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hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
, the Klein quartic, named after
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
, is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
with the highest possible order
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
for this genus, namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed. As such, the Klein quartic is the
Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by vir ...
of lowest possible genus; see
Hurwitz's automorphisms theorem In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms ...
. Its (orientation-preserving) automorphism group is isomorphic to , the second-smallest non-abelian
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
after the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
A5. The quartic was first described in . Klein's quartic occurs in many branches of mathematics, in contexts including
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, homology theory,
octonion multiplication In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
,
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see for a survey of properties. Originally, the "Klein quartic" referred specifically to the subset of the
complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
defined by an algebraic equation. This has a specific
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
(that makes it a minimal surface in ), 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
by a certain
cocompact Cocompact may refer to: * Cocompact group action * Cocompact Coxeter group * Cocompact embedding * Cocompact lattice {{dab ...
group that acts freely on by isometries. This gives the Klein quartic a Riemannian metric of constant curvature that it inherits from . 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 , and also as the isomorphic group . By
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
theory, the group mentioned above is isomorphic to the fundamental group of the compact surface of genus .


Closed and open forms

It is important to distinguish two different forms of the quartic. The '' closed'' quartic is what is generally meant in geometry; topologically it has genus 3 and is a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
. The ''
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
'' or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.


As an algebraic curve

The Klein quartic can be viewed as a projective
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s , defined by the following quartic equation in
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
on : :x^3y + y^3z + z^3x = 0. The locus of this equation in is the original Riemannian surface that Klein described.


Quaternion algebra construction

The compact Klein quartic can be constructed as the quotient of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
by the action of a suitable
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
which is the principal
congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...
associated with the ideal I=\langle \eta-2\rangle in the ring of algebraic integers of the field where . Note the identity :(2-\eta)^3= 7(\eta-1)^2, exhibiting as a prime factor of 7 in the ring of algebraic integers. The group is a subgroup of the (2,3,7) hyperbolic
triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangl ...
. Namely, is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators and relations :i^2=j^2=\eta, \qquad ij=-ji. One chooses a suitable
Hurwitz quaternion order The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. ...
\mathcal Q_ in the quaternion algebra, is then the group of norm 1 elements in 1+I\mathcal Q_. The least absolute value of a trace of a hyperbolic element in is \eta^2+3\eta+2, corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this genus.


Tiling

The Klein quartic admits tilings connected with the symmetry group (a " regular map"), and these are used in understanding the symmetry group, dating back to Klein's original paper. Given a fundamental domain for the group action (for the full, orientation-reversing symmetry group, a (2,3,7) triangle), the reflection domains (images of this domain under the group) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group (reflections in the lines of a given fundamental triangle give a set of 3 generating reflections). This tiling is a quotient of the order-3 bisected heptagonal tiling of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
(the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of the quartic), and all Hurwitz surfaces are tiled in the same way, as quotients. This tiling is uniform but not regular (it is by
scalene triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
s), and often regular tilings are used instead. A quotient of any tiling in the (2,3,7) family can be used (and will have the same automorphism group); of these, the two regular tilings are the tiling by 24 regular hyperbolic
heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...
s, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
s, each of degree 7 (meeting at 24 vertices). The order of the automorphism group is related, being the number of polygons times the number of edges in the polygon in both cases. :24 × 7 = 168 :56 × 3 = 168 The covering tilings on the hyperbolic plane are the order-3 heptagonal tiling and the
order-7 triangular tiling In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . Hurwitz surfaces The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is ...
. The automorphism group can be augmented (by a symmetry which is not realized by a symmetry of the tiling) to yield the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M24. Corresponding to each ''tiling'' of the quartic (partition of the quartic variety into subsets) is an
abstract polyhedron In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...
, which abstracts from the geometry and only reflects the combinatorics of the tiling (this is a general way of obtaining an
abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...
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. In this way the geometry reduces to combinatorics.


