31:
2444:
2405:
1073:
2507:
2503:{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 â 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.
865:
2479:, 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 (
1782:
291:. 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 â geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.
1412:
1225:
1760:
2439:
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
1801:
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
1415:
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
2458:
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
884:
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 â reflections in the
2599:), 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).
1080:
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
961:
from a tiling) â 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
974:
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.
1087:
1492:
239:. 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
1331:
2578:
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
1406:
2518:
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
1802:
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.
201:
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
2641:
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.
1769:
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
885:
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
800:
699:
744:
1081:
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
584:
2412:
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.
478:
1336:
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" (
399:
1220:{\displaystyle 16\sinh ^{-1}\left(\left({\tfrac {1}{2}}{\sqrt {\csc ^{2}\left({\tfrac {\pi }{7}}\right)-4}}\right)\sin \left({\tfrac {\pi }{7}}\right)\right)\approx 3.93594624883.}
916:, 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.
850:
3284:
1755:{\displaystyle \left\{l(S),{\tfrac {l(S)}{8}};l(S),{\tfrac {l(S)}{8}};l(S),{\tfrac {l(S)}{8}};l(S),{\tfrac {l(S)}{8}};l(S),{\tfrac {l(S)}{8}};l(S),{\tfrac {l(S)}{8}}\right\}.}
1455:
2386:
2350:
2314:
2278:
2242:
2206:
2170:
2134:
2098:
2062:
2026:
1990:
1954:
1918:
1882:
1846:
3356:
1785:
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
1236:
2543:
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 by
2672:
PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, 660) are analogous. Note that 4 Ă 5 Ă 6/2 = 60, 6 Ă 7 Ă 8/2 = 168, and 10 Ă 11 Ă 12/2 = 660. These correspond to
1063:
1484:
2483:), detailing properties of the quartic and containing the first English translation of Klein's paper. Polyhedral models with tetrahedral symmetry most often have
2472:
2943:
1069:. 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.
3972:
1346:
287:
or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are
3779:
2547:
3508:
3871:
3468:
3349:
3139:
3112:
3072:
2650:
2440:
octahedral (order 24) symmetries; the remaining order 7 symmetry cannot be as easily visualized, and in fact is the title of Klein's paper.
2622:(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
2693:
3153:
905:
3937:
3559:
3458:
886:
105:
3927:
3216:
2884:
623:
is a subgroup of the group of elements of unit norm in the quaternion algebra generated as an associative algebra by the generators
1797:
of the Klein quartic. Because the Klein quartic has the largest symmetry group of surfaces in its topological class, much like the
908:
can be used (and will have the same automorphism group); of these, the two regular tilings are the tiling by 24 regular hyperbolic
3246:
3637:
3342:
150:
2669:
2555:
3784:
3695:
3148:
2566:
by adding to PSL(2,7) the permutation which interchanges opposite points of the bisecting lines of the squares and octagons.
957:, which abstracts from the geometry and only reflects the combinatorics of the tiling (this is a general way of obtaining an
3705:
3632:
3382:
3602:
3498:
987:
757:
154:
3861:
3825:
938:
The automorphism group can be augmented (by a symmetry which is not realized by a symmetry of the tiling) to yield the
3524:
3437:
1425:
1421:
633:
710:
3328:
1428:, where the length parameters are all equal to the length of the systole, and the twist parameters are all equal to
3977:
3835:
3473:
3131:
3104:
877:
519:
220:
146:
3967:
3881:
2949:
1038:
The Klein quartic can be obtained as the quotient of the hyperbolic plane by the action of a
Fuchsian group. The
932:
928:
52:
3794:
3774:
3710:
3627:
3488:
2499:) for examples and illustrations. Some of these models consist of 20 triangles or 56 triangles (abstractly, the
3529:
705:
445:
165:
3493:
2443:
1000:
142:
1076:
The fundamental domain of the Klein quartic. The surface is obtained by associating sides with equal numbers.
