Klein quartic tiled by heptagons

A model of the Klein quartic. This fascinating object is at the same time:

• a hyperbolic surface tiled by 24 regular heptagons, with three of them meeting at every vertex (or dually: by 56 equilateral triangles, seven of them around each vertex);

• the surface with equation x³ y + y³ z + z³ x = 1, which has the largest possible automorphism group for a genus-3 surface (this is the simple group with order 168);

• the modular curve X(7), which is the quotient of the Poincaré half-plane by the group of matrices ≡ 1 (mod 7); each point of this curve defines an elliptic curve with marked 7-torsion.

In this model, each heptagon is labeled by a fraction, according to the Stern-Brocot tree (modulo 7).