The Klein quartic

: x 3 y + y 3 z + z 3 x = 0,

named after Felix Klein, is a Riemann surface, and an algebraic curve of genus (mathematics) 3 over the complex numbers C.

The Klein quartic has automorphism group isomorphic to the general linear group G = PSL(2,7). The order 168 of G is the Hurwitz s automorphisms theorem 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.

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