Klein quartic tiled by heptagons

thingiverse

A stunning representation of the Klein quartic showcases its multifaceted nature: - It takes the form of a hyperbolic surface, intricately tiled by 24 regular heptagons that converge at every vertex in a precise pattern, with three of them meeting simultaneously; conversely, it can be envisioned as composed of 56 equilateral triangles, each one encircling its neighboring vertices. - The quartic's surface is defined by the equation x³y + y³z + z³x = 1, boasting an exceptional automorphism group size for a genus-3 surface - this extraordinary group boasts a mere 168 members, representing the largest possible group of its kind. - This remarkable model also serves as the modular curve X(7), resulting from dividing the Poincaré half-plane by a specific group of matrices equivalent to 1 (mod 7); each point on this curve is associated with an elliptic curve that features a distinct 7-torsion mark. In this rendition, each heptagon bears a label derived from the Stern-Brocot tree (modulo 7), adding another layer of intricate detail to this already complex and captivating model.