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-pla