Related Surfaces
The Klein quartic is related to various other surfaces.
Geometrically, it is the smallest Hurwitz surface (lowest genus); the next is the Macbeath surface (genus 7), and the following is the First Hurwitz triplet (3 surfaces of genus 14). More generally, it is the most symmetric surface of a given genus (being a Hurwitz surface); in this class, the Bolza surface is the most symmetric genus 2 surface, while Bring's surface is a highly symmetric genus 4 surface – see isometries of Riemann surfaces for further discussion.
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 surface of genus 7 and 14), and as such parametrizes principally polarized abelian varieties of dimension 6.
There are also other quartic surfaces of interest – see special quartic surfaces.
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. In this collection, the projective special linear groups PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, 660) are analogous, corresponding to icosahedral symmetry (genus 0), the symmetries of the Klein quartic (genus 2), and the buckyball surface (genus 70). These are further connected to many other exceptional phenomena, which is elaborated at "trinities".
Read more about this topic: Klein Quartic
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