Affine quartic

The above is a tiling of the ''projective'' quartic (a closed manifold); the affine quartic has 24 cusps (topologically, punctures), which correspond to the 24 vertices of the regular triangular tiling, or equivalently the centers of the 24 heptagons in the heptagonal tiling, and can be realized as follows. Considering the action of on the upper half-plane model of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
by Möbius transformations, the affine Klein quartic can be realized as the quotient . (Here is the
congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...
of consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)


Fundamental domain and pants decomposition

The Klein quartic can be obtained as the quotient of the hyperbolic plane by the action of a Fuchsian group. The fundamental domain is a regular 14-gon, which has area 8\pi by the Gauss-Bonnet theorem. 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. Within the tessellation by (2,3,7) triangles is a tessellation by 24 regular heptagons. The systole of the surface passes through the midpoints of 8 heptagon sides; for this reason it has been referred to as an "eight step geodesic" in the literature, and is the reason for the title of the book in the section below. All the coloured curves in the figure showing the pants decomposition are systoles, however, this is just a subset; there are 21 in total. The length of the systole is :16\sinh^\left(\left(\tfrac\sqrt\right)\sin\left(\tfrac\right)\right)\approx3.93594624883. An equivalent closed formula is :8\cosh^\left(\tfrac-2\sin^2\left(\tfrac\right)\right). Whilst the Klein quartic maximises the symmetry group for surfaces of genus 3, it does not maximise the systole length. The conjectured maximiser is the surface referred to as "M3" . M3 comes from a tessellation of (2,3,12) triangles, and its systole has multiplicity 24 and length :2\cosh^\left(2+\sqrt\right)\approx3.9833047820988736. The Klein quartic can be decomposed into four pairs of pants by cutting along six of its systoles. This decomposition gives a symmetric set of Fenchel-Nielsen coordinates, where the length parameters are all equal to the length of the systole, and the twist parameters are all equal to \tfrac of the length of the systole. In particular, taking l(S) to be the systole length, the coordinates are :\left\. The
cubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bi ...
corresponding to this pants decomposition is the tetrahedral graph, that is, the graph of 4 nodes, each connected to the other 3. The tetrahedral graph is similar to the graph for the projective
Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
; indeed, the automorphism group of the Klein quartic is isomorphic to that of the Fano plane.


Spectral theory

Little has been proved about the
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
of the Klein quartic. Because the Klein quartic has the largest symmetry group of surfaces in its topological class, much like the
Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
in genus 2, it has been conjectured that it maximises the first positive eigenvalue of the Laplace operator among all compact Riemann surfaces of genus 3 with constant negative curvature. It also maximizes mutliplicity of the first positive eigenvalue (8) among all such surfaces, a fact that has been recently proved. Eigenvalues of the Klein quartic have been calculated to varying degrees of accuracy. The first 15 distinct positive eigenvalues are shown in the following table, along with their multiplicities.


