The Klein quartic with the two dual Klein graphs (14-gon edges marked with the same number are equal.)
The Klein quartic is a quotient of the heptagonal tiling (compare the 3-regular graph in green) and its dual triangular tiling (compare the 7-regular graph in violet).
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to PSL(2, 7), the second-smallest non-abelian simple group after the alternating group A5. The quartic was first described in (Klein 1878b).
Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties.
Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane P2(C) defined by an algebraic equation. This has a specific Riemannian metric (that makes it a minimal surface in P2(C)), under which its Gaussian curvature is not constant. But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane H2 by a certain cocompact group G that acts freely on H2 by isometries. This gives the Klein quartic a Riemannian metric of constant curvature −1 that it inherits from H2. This set of conformally equivalent Riemannian surfaces is precisely the same as all compact Riemannian surfaces of genus 3 whose conformal automorphism group is isomorphic to the unique simple group of order 168. This group is also known as PSL(2, 7), and also as the isomorphic group PSL(3, 2). By covering space theory, the group G mentioned above is isomorphic to the fundamental group of the compact surface of genus 3.
simple group after the alternating group A5. The quartic was first described in (Klein 1878b). Klein'squartic occurs in many branches of mathematics, in contexts...
algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: A x 4 + B...
degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero...
as the Kleinquartic), where it forms the Klein map with 24 heptagonal faces, Schläfli symbol {7,3}8. According to the Foster census, the Klein graph,...
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is a x 4...
In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under...
(order 3 biplane). Of these, the icosahedron dates to antiquity, the Kleinquartic to Klein in the 1870s, and the buckyball surface to Pablo Martin and David...
geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine...
obtained an explicit representation of a Riemann surface now termed the Kleinquartic. He showed that it was a complex curve in projective space, that its...
line, R Real projective line, RP1 ≅ S1 Cylinder, S1 × R Klein bottle, RP2 # RP2 Kleinquartic (a genus 3 surface) Möbius strip Real projective plane,...
Bolza surface, with order 48. For genus 3 the order is maximized by the Kleinquartic, with order 168; this is the first Hurwitz surface, and its automorphism...
triple torus is also occasionally used to denote a genus 3 surface. The Kleinquartic is a compact Riemann surface of genus 3 with the highest possible order...
degree 7. This regular tiling is significant as it is a tiling of the Kleinquartic, the genus 3 surface with the most symmetric metric (automorphisms of...
four-group Klein geometry Klein graphs Klein's inequality Klein model Klein polyhedron Klein surface Klein quadric Kleinquartic Kleinian group Kleinian...
Uniformization of the Kleinquartic, a surface of genus three and Euler characteristic −4, as a quotient of the hyperbolic plane by the symmetry group...
genus 0), the order 2 biplane (complementary Fano plane) inside the Kleinquartic (genus 3), and the order 3 biplane (Paley biplane) inside the buckyball...
hyperbolic geometry structure. Examples are the hyperelliptic curves, the Kleinquartic curve, and the Fermat curve xn + yn = zn when n is greater than three...
Elliptic integral Complex multiplication Weil pairing Hyperelliptic curve Kleinquartic Modular curve Modular equation Modular function Modular group Supersingular...
generation. M24 can be constructed starting from the symmetries of the Kleinquartic (the symmetries of a tessellation of the genus three surface), which...
automorphism group as Riemann surfaces. The smallest Hurwitz surface is the Kleinquartic (genus 3, automorphism group of order 168), and the induced tiling has...
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