336:
3962:
3685:
2540:
2511:
2500:
316:
880:"), and these are used in understanding the symmetry group, dating back to Klein's original paper. Given a
3478:
2420:
as a 3-dimensional figure, in the sense that no 3-dimensional figure has (rotational) symmetries equal to
608:
3856:
3592:
2627:
3392:
2723:
2619:
2488:
813:
134:
3554:
3503:
3127:
2544:
1340:). M3 comes from a tessellation of (2,3,12) triangles, and its systole has multiplicity 24 and length
3932:
3804:
3715:
3463:
2680:(genus 70). These are further connected to many other exceptional phenomena, which is elaborated at "
2673:
2665:
2527:(have self-intersections), not embedded. Such polyhedra may have various convex hulls, including the
2476:
1431:
1066:
913:
3769:
3647:
3612:
3569:
3549:
2536:
2364:
2328:
2292:
2256:
2220:
2184:
1326:{\displaystyle 8\cosh ^{-1}\left({\tfrac {3}{2}}-2\sin ^{2}\left({\tfrac {\pi }{7}}\right)\right).}
1015:
996:
954:
890:
439:
424:
183:
82:
63:
3166:
2928:
2148:
2112:
2076:
2040:
2004:
1968:
1932:
1896:
1860:
1824:
30:
3899:
3483:
3322:
3266:
3233:
3086:
3050:
3025:
2681:
2657:
1039:
1027:
881:
300:
198:
89:
3690:
3670:
3642:
2797:"The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian"
2575:
2436:) â it does not have a (non-trivial) 3-dimensional linear representation over the real numbers.
1773:; indeed, the automorphism group of the Klein quartic is isomorphic to that of the Fano plane.
3799:
3746:
3617:
3432:
3427:
3278:
3135:
3108:
3068:
2890:
2880:
2677:
2523:
meeting on orthogonal axes, while it can also be modeled as a closed polyhedron which must be
958:
853:
288:
256:
119:
45:
3098:
2869:
3789:
3675:
3652:
3258:
3225:
3178:
3060:
3017:
2978:
2718:
2713:
2615:
2468:
2451:
1026:
consisting of matrices that are congruent to the identity matrix when all entries are taken
901:
869:
203:
3122:
3082:
2404:
1045:
3904:
3720:
3662:
3564:
3387:
3366:
3144:
3118:
3078:
2703:
2661:
2631:
2611:
2551:
1794:
1460:
894:
303:
273:
138:
101:
78:
3587:
2834:
2506:
1072:
3412:
3397:
3374:
2588:
2528:
612:
428:
307:
248:
3049:, Lecture Notes in Computer Science, vol. 1423, Berlin: Springer, pp. 1â47,
2514:
is a polyhedral immersion of the tiling of the Klein quartic with octahedral symmetry.
2463:
of the shape in 3 dimensions. The most notable smooth model (tetrus) is the sculpture
277:
quartic is what is generally meant in geometry; topologically it has genus 3 and is a
229:
by isometries. This gives the Klein quartic a
Riemannian metric of constant curvature
3956:
3919:
3680:
3607:
3402:
3293:
3270:
3237:
3029:
2708:
2698:
2646:
2638:
2623:
2559:
1798:
939:
283:
278:
212:
75:
24:
3090:
3866:
3840:
3830:
3820:
3622:
3442:
3196:
3162:
2808:
897:
of the quartic), and all
Hurwitz surfaces are tiled in the same way, as quotients.
115:
35:
2877:
BRIDGES Mathematical
Connections in Art, Music, and Science Conference Proceedings
1424:
by cutting along six of its systoles. This decomposition gives a symmetric set of
3214:
Schmutz, P. (1993). "Riemann surfaces with shortest geodesic of maximal length".