3-dimensional models

The Klein quartic cannot be ''realized'' as a 3-dimensional figure, in the sense that no 3-dimensional figure has (rotational) symmetries equal to , since does not embed as a subgroup of (or ) – it does not have a (non-trivial) 3-dimensional linear representation over the real numbers. However, many 3-dimensional models of the Klein quartic have been given, starting in Klein's original paper, which seek to demonstrate features of the quartic and preserve the symmetries topologically, though not all geometrically. The resulting models most often have either tetrahedral (order 12) or octahedral (order 24) symmetries; the remaining order 7 symmetry cannot be as easily visualized, and in fact is the title of Klein's paper. 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", or by polyhedral approximations, which have been dubbed "tetroids"; 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 Helaman Rolfe Pratt Ferguson (born 1940 in Salt Lake City, Utah) is an American sculptor and a digital artist, specifically an algorist. He is also well known for his development of the PSLQ algorithm, an integer relation detection algorithm. E ...
at the
Mathematical Sciences Research Institute The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, Califo ...
in
Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and E ...
, made of marble and serpentine, and unveiled on November 14, 1993. The title refers to the fact that starting at any vertex of the triangulated surface and moving along any edge, if you alternately turn left and right when reaching a vertex, you always return to the original point after eight edges. The acquisition of the sculpture led in due course to the publication of a book of papers , detailing properties of the quartic and containing the first English translation of Klein's paper. Polyhedral models with tetrahedral symmetry most often have convex hull a
truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncation (geometry), truncating all 4 vertices of ...
– see and for examples and illustrations. Some of these models consist of 20 triangles or 56 triangles (abstractly, the
regular skew polyhedron In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra ...
, 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, 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. Alternatively, the quartic can be modeled by a polyhedron with octahedral symmetry: Klein modeled the quartic by a shape with octahedral symmetries and with points at infinity (an "open polyhedron"), namely three hyperboloids meeting on orthogonal axes, while it can also be modeled as a closed polyhedron which must be ''immersed'' (have self-intersections), not embedded. Such polyhedra may have various convex hulls, including the
truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
, the
snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...
, or the
rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...
, as in the
small cubicuboctahedron In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral. The small cubicuboctahedro ...
at right. The small cubicuboctahedron immersion is obtained by joining some of the triangles (2 triangles form a square, 6 form an octagon), which can be visualized b
coloring the triangles
(the corresponding tiling is topologically but not geometrically the 3 4 | 4 tiling). This immersion can also be used to geometrically construct the
Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
M24 by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons.


Dessin d'enfants

The
dessin d'enfant In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French ...
on the Klein quartic associated with the quotient map by its automorphism group (with quotient the Riemann sphere) is precisely the 1-skeleton of the order-3 heptagonal tiling. That is, the quotient map is ramified over the points , and ; dividing by 1728 yields a
Belyi function In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at ...
(ramified at , and ), where the 56 vertices (black points in dessin) lie over 0, the midpoints of the 84 edges (white points in dessin) lie over 1, and the centers of the 24 heptagons lie over infinity. The resulting dessin is a "platonic" dessin, meaning edge-transitive and "clean" (each white point has valence 2).


Related Riemann surfaces

The Klein quartic is related to various other Riemann surfaces. Geometrically, it is the smallest
Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by vir ...
(lowest genus); the next is the
Macbeath surface In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbeath surface is the simple group PSL(2,8), con ...
(genus 7), and the following is the
First Hurwitz triplet In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, res ...
(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 In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
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 In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
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.Elkies, section 4.4 (pp. 94–97) in . More exceptionally, the Klein quartic forms part of a "
trinity The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God th ...
" in the sense of
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
, which can also be described as a
McKay correspondence In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducibl ...
. In this collection, the
projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
s PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, 660) are analogous. Note that 4 × 5 × 6/2 = 60, 6 × 7 × 8/2 = 168, and 10 × 11 × 12/2 = 660. These correspond to
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
(genus 0), the symmetries of the Klein quartic (genus 3), and the
buckyball surface In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, ...
(genus 70). These are further connected to many other exceptional phenomena, which is elaborated at " trinities".


See also

* Grünbaum–Rigby configuration * Shimura curve *
Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by vir ...
*
Bolza surface In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
*
Bring's curve In mathematics, Bring's curve (also called Bring's surface) is the curve given by the equations :v+w+x+y+z=v^2+w^2+x^2+y^2+z^2=v^3+w^3+x^3+y^3+z^3=0. It was named by after Erland Samuel Bring who studied a similar construction in 1786 in a Promot ...
*
Macbeath surface In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface. The automorphism group of the Macbeath surface is the simple group PSL(2,8), con ...
*
First Hurwitz triplet In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, res ...


References


Literature

* Translated in . * *
Paperback edition
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...
, 2001, . Reviewed by: * * * * * *


External links


Klein's Quartic Curve
John Baez, July 28, 2006

by Greg Egan – illustrations

by Greg Egan – illustrations {{Algebraic curves navbox Algebraic curves Riemann surfaces Differential geometry of surfaces Systolic geometry