I have always been fascinated by geometry in three dimensions.
I can still remember an eighth-grade project in my algebra class where we built polyhedra — mine was a white icosahedron with smaller, orange equilateral triangles connecting the midpoints of the edges. This is what an icosahedron looks like:
I even naively tried to build an icosahedron by trying to glue twenty regular tetrahedra (triangular pyramids) together, thinking that if you took a face of the icosahedron and connected its vertices to the center of the polyhedron, the result would be a regular tetrahedron. It’s pretty close — but not quite there. I wondered why my model didn’t close up.
Next, I remember walking down the aisles of the mathematics and science library at Carnegie Mellon, looking at mathematical “picture books.” I don’t know how many times I checked out Polyhedron Models by Magnus Wenninger, just looking at all the photos of the paper polyhedron models, flipping back and forth between them, trying to see how they were related to each other.
With each model, there was a net — a set of connected polygons you could use to create the model. The net you see here is for a polyhedron called the rhombic dodecahedron. I wondered how you could make each net. Later, I found out that there were lots of data published about various lengths and angles in different polyhedra — but those data were often numerical approximations, not exact values. What were the exact values?
These questions stimulated me to study more deeply — using tools such as coordinate geometry, linear algebra, and spherical trigonometry. I eventually answered many of the questions I asked so far, but of course generated many more questions, which were usually more difficult to answer.
Once I finished graduate school, I started writing a book on the mathematics of polyhedra, and eventually used it in a university-level geometry course. As I gained experience, I was asked to help design a senior capstone course for a local high school which used my text. A few years into this, my colleagues Todd Klauser and Sandy Spalt-Fulte helped organize a project where students in this senior course — as well as Todd’s other geometry students — went to a local middle school and taught the younger students how to make three-dimensional models of dodecahedra. And so Dodecahedron Day was born.
Dodecahedron Day is celebrated on December 5 of each year (for the 12 pentagons on a dodecahedron), and was first celebrated in 2005. Perhaps it’s a bit early in the year to talk about it — but just yesterday, I ran a booth at a fundraiser for the San Francisco Math Circle where we had students build three-dimensional models of different types of dodecahedra.
What I love about this type of activity is how much the students love it as well. The dodecahedron you see here was created by a young girl of about eight or nine years old — she worked painstakingly with her glue stick for close to an hour getting it together. Her focus was intense. No, it wasn’t perfect. But it was hers.
Interestingly, the parents of several students also built models — and took home nets to build more! And one of my student assistants, Simon, got very creative with decorating a net for a small stellated dodecahedron.
Students have fun building polyhedra. Frankly, I think mathematical activities which are just fun are very useful activities — improving students’ attitudes about math is really critical to their success. Students perform better in subjects they like.
But in addition to being fun, building models requires focus and attention to detail, and also develops spatial abilities. In fact, an undergraduate in my 3D geometry course who later went on to get a Ph.D. in chemistry told me that my geometry course helped her in graduate school more than any of her chemistry courses! Just think about the geometry inherent in studying orbitals.
Over the years, I’ve developed many activities for Dodecahedron Day, and include some on the Day’s website www.dodecahedronday.org. For younger students, I’ve created activities involving pentominoes, since there are 12 pentominoes, each made up of five connected unit squares (keeping with the 12/5 theme). I’m sure others have and will continue to develop different activities — the important thing is that students take a day to truly enjoy doing geometry.
One thing I do insist upon, though, is that teachers don’t create a contest out of who makes the “best” dodecahedron. There’s too much competition in schools anyway, and it defeats the purpose of Dodecahedron Day to have a student who is genuinely proud of his model to leave the day thinking, “Mine was really nice, but hers was better. The teacher said so.”
The great thing is that once students learn basic model-making skills, they can search the internet for printable nets of almost any polyhedron they can think of. Pieces can be made from different colors, and particular color arrangmements of pieces can create really beautiful models.
Another reason for writing about Dodecahedron Day a bit early is that it really takes some planning. If a school or school district wants to set aside December 5 for activities, it’s best to make that decision before the school year starts — otherwise there may be time pressure to coordinate with course syllabi, school leadership and other teachers. Further, if teachers want to take their students to other schools so that their students can teach younger pupils how to build models, it helps to develop relationships with local schools if they don’t already exist. I can tell you from experience, the earlier you start, the better.
So if you’re a student reading this, ask your teacher to celebrate Dodecahedron Day this year! If you’re a teacher or school principal, think about it — and feel free to comment with questions, concerns, or ideas. I’m happy to help in any way I can.
And if you’re not from the US, consider introducing Dodecahedron Day into schools in your country! You’ll notice the title of today’s post — I’m hoping to make Dodecahedron Day 2016 an international event, and I can’t do it alone. Perhaps materials need to be translated, or simply reformatted for A4 paper…. By starting early, it’s possible to create an enjoyable experience for everyone involved.
So in anticipation, Happy Dodecahedron Day 2016! Let’s make this a day to introduce a passion for geometry to students all around the world!