1806:
Numerical computations of the first 15 positive eigenvalues of the Klein quartic
1786:
1401:{\displaystyle 2\cosh ^{-1}\left(2+{\sqrt {3}}\right)\approx 3.9833047820988736.}
100:
automorphisms if orientation may be reversed. As such, the Klein quartic is the
3741:
3579:
3042:
3001:
2520:
2484:
1766:
71:
3182:
904:), and often regular tilings are used instead. A quotient of any tiling in the
864:
3736:
1781:
1770:
133:
Klein's quartic occurs in many branches of mathematics, in contexts including
3334:
2894:
3597:
2924:
2812:
2532:
2409:
3316:
164:
Originally, the "Klein quartic" referred specifically to the subset of the
2649:(as are the Hurwitz surfaces of genus 7 and 14), and as such parametrizes
909:
110:
3055:
3006:"Ueber die Transformation siebenter Ordnung der elliptischen Functionen"
3909:
3894:
3262:
3229:
3103:, Mathematical Sciences Research Institute Publications, vol. 35,
3064:
3021:
1411:
912:, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56
876:
The Klein quartic admits tilings connected with the symmetry group (a "
271:
It is important to distinguish two different forms of the quartic. The
3889:
2554:(the corresponding tiling is topologically but not geometrically the
953:
of the quartic (partition of the quartic variety into subsets) is an
868:
The tiling of the quartic by reflection domains is a quotient of the
3005:
2849:
423:
The compact Klein quartic can be constructed as the quotient of the
108:. Its (orientation-preserving) automorphism group is isomorphic to
3008:[On the order-seven transformation of elliptic functions].
2796:
2676:(genus 0), the symmetries of the Klein quartic (genus 3), and the
2558:). This immersion can also be used to geometrically construct the
2505:
2442:
2403:
1780:
1410:
1071:
863:
802:. The least absolute value of a trace of a hyperbolic element in
2607:
The Klein quartic is related to various other
Riemann surfaces.
3338:
773:
717:
1003:, the affine Klein quartic can be realized as the quotient
3245:
Scholl, P.; SchĂŒrmann, A.; Wills, J. M. (September 2002),
596:
as a prime factor of 7 in the ring of algebraic integers.
415:
is the original
Riemannian surface that Klein described.
3149:"The Eightfold Way: The Beauty of Klein's Quartic Curve"
856:
of the Klein quartic, one of the highest in this genus.
2879:. Bridges 2006. London, UK: Tarquin. pp. 245â254.
2656:
More exceptionally, the Klein quartic forms part of a "
1722:
1681:
1640:
1599:
1558:
1517:
1436:
1300:
1265:
1191:
1152:
1121:
3167:"A Polyhedral Realization of Felix Klein's Map {3, 7}
2367:
2331:
2295:
2259:
2223:
2187:
2151:
2115:
2079:
2043:
2007:
1971:
1935:
1899:
1863:
1827:
1495:
1463:
1434:
1349:
1239:
1090:
1048:
927:
The covering tilings on the hyperbolic plane are the
816:
760:
713:
636:
522:
448:
339:
40:(14-gon edges marked with the same number are equal.)
2980:
From Biplanes to the Klein quartic and the Buckyball
2765:
2496:
1457:
of the length of the systole. In particular, taking
3918:
3880:
3849:
3813:
3762:
3755:
3729:
3661:
3578:
3542:
3517:
3451:
3420:
3411:
3373:
2977:Martin, David; Singerman, Pablo (April 17, 2008),
2380:
2344:
2308:
2272:
2236:
2200:
2164:
2128:
2092:
2056:
2020:
1984:
1948:
1912:
1876:
1840:
1754:
1478:
1449:
1400:
1325:
1219:
1057:
844:
795:{\displaystyle 1+I{\mathcal {Q}}_{\mathrm {Hur} }}
794:
738:
693:
578:
472:
393:
2645:More subtly, the (projective) Klein quartic is a
2637:Algebraically, the (affine) Klein quartic is the
900:This tiling is uniform but not regular (it is by
3283:: CS1 maint: bot: original URL status unknown (
962:this way the geometry reduces to combinatorics.
