(Click for larger.) As you can see, I ran out of blue yarn and continued with the light green, and I haven’t finished working on it. That’s a problem with a shape like this, in contrast to things like cubes or footballs: it doesn’t naturally stop somewhere…
The idea for this piece came from the tetrahedrons that I had made before. In a tetrahedron, in every corner three triangles meet. If you let four triangles meet at a corner, you get an octahedron – with eight faces. Five triangles meeting at a corner form part of an even larger structure, with twenty faces (five forming the top, five the bottom, and a strand of ten in between). Six regular triangles meeting at a corner might come as a disappointment: they fit into a flat circle, and no matter how many triangles you add, you’ll never get a closed surface.
So I decided to go a step further. The building block of this surface is a circle containing seven regular triangles, as you can see in more detail below the fold.
Since six regular triangles make up a circle, seven of those triangles do not fit into a flat circle: it has to bend into a saddle-shape to make room for them all. Because seven triangles meet at every corner of the surface, it’s easy to see that it has to have a saddle-shape – have negative curvature – everywhere.