Now that I’ve moved on to semi-retirement, there is time to take on a 25-year-old project: my textbook on polyhedra!
I became interested in polyhedra during graduate school, when I was fascinated by the trio of books by Magnus Wenninger: Polyhedron Models, Dual Models, and Spherical Models. I don’t know how many times I checked out Polyhedron Models from the mathematics library at Carnegie Mellon University, but I have clear memories of flipping pages back and forth over and over again, trying to understand the various and subtle relationships among the 100+ models shown in the book.
Of course this prompted me to build my own models — but at a time when there weren’t nets available online. So I designed my own nets using Postscript. Since I loved coding, this was no problem at all. I essentially wrote my own turtle graphics package in Postscript, and used this to create any net I wanted.
Having moved around several times in recent years, I have very few models that I’ve built. And like many model builders, I’ve given most of them away, anyway. But here are a few I made for my friend Sandy (whom I’m visiting as I write this post).
Eventually I finished graduate school and went on to my first university teaching position, where I stayed for fourteen years. I was at a small, liberal arts school, where many of the mathematics majors were destined to be middle school or high school mathematics teachers. Moreover, I was to replace a retiring faculty member who had taught a course entitled, “Higher Geometry.”
I eagerly agreed to take on this mantle, but was interested in shifting the focus. In particular, I wanted to make the course about polyhedra rather than the usual content of a Higher Geometry course, which often included a lengthy discussion of hyperbolic geometry.
Allow me a moment to step on my pedagogical soapbox here. Yes, I understand the importance of introducing students to a non-Euclidean geometry. But as many of my students were prospective teachers, I knew there was really no way they would be able to introduce hyperbolic geometry to their students.
But spherical geometry is also an example of a non-Euclidean geometry, and further, you can actually build physical models of non-Euclidean objects by building geodesic models. So while you can’t really see that a triangle in hyperbolic geometry has an angle sum less than 180°, you can actually see that a spherical triangle has an angle sum greater than 180°.
You can also look at axiomatics in spherical geometry, with the added bonus that you expose students to the important concept of duality. Finally, you can ramp up the mathematical content of such a study by introducing students to spherical trigonometry. I should remark that, very likely, fewer than 1 in 10 (or perhaps even 1 in 100) mathematicians can rattle off the cosine law for spherical triangles — so exposing students to spherical trigonometry is significant. It’s practical as well — think of flight paths — but I never went into this application as there just wasn’t enough time.
Stepping off my soapbox now…suffice it to say that I was given free reign to retool the Higher Geometry course.
I decided to have the course be centered on spherical trigonometry. Why? First, the course needed some substantial mathematical content; spherical trigonometry can be quite challenging, especially some of the more involved derivations. This also allowed for a fairly detailed study of polyhedra, as the edge and dihedral angles of all the uniform polyhedra can be found using spherical trigonometry.
Now it is possible to find edge and dihedral angles of polyhedra in other ways, but these usually involve linear algebra applied to Cartesian coordinates in three dimensions. And in the typical undergraduate curriculum, linear algebra follows the calculus sequence.
So if I wanted the course to be accessible to other students — such as those needing a mathematics elective but were too advanced for, say, college algebra — I couldn’t have linear algebra as a prerequisite.
And so a new “Higher Geometry” was born. I did eventually rename the course to “Polyhedra and Geodesic Structures,” as it was more apt — one main application of spherical trigonometry I introduced was building spherical models, like those described extensively in Wenninger’s Spherical Models. It was a highly successful course, which I taught off and on at various institutions for about twenty years. I also conducted any number of workshops for both teachers and students of all ages over the same time span.
Essentially, students and teachers of all ages just loved the hands-on aspect of building polyhedra and spherical models. They often commented on how building their own models made mathematics “real.” There was always the added bonus that they got to take their work home with them!
Yes, model building is a fun activity. But I always made sure to balance content with the hands-on laboratory experience. We never built any models without understanding some aspect of the geometry underlying the models.
Naturally, that geometry varied with the students involved. For middle school students, working with spherical trigonometry was far too advanced. But we could always see how Euler’s formula applies to convex polyhedra.
In my university-level course, we actually proved Euler’s formula using spherical geometry with the method attribute to Legendre; despite others’ claims to the contrary, it is in fact the most elegant proof of Euler’s formula….
And this is just the first part of the book! In my next post, I’ll say a little more about the genesis of the first part, and then go on to describe the second part of the book. Expect a long thread about polyhedra and three-dimensional geometry in the upcoming months….
Creativity in Mathematics 