694:{\displaystyle i^{2}=j^{2}=\eta ,\qquad ij=-ji.}
739:{\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }}
315:, defined by the following quartic equation in
3047:Algorithmic number theory (Portland, OR, 1998)
1486:to be the systole length, the coordinates are
1420:The Klein quartic can be decomposed into four
3350:
3292:Singerman, David; Syddall, Robert I. (2003),
2626:is the most symmetric genus 2 surface, while
2473:Simons Laufer Mathematical Sciences Institute
579:{\displaystyle (2-\eta )^{3}=7(\eta -1)^{2},}
8:
2911:
2828:
2826:
2630:is a highly symmetric genus 4 surface â see
2492:
467:
455:
3759:
3417:
3357:
3343:
3335:
3273:, archived from the original on 2007-06-11
3247:"Polyhedral Models of Felix Klein's Group"
1034:Fundamental domain and pants decomposition
179:
96:orientation-preserving automorphisms, and
3294:"The Riemann Surface of a Uniform Dessin"
3054:
2906:
2904:
2875:. In Sarhangi, Reza; Sharp, John (eds.).
2778:
2776:
2774:
2372:
2366:
2336:
2330:
2300:
2294:
2264:
2258:
2228:
2222:
2192:
2186:
2156:
2150:
2120:
2114:
2084:
2078:
2048:
2042:
2012:
2006:
1976:
1970:
1940:
1934:
1904:
1898:
1868:
1862:
1832:
1826:
1721:
1680:
1639:
1598:
1557:
1516:
1494:
1462:
1435:
1433:
1380:
1357:
1348:
1299:
1286:
1264:
1247:
1238:
1190:
1151:
1138:
1132:
1120:
1098:
1089:
1047:
821:
815:
779:
778:
772:
771:
759:
723:
722:
716:
715:
712:
654:
641:
635:
567:
539:
521:
473:{\displaystyle I=\langle \eta -2\rangle }
447:
376:
360:
344:
338:
2863:
2861:
2859:
2760:
2758:
2756:
2754:
2752:
1804:
852:, corresponding the value 3.936 for the
754:is then the group of norm 1 elements in
29:
2870:"Patterns on the Genus-3 Klein Quartic"
2735:
2651:principally polarized abelian varieties
1337:
394:{\displaystyle x^{3}y+y^{3}z+z^{3}x=0.}
56:(compare the 7-regular graph in violet)
44:The Klein quartic is a quotient of the
3780:Clifford's theorem on special divisors
3276:
3045:(1998), "Shimura curve computations",
2835:"Platonic tilings of Riemann surfaces"
127:
126:. The quartic was first described in (
49:(compare the 3-regular graph in green)
299:The Klein quartic can be viewed as a
255:mentioned above is isomorphic to the
7:
3035:
2964:
2963:Elkies, section 4.4 (pp. 94â97) in (
2795:Maxime Fortier Bourque, Bram Petri.
2743:
2618:(genus 7), and the following is the
2480:
1042:is a regular 14-gon, which has area
186:(that makes it a minimal surface in
158:
34:The Klein quartic with the two dual
3154:Mathematical Association of America
2783:
2664:, which can also be described as a
243:, and also as the isomorphic group
3938:Vector bundles on algebraic curves
3872:Weber's theorem (Algebraic curves)
3469:Hasse's theorem on elliptic curves
3459:Counting points on elliptic curves
3298:BeitrÀge zur Algebra und Geometrie
2766:Scholl, SchĂŒrmann & Wills 2002
2610:Geometrically, it is the smallest
2497:Scholl, SchĂŒrmann & Wills 2002
887:order-3 bisected heptagonal tiling
845:{\displaystyle \eta ^{2}+3\eta +2}
786:
783:
780:
730:
727:
724:
480:in the ring of algebraic integers
114:, the second-smallest non-abelian
16:Compact Riemann surface of genus 3
14:
3973:Differential geometry of surfaces
2942:le Bruyn, Lieven (7 March 2007),
1793:Little has been proved about the
151:imaginary quadratic number fields
3171:on a Riemann Surface of Genus 3"
2670:projective special linear groups
2614:(lowest genus); the next is the
2428:does not embed as a subgroup of
1230:An equivalent closed formula is
259:of the compact surface of genus
88:with the highest possible order
3560:Hurwitz's automorphisms theorem
3198:How to Make the Mathieu Group M
2945:The best rejected proposal ever
1450:{\displaystyle {\tfrac {1}{8}}}
669:
419:Quaternion algebra construction
106:Hurwitz's automorphisms theorem
3785:Gonality of an algebraic curve
3696:Differential of the first kind
3331:, by Greg Egan â illustrations
3325:, by Greg Egan â illustrations
3251:The Mathematical Intelligencer
2632:isometries of Riemann surfaces
1734:
1728:
1715:
1709:
1693:
1687:
1674:
1668:
1652:
1646:
1633:
1627:
1611:
1605:
1592:
1586:
1570:
1564:
1551:
1545:
1529:
1523:
1510:
1504:
1473:
1467:
564:
551:
536:
523:
404:The locus of this equation in
161:) for a survey of properties.
