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[[File:Uniform tiling 73-t2.png|thumb|The Klein quartic is a quotient of the [[order-7 triangular tiling]].]]
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[[File:Uniform tiling 73-t0.png|thumb|Dually, the Klein quartic is a quotient of the dual tiling, the [[order-3 heptagonal tiling]].]]


In [[hyperbolic geometry]], the '''Klein quartic''', named after [[Felix Klein]], is a [[compact space|compact]] [[Riemann surface]] of [[genus (mathematics)|genus]] 3 with the highest possible order [[automorphism group]] for this genus, namely order 168 orientation-preserving automorphisms, and 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]]. The quartic was first described in {{Harv|Klein|1878b}}.
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Klein's quartic occurs in many branches of mathematics, in contexts including [[representation theory]], [[homology theory]], [[octonion multiplication]], [[Fermat's last theorem]], and the [[Stark-Heegner theorem]] on [[imaginary quadratic number field]]s of [[class number (number theory)|class number]] one; see {{Harv|Levy|1999}} for a survey of properties.
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Originally, the "Klein quartic" referred specifically to the subset of the [[complex projective plane]] '''CP'''<sup>2</sup> defined by the equation given in the '''As an algebraic curve''' section. This has a specific  [[Riemannian manifold|Riemannian metric]] (that makes it a minimal surface in '''CP'''<sup>2</sup>), 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]] '''H'''<sup>2</sup> by a certain [[cocompact]] group '''G''' that acts  [[Group_action#Types_of_actions|freely]] on '''H'''<sup>2</sup> by isometries. This gives the Klein quartic a Riemannian metric of constant  negative curvature = −1  that it inherits from '''H'''<sup>2</sup>. 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, '''Z'''/7'''Z'''), and also as the isomorphic group '''PSL'''(3, '''Z'''/2'''Z'''). By [[covering space]] theory, the group '''G''' mentioned above is isomorphic to the [[fundamental group]] of the compact surface of genus 3.
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== Closed and open forms ==
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It is important to distinguish two different forms of the quartic. The ''[[Closed manifold|closed]]'' quartic is what is generally meant in geometry; topologically it has genus 3 and is a [[compact space]]. The ''[[open manifold|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 [[Cusp neighborhood|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<ref>{{Harv|Levy|1999|loc=p. 24}}</ref> – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.


==As an algebraic curve==
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The Klein quartic can be viewed as a [[projective variety|projective]] [[algebraic curve]] over the [[complex number]]s '''C''', defined by the following quartic equation in [[homogeneous coordinates]] [x:y:z] on '''CP'''<sup>2</sup>:
 
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:<math>x^3y + y^3z + z^3x = 0.\ </math>
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The locus of this equation in '''CP'''<sup>2</sup> is the original Riemannian surface that Klein described.
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The compact Klein quartic can be constructed as the quotient of the [[Hyperbolic geometry|hyperbolic plane]] by the action of a suitable [[Fuchsian group]] Γ(I) which is the principal [[congruence subgroup]] associated with the ideal <math>I=\langle \eta-2\rangle</math> in the ring of integers <math>\mathbb{Z}(\eta)</math> of the field <math>\mathbb Q(\eta)</math> where <math>\eta=2\cos (2\pi/7)</math>. Note the identity
 
 
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:<math>(2-\eta)^3= 7(\eta-1)^2,\ </math>
 
 
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exhibiting <math>2-\eta</math> as a prime factor of 7 in the ring of integers.
 
 
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The group Γ(I) is a subgroup of the [[(2,3,7) triangle group|(2,3,7)]] hyperbolic [[triangle group]].  Namely, Γ(I) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i,j and relations <math>i^2=j^2=\eta, \quad ij=-ji</math>. One chooses a suitable [[Hurwitz quaternion order]] <math>\mathcal Q_{\mathrm{Hur}}</math> in the quaternion algebra, Γ(I) is then the group of norm 1 elements in <math>1+I\mathcal Q_{\mathrm{Hur}}</math>. The least absolute value of a trace of a hyperbolic element in Γ(I) is <math>\eta^2+3\eta+2</math>, corresponding the value 3.936 for the [[systolic geometry|systole]] of the Klein quartic, one of the highest in this genus.
 
