesseract pop out at you, although, as Baez says, I’m not sure how much insight that gives. “Crossing my eyes and looking at this 3d image puts me into an altered state of mind which is fun but not good for doing mathematics!”

Image: Maninthemasterplan, via Wikimedia Commons.

Image: Maninthemasterplan, via Wikimedia Commons.

The series has also introduced me to a blog that is right up my alley but I had never heard of before: The Fairyland of Geometry, a “cultural history of higher space, 1853-1907,” by Mark Blacklock, a PhD student at Birkbeck College. I haven’t explored it much yet, but it looks like it has a lot to offer someone interested in the history of geometry.

As of the time I wrote this post (I’m on vacation right now!), there were 12 posts in Baez’s series. Part 9 contains links to the first 8 parts, and here are parts 10, 11, and 12.

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1 Response to Platonic Solids, Symmetry, and the Fourth Dimension

  1. Avatar Glenn Westmore says:

    The ancient Greeks performed geometric constructions with only a simple compass and straightedge, and a Persian mathematician Abul Wafa proposed the “rusty compass” (fixed radius compass) principle. This set of drawings illustrates it:

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