Klein quartic

The Klein quartic x³y + y³z + z³x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C.

The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). The order 168 of G is the maximum allowed for this genus 3; and this curve is uniquely determined by requiring that the symmetry is as large as this.

The Klein quartic can be given a metric of constant negative curvature and then tiled with 24 regular heptagons. The order of G is thus related to the fact that 24 x 7 = 168.

Klein's quartic occurs all over mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and Stark's theorem on imaginary quadratic number fields of class number 1.

External links

• Klein's quartic curve (John Baez)
• Polyhedral models of Felix Klein's quartic
• The Eightfold Way: The Beauty of Klein's Quartic Curve (Silvio Levy, ed.)