 
</ul>
==Tiling==
[[File:Order-3 heptakis heptagonal tiling.png|thumb|The tiling of the quartic by reflection domains is a quotient of the [[order-3 bisected heptagonal tiling]].]]
The Klein quartic admits tilings connected with the symmetry group (a "[[Regular map (graph theory)|regular map]]"<ref name="scholl"/>), 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 geometry|hyperbolic plane]] (the [[universal cover]] 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]]s), and often regular tilings are used instead. A quotient of any tiling in the [[Uniform_tilings_in_hyperbolic_plane#The_.5B7.2C3.5D_.287_3_2.29_group_family|(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]]s, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56 [[equilateral triangle]]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 &times; 7 = 168
:56 &times; 3 = 168
The covering tilings on the hyperbolic plane are the [[order-3 heptagonal tiling]] and the [[order-7 triangular tiling]].
 
The automorphism group can be augmented (by a symmetry which is not realized by a symmetry of the tiling) to yield the [[Mathieu group]] M<sub>24</sub>.<ref name="richter"/>
 
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. 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 SL(2,'''R''') on the [[Poincaré half-plane model|upper half-plane model]] '''H'''<sup>2</sup> of the [[Hyperbolic geometry|hyperbolic plane]] by [[Möbius transformation]]s, the affine Klein quartic can be realized as the quotient Γ(7)\'''H'''<sup>2</sup>. (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 [[modular arithmetic|modulo]] 7.)
 
== 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 PSL(2,7), since PSL(2,7) does not embed as a subgroup of SO(3) (or O(3)) – it does not have a (non-trivial) 3-dimensional linear representation.
 
However, many 3-dimensional models of the Klein quartic have been given, starting in Klein's original paper,<ref name="scholl">{{Harv|Scholl|Schürmann|Wills|2002}}</ref><ref name="baez">[http://math.ucr.edu/home/baez/klein.html Klein's Quartic Curve], John Baez, July 28, 2006</ref><ref name="westendorp">[http://www.xs4all.nl/~westy31/Geometry/Geometry.html Platonic tilings of Riemann surfaces], [http://www.xs4all.nl/~westy31/ Gerard Westendorp]</ref><ref>[http://www.cs.auckland.ac.nz/~mike/klein/ Paper models of the Klein quartic], [http://www.cs.auckland.ac.nz/~mike/ Mike Stay]</ref><ref name="sequin">[http://www.cs.berkeley.edu/~sequin/PAPERS/Bridges06_PatternsOnTetrus.pdf Patterns on the Genus-3 Klein Quartic], by Carlo H. Séquin, accompanying [http://www.bridgesmathart.org/carlo-mirror/BRIDGES2006/ArtExhibit_Sequin/index.html Pieces at the Bridges Art-Exhibit, London, August 4-8, 2006], with "Klein Quartic Quilt", by Eveline Séquin, based on a pattern by Bill Thurston</ref> 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.
 
[[File:The Eightfold Way - Silvio Levy - cover.jpg|thumb|upright|''The Eightfold Way'' – sculpture by [[Helaman Ferguson]] and accompanying book.]]
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",<ref name="sequin" /> or by polyhedral approximations, which have been dubbed "tetroids";<ref name="sequin" /> 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 [[Mathematical Sciences Research Institute]] in [[Berkeley, California]], 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 {{Harv|Levy|1999}}, 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]] – see {{Harv|Schulte|Wills|1985}} and {{Harv|Scholl|Schürmann|Wills|2002}} for examples and illustrations. 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,<ref name="schulte">{{Harv|Schulte|Wills|1985}}</ref> 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.<ref name="scholl"/>
 
[[File:Small cubicuboctahedron.png|thumb|The [[small cubicuboctahedron]] is a polyhedral immersion of the tiling of the Klein quartic with octahedral symmetry.]]
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"),<ref name="westendorp" /> namely three [[hyperboloid]]s meeting on orthogonal axes,<ref name="scholl"/> while it can also be modeled as a closed polyhedron which must be ''immersed'' (have self-intersections), not embedded.<ref name="scholl"/> Such polyhedra may have various convex hulls, including the [[truncated cube]],<ref name="egan">[http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html Klein's Quartic Curve], by Greg Egan</ref> the [[snub cube]],<ref name="schulte"/> or the [[rhombicuboctahedron]], as in the [[small cubicuboctahedron]] at right.<ref name="richter">{{Harv|Richter}}</ref> 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 by [http://homepages.wmich.edu/~drichter/images/mathieu/hypercolors.jpg coloring the triangles] (the corresponding tiling is topologically but not geometrically the [[:File:Uniform tiling 443-t01.png|t<sub>0,1</sub>{4, 3, 3} tiling]]). This immersion can also be used to geometrically construct the [[Mathieu group]] M<sub>24</sub> by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons.<ref name="richter" />
 
==Dessin d'enfants==
The [[dessin d'enfant]] 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.<ref>{{citation
| last = le Bruyn | first = Lieven
| title = The best rejected proposal ever
| date = 7 March 2007
| url = http://www.neverendingbooks.org/index.php/the-best-rejected-proposal-ever.html}}.</ref> That is, the quotient map is ramified over the points 0, 1728, and ∞; dividing by 1728 yields a [[Belyi function]] (ramified at 0, 1, 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 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 [[Riemann_surface#Isometries_of_Riemann_surfaces|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 variety|principally polarized abelian varieties]] of dimension 6.<ref>Elkies, section 4.4 (pp. 94–97) in {{Harv|Levy|1999}}.</ref>
 
There are also other [[quartic surface]]s of interest – see [[Quartic surface#Special quartic surfaces|special quartic surfaces]].
 