104:of lowest possible genus; see
1:
3928:BirkhoffâGrothendieck theorem
3638:Nagata's conjecture on curves
3509:SchoofâElkiesâAtkin algorithm
3383:Five points determine a conic
3165:; Wills, J. M. (1985-12-01),
2381:{\displaystyle \lambda _{15}}
2345:{\displaystyle \lambda _{14}}
2309:{\displaystyle \lambda _{13}}
2273:{\displaystyle \lambda _{12}}
2237:{\displaystyle \lambda _{11}}
2201:{\displaystyle \lambda _{10}}
970:The above is a tiling of the
92:for this genus, namely order
3499:Supersingular elliptic curve
2694:GrĂŒnbaumâRigby configuration
2587:; dividing by 1728 yields a
2416:The Klein quartic cannot be
2165:{\displaystyle \lambda _{9}}
2129:{\displaystyle \lambda _{8}}
2093:{\displaystyle \lambda _{7}}
2057:{\displaystyle \lambda _{6}}
2021:{\displaystyle \lambda _{5}}
1985:{\displaystyle \lambda _{4}}
1949:{\displaystyle \lambda _{3}}
1913:{\displaystyle \lambda _{2}}
1877:{\displaystyle \lambda _{1}}
1841:{\displaystyle \lambda _{0}}
427:by the action of a suitable
3706:Riemann's existence theorem
3633:Hilbert's sixteenth problem
3525:Elliptic curve cryptography
3438:Fundamental pair of periods
1426:Fenchel-Nielsen coordinates
746:in the quaternion algebra,
3994:
3836:Moduli of algebraic curves
3319:, John Baez, July 28, 2006
3132:Cambridge University Press
3105:Cambridge University Press
3097:Levy, Silvio, ed. (1999),
2668:. In this collection, the
978:Considering the action of
442:associated with the ideal
18:
3329:Klein's Quartic Equations
2868:SĂ©quin, Carlo H. (2006).
933:order-7 triangular tiling
929:order-3 heptagonal tiling
3603:CayleyâBacharach theorem
3530:Elliptic curve primality
3183:10.1112/jlms/s2-32.3.539
2912:Schulte & Wills 1985
2634:for further discussion.
2603:Related Riemann surfaces
2493:Schulte & Wills 1985
706:Hurwitz quaternion order
166:complex projective plane
3862:RiemannâHurwitz formula
3826:GromovâWitten invariant
3686:Compact Riemann surface
3474:Mazur's torsion theorem
2929:"Klein's Quartic Curve"
2813:"Klein's Quartic Curve"
2541:small cubicuboctahedron
2512:small cubicuboctahedron
2501:regular skew polyhedron
704:One chooses a suitable
438:which is the principal
317:homogeneous coordinates
3479:Modular elliptic curve
3177:, s2-32 (3): 539â547,
2545:coloring the triangles
2515:
2455:
2454:and accompanying book.
