The Small Triambic Icosahedron, or Why My Friends Suck

Math is cool. Geometry is even cooler, because it's math that you can see and touch and make off-color jokes about. Wait, scratch that last one. That's terrible. Why would I say that.

Back to geometry though. I had accidentally discovered how to knit a tetrahedron (sans base), and wanted to make something out of a bunch of them. The first thing that came to mind was a big spiky ball, or as Wikipedia calls it, a small triambic icosahedron. I dove right in. How about several colors, so that no tetrahedron is next to another one of the same color? How many colors would I need? More importantly, how many colors would I need to make it look good?


Well, how many?

Mathematically speaking, it takes a minimum of three colors:

The green triangle can't be red or blue

.

But it's ugly. There are twenty tetrahedra; three doesn't divide evenly into it, so I'd have unequal numbers of each color.

Four colors, maybe? That divides evenly into twenty, right? Well yes, but it's still ugly:

There are too many blue triangles

.

 
At a cluster of five tetrahedra, having only four colors means that one of them will occur twice, asymmetrically.

By that criterion, I'd need five colors to do it right. But is it possible to have no two adjacent tetrahedra, and ensure that every cluster-of-five has one of every color?


Well, is it?

Yes, it is! I came up with a five-color solution that not only fits the rules mentioned above, but has some additional nifty properties as well:

• No tetrahedron has two neighbors of the same color
• The four tetrahedra of each color are spaced as far apart from each other as possible
• The set of a tetrahedron, its three neighbors, and the tetrahedron exactly on the opposite side from it, contains all five colors

What's especially cool is that I wasn't trying to get those extra properties, they just happened. Mathemagically. I was super psyched to show my knitting friends all these emergent mathematical properties that I didn't even realize existed until after I'd finished the project!


Well then.

They think it looks like a ball of nipples. The first thing they said about it was, "It's a nipple ball!" Nothing I say or do can convince them otherwise. My desperate appeals to the mathematical beauty of symmetry groups and set theory fall on deaf ears. But the worst part?

The worst part is when I realize they're right.

It looks like nipples. It looks like fuzzy, colorful, triangular, shiny metallic nipples. It looks like there was an error on the assembly line that made robotic stripper outfits. What was once a triumph of the elegant beauty of pure mathematical form is now a boob joke. And nothing can ever change that.

Jerks.