More exceptionally, the Klein quartic forms part of a "[[ADE classification#Trinities|trinity]]" in the sense of [[Vladimir Arnold]], which can also be described as a [[McKay correspondence]]. In this collection, the [[projective special linear group]]s 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 3), and the [[buckyball surface]] (genus 70).<ref name="martin">{{citation | last1 = Martin | first2 = Pablo | last2 = Singerman | first1 = David | title = From Biplanes to the Klein quartic and the Buckyball | url = http://www.neverendingbooks.org/DATA/biplanesingerman.pdf | date = April 17, 2008 }}</ref> These are further connected to many other exceptional phenomena, which is elaborated at "[[ADE classification#Trinities|trinities]]".
 
==See also==
*[[Bolza surface]]
*[[Macbeath surface]]
*[[First Hurwitz triplet]]
 
==References==
{{reflist}}
{{refbegin}}
*{{cite doi|10.1007/BF01677143}}
*{{citation|authorlink=Noam Elkies|last=Elkies|first= N.|chapter= Shimura curve computations|title=  Algorithmic number theory (Portland, OR, 1998)|pages= 1–47|series= Lecture Notes in Computer Science|volume= 1423|publisher= Springer|place= Berlin|year=1998|arxiv=math.NT/0005160}}
*{{Citation | editor1-last=Levy | editor1-first=Silvio | title= The eightfold way | url=http://www.msri.org/communications/books/Book35/index.html | publisher=[[Cambridge University Press]] | series=Mathematical Sciences Research Institute Publications | isbn=978-0-521-66066-2 | mr=1722410 | year=1999 | volume=35 | postscript =, [http://www.cambridge.org/catalogue/catalogue.asp?ISBN=9780521004190 paperback edition] by [[Cambridge University Press]], 2001, ISBN 978-0-521-00419-0. [http://www.maa.org/reviews/eightfold.html Read This: The Eightfold Way, reviewed by Ruth Michler].}}
* {{cite doi|10.1112/jlms/s2-32.3.539}}
* {{Citation
| last1 = Karcher
| first1 = H.
| first2 = M. | last2 = Weber
| title = On Klein's Riemann Surface
| accessdate = 2010-04-17
| year = 1996
| url = http://www.math.uni-bonn.de/people/weber/zoo/klein.ps.gz
| id = {{citeseerx|10.1.1.47.1879}}
}} {{Dead link|date=October 2010|bot=H3llBot}}
 
* {{citation | ref = {{harvid|Richter}} | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/mathieu.htm | title = How to Make the Mathieu Group M<sub>24</sub> | accessdate = 2010-04-15 }}
* {{cite doi|10.1007/BF03024730}}
* {{citation | journal = [http://www.heldermann.de/BAG/bagcon.htm Beiträge zur Algebra und Geometrie] (Contributions to Algebra and Geometry) | volume = [http://www.heldermann.de/BAG/bag44.htm 44] | year = 2003 | issue = [http://www.heldermann.de/BAG/bag44.htm#bag442 2] | pages = 413–430 | title = The Riemann Surface of a Uniform Dessin | url = http://www.emis.de/journals/BAG/vol.44/no.2/9.html | first1 = David | last1 = Singerman | first2 = Robert I. | last2 = Syddall | postscript =, [http://www.emis.de/journals/BAG/vol.44/no.2/b44h2sin.pdf PDF] }}
{{refend}}
 
== External links ==
* [http://math.ucr.edu/home/baez/klein.html Klein's Quartic Curve], John Baez, July 28, 2006
* [http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html Klein's Quartic Curve], by Greg Egan – illustrations
* [http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html Klein's Quartic Equations], by Greg Egan – illustrations
 
{{Algebraic curves navbox}}
 
[[Category:Algebraic curves]]
[[Category:Riemann surfaces]]
[[Category:Differential geometry of surfaces]]
[[Category:Systolic geometry]]

Latest revision as of 15:01, 1 December 2014

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