2413:
2382:
2346:
2310:
2274:
2238:
2202:
2166:
2130:
2094:
2058:
2022:
1986:
1950:
1914:
1878:
1842:
1790:
1756:
1480:
1451:
1417:
1402:
1327:
1221:
1077:
1059:
1001:Möbius transformations
988:upper half-plane model
949:Corresponding to each
873:
846:
796:
740:
695:
580:
474:
395:
233:that it inherits from
182:. This has a specific
59:
3393:Rational normal curve
3323:Klein's Quartic Curve
3317:Klein's Quartic Curve
3010:Mathematische Annalen
2724:First Hurwitz triplet
2620:First Hurwitz triplet
2569:
2509:
2489:truncated tetrahedron
2446:
2407:
2383:
2347:
2311:
2275:
2239:
2203:
2167:
2131:
2095:
2059:
2023:
1987:
1951:
1915:
1879:
1843:
1784:
1757:
1481:
1452:
1414:
1403:
1328:
1222:
1075:
1060:
1058:{\displaystyle 8\pi }
914:equilateral triangles
867:
847:
797:
741:
696:
607:is a subgroup of the
581:
513:. Note the identity
475:
396:
295:As an algebraic curve
267:Closed and open forms
180:an algebraic equation
147:StarkâHeegner theorem
143:Fermat's Last Theorem
135:representation theory
33:
3933:Stable vector bundle
3805:Weil reciprocity law
3795:RiemannâRoch theorem
3775:BrillâNoether theory
3711:RiemannâRoch theorem
3628:Genusâdegree formula
3489:MordellâWeil theorem
3464:Division polynomials
3175:J. London Math. Soc.
2833:Westendorp, Gerard.
2674:icosahedral symmetry
2666:McKay correspondence
2477:Berkeley, California
2400:3-dimensional models
2365:
2329:
2293:
2257:
2221:
2185:
2149:
2113:
2077:
2041:
2005:
1969:
1933:
1897:
1861:
1825:
1493:
1479:{\displaystyle l(S)}
1461:
1432:
1347:
1237:
1088:
1067:Gauss-Bonnet theorem
1046:
814:
758:
711:
634:
520:
446:
337:
3756:Structure of curves
3648:Quartic plane curve
3570:Hyperelliptic curve
3550:De Franchis theorem
3494:NagellâLutz theorem
3195:Richter, David A.,
2952:on 27 February 2014
2556:3 4 | 4 tiling
2537:rhombicuboctahedron
1807:
1396:3.9833047820988736.
1016:congruence subgroup
955:abstract polyhedron
440:congruence subgroup
197:), under which its
64:hyperbolic geometry
3763:Divisors on curves
3555:Faltings's theorem
3504:Schoof's algorithm
3484:Modularity theorem
3263:10.1007/BF03024730
3230:10.1007/BF01896258
3065:10.1007/BFb0054850
3022:10.1007/BF01677143
2660:" in the sense of
2550:2016-03-03 at the
2516:
2456:
2414:
2378:
2342:
2306:
2270:
2234:
2198:
2162:
2126:
2090:
2054:
2018:
1982:
1946:
1910:
1874:
1838:
1805:
1791:
1752:
1742:
1701:
1660:
1619:
1578:
1537:
1476:
1447:
1445:
1418:
1398:
1323:
1309:
1274:
1217:
1200:
1161:
1130:
1078:
1055:
1040:fundamental domain
882:fundamental domain
874:
842:
792:
736:
691:
576:
470:
391:
251:theory, the group
199:Gaussian curvature
90:automorphism group
60:
3978:Systolic geometry
3950:
3949:
3946:
3945:
3857:HasseâWitt matrix
3800:Weierstrass point
3747:Smooth completion
3716:TeichmĂŒller space
3618:Cubic plane curve
3538:
3537:
3452:Arithmetic theory
3433:Elliptic integral
3428:Elliptic function
3140:978-0-521-00419-0
3128:Paperback edition
3114:978-0-521-66066-2
3100:The Eightfold Way
3074:978-3-540-64657-0
2850:"Klein's quartic"
2817:John Baez's stuff
2678:buckyball surface
2465:The Eightfold Way
2448:The Eightfold Way
2397:
2396:
1741:
1700:
1659:
1618:
1577:
1536:
1444:
1385:
1308:
1273:
1199:
1173:
1160:
1129:
959:abstract polytope
902:scalene triangles
257:fundamental group
184:Riemannian metric
120:alternating group
57:
53:triangular tiling
50:
46:heptagonal tiling
41:
3985:
3968:Riemann surfaces
3790:Jacobian variety
3760:
3663:Riemann surfaces
3653:Real plane curve
3613:Cramer's paradox
3593:BĂ©zout's theorem
3418:
3367:algebraic curves
3359:
3352:
3345:
3336:
3305:
3288:
3282:
3274:
3241:
3210:
3209:
3208:
3191:
3190:
3189:
3158:
3147:(31 July 2000).
