Belyi Triangulations

Every orientable hypermap is a associated with a Belyi triangulation—and thereby with a conformal mapping to the Riemann sphere. (Ordinary maps become hypermaps in their Walsh representation, that is, with a white vertex subdividing every original edge of the map.)

In the illustrations below, the darkened triangles map to the lower hemisphere of the Riemann sphere. The black, white, pink vertices map respectively to the points 0, 1, ∞ on the real equatorial circle of the Riemann sphere.

If the Riemann sphere is imagined flattened and stretched into a triangle with vertices at 0, 1, and ∞, these conformal mappings onto the sphere also describe ways to origami fold the topological surface into a triangle. [Di Francesco and Guitter]