Tessellation of a Sphere by an Icosahedron
- Jordan Schettler
Construct an icosahedron. Tessellate its circumsphere. Then stereographically project this tessellation into the complex plane. See how each object transforms under a mid-edge to mid-edge rotation of the icosahedron. Display the fixed points in the complex plane.
If the circumsphere has radius 1 and we orient the complex plane appropriately, then the positive fixed point of this rotation is the special value of the Rogers-Ramaujan continued fraction (viewed as a function on the upper half plane) evaluated at the imaginary unit. In general, this function is modular up to the rotational symmetries of the icosahedron and computing fixed points allows one to recover the famous identities found in Ramanujan's first letter to Hardy and in the so-called "lost" notebook.