Klein quartic

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.

The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space.

The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal

Note the identity exhibiting 2 – η as a prime factor of 7 in the ring of algebraic integers.

[3] Corresponding to each tiling of the quartic (partition of the quartic variety into subsets) is an abstract polyhedron, which abstracts from the geometry and only reflects the combinatorics of the tiling (this is a general way of obtaining an abstract polytope from a tiling) – the vertices, edges, and faces of the polyhedron are equal as sets to the vertices, edges, and faces of the tiling, with the same incidence relations, and the (combinatorial) automorphism group of the abstract polyhedron equals the (geometric) automorphism group of the quartic.

Considering the action of SL(2, R) on the upper half-plane model H2 of the hyperbolic plane by Möbius transformations, the affine Klein quartic can be realized as the quotient Γ(7)\H2.

(Here Γ(7) is the congruence subgroup of SL(2, Z) consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)

The Klein quartic can be obtained as the quotient of the hyperbolic plane by the action of a Fuchsian group.

This can be seen in the adjoining figure, which also includes the 336 (2,3,7) triangles that tessellate the surface and generate its group of symmetries.

All the coloured curves in the figure showing the pants decomposition are systoles, however, this is just a subset; there are 21 in total.

Most often, the quartic is modeled either by a smooth genus 3 surface with tetrahedral symmetry (replacing the edges of a regular tetrahedron with tubes/handles yields such a shape), which have been dubbed "tetruses",[8] or by polyhedral approximations, which have been dubbed "tetroids";[8] in both cases this is an embedding of the shape in 3 dimensions.

The most notable smooth model (tetrus) is the sculpture The Eightfold Way by Helaman Ferguson at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, made of marble and serpentine, and unveiled on November 14, 1993.

Some of these models consist of 20 triangles or 56 triangles (abstractly, the regular skew polyhedron {3,7|,4}, with 56 faces, 84 edges, and 24 vertices), which cannot be realized as equilateral, with twists in the arms of the tetrahedron; while others have 24 heptagons – these heptagons can be taken to be planar, though non-convex,[9] and the models are more complex than the triangular ones because the complexity is reflected in the shapes of the (non-flexible) heptagonal faces, rather than in the (flexible) vertices.

This immersion can also be used to geometrically construct the Mathieu group M24 by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons.

Algebraically, the (affine) Klein quartic is the modular curve X(7) and the projective Klein quartic is its compactification, just as the dodecahedron (with a cusp in the center of each face) is the modular curve X(5); this explains the relevance for number theory.

More subtly, the (projective) Klein quartic is a Shimura curve (as are the Hurwitz surfaces of genus 7 and 14), and as such parametrizes principally polarized abelian varieties of dimension 6.

[12] More exceptionally, the Klein quartic forms part of a "trinity" in the sense of Vladimir Arnold, which can also be described as a McKay correspondence.

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) .
The tiling of the quartic by reflection domains is a quotient of the 3-7 kisrhombille .
The fundamental domain of the Klein quartic. The surface is obtained by associating sides with equal numbers.
A pants decomposition of the Klein quartic. The figure on the left shows the boundary geodesics in the (2,3,7) tessellation of the fundamental domain. In the figure to the right, the pants have each been coloured differently to make it clear which part of the fundamental domain belongs to which pair of pants.
The eight functions corresponding to the first positive eigenvalue of the Klein quartic. The functions are zero along the light blue lines. These plots were produced in FreeFEM++ .
An animation by Greg Egan showing an embedding of Klein's Quartic Curve in three dimensions, starting in a form that has the symmetries of a tetrahedron, and turning inside out to demonstrate a further symmetry.
The Eightfold Way – sculpture by Helaman Ferguson and accompanying book.
The small cubicuboctahedron is a polyhedral immersion of the tiling of the Klein quartic with octahedral symmetry.