Suppose you lived on a dodecahedron. How would you avoid your neighbors?

“Suppose you lived on a dodecahedron,” I say. “How would you avoid your neighbors?”

“But I don’t live on a dodecahedron,” you claim. “Anyway, what if I don’t want to avoid my neighbors?”

Stop! This is mathematics. We make abstractions away from the real world; later on we can apply to those to real-world situations—maybe tomorrow, maybe next year, maybe in 2000 years. That’s how math works.

So bear with me as we walk down the yellow-brick road to discuss an intriguing Numberphile video in which Jayadev Athreya takes you on the shortest possible journey (subject to a couple of constraints) if you lived on the surface of a dodecahedron rather than the surface of a sphere. To keep it from being too abstract, he tells you stories, starting with one about a jogger who lives on a dodecahedron and wants to visit a mathematician who lives at a vertex. And then there’s the problem of how the Little Prince can prevent his sheep from eating a rose—again assuming that they all lived on a dodecahedron.

Athreya is remarkably clear—incredibly clear—about all of this. It probably helps if you recall that there are only five regular (“Platonic”) solids—but, if you don’t, he even states this explicitly, along with 3-D and 2-D illustrations, a.k.a. nets. Athreya is clearly a master teacher (at the University of Washington and University of Illinois). Watch for the surprise at the 12-minute mark!

And learn about the yellow brick road, with a significance that Baum never knew!




Categories: Math