Triangulations of the Sphere

If you cut out the yellow shape here, you can fold it up along the red lines, and all 11 sharp tips will meet at one point! You’ll get a polyhedron with 12 corners, tiled by equilateral triangles. 5 triangles meet at each corner. There are also places where 6 triangles meet: these places are flat. Some triangles get folded along the red lines.

This may seem like a curiosity. But it shows up in a paper by the great mathematician Bill Thurston, and there’s a lot more to it.

Here’s how he dr...