Klein quartic
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The Klein quartic
- x3y + y3z + z3x = 0,
named after Felix Klein, is a Riemann surface, and an algebraic 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 the Stark-Heegner theorem on imaginary quadratic number fields of class number one.