3145:Michler, Ruth I.
3125:
3093:
3058:
3033:
2987:
2986:
2985:
2974:
2968:
2961:
2955:
2953:
2948:, archived from
2939:
2933:
2932:
2931:. Science Notes.
2921:
2915:
2908:
2899:
2898:
2874:
2865:
2854:
2853:
2845:
2839:
2838:
2830:
2821:
2820:
2805:
2799:
2793:
2787:
2780:
2769:
2762:
2747:
2740:
2719:Macbeath surface
2653:of dimension 6.
2616:Macbeath surface
2598:
2594:
2586:
2582:
2570:Dessin d'enfants
2469:Helaman Ferguson
2452:Helaman Ferguson
2435:
2431:
2427:
2423:
2408:An animation by
2387:
2385:
2384:
2379:
2377:
2376:
2351:
2349:
2348:
2343:
2341:
2340:
2315:
2313:
2312:
2307:
2305:
2304:
2279:
2277:
2276:
2271:
2269:
2268:
2243:
2241:
2240:
2235:
2233:
2232:
2207:
2205:
2204:
2199:
2197:
2196:
2171:
2169:
2168:
2163:
2161:
2160:
2135:
2133:
2132:
2127:
2125:
2124:
2099:
2097:
2096:
2091:
2089:
2088:
2063:
2061:
2060:
2055:
2053:
2052:
2027:
2025:
2024:
2019:
2017:
2016:
1991:
1989:
1988:
1983:
1981:
1980:
1955:
1953:
1952:
1947:
1945:
1944:
1919:
1917:
1916:
1911:
1909:
1908:
1883:
1881:
1880:
1875:
1873:
1872:
1847:
1845:
1844:
1839:
1837:
1836:
1814:Numerical value
1808:
1761:
1759:
1758:
1753:
1748:
1744:
1743:
1737:
1723:
1702:
1696:
1682:
1661:
1655:
1641:
1620:
1614:
1600:
1579:
1573:
1559:
1538:
1532:
1518:
1485:
1483:
1482:
1477:
1456:
1454:
1453:
1448:
1446:
1437:
1407:
1405:
1404:
1399:
1391:
1387:
1386:
1381:
1365:
1364:
1332:
1330:
1329:
1324:
1319:
1315:
1314:
1310:
1301:
1291:
1290:
1275:
1266:
1255:
1254:
1226:
1224:
1223:
1218:
1210:
1206:
1205:
1201:
1192:
1179:
1175:
1174:
1166:
1162:
1153:
1143:
1142:
1133:
1131:
1122:
1106:
1105:
1064:
1062:
1061:
1056:
1025:
1013:
1009:
997:hyperbolic plane
994:
985:
891:hyperbolic plane
870:3-7 kisrhombille
851:
849:
848:
843:
826:
825:
809:
801:
799:
798:
793:
791:
790:
789:
777:
776:
753:
745:
743:
742:
737:
735:
734:
733:
721:
720:
700:
698:
697:
692:
659:
658:
646:
645:
626:
622:
606:
595:
585:
583:
582:
577:
572:
571:
544:
543:
512:
501:
490:
479:
477:
476:
471:
437:
425:hyperbolic plane
414:
400:
398:
397:
392:
381:
380:
365:
364:
349:
348:
329:
314:
262:
254:
246:
242:
238:
232:
228:
218:
210:
204:hyperbolic plane
196:
177:
113:
99:
95:
87:
55:
48:
39:
3993:
3992:
3988:
3987:
3986:
3984:
3983:
3982:
3953:
3952:
3951:
3942:
3914:
3905:Delta invariant
3876:
3845:
3809:
3770:AbelâJacobi map
3751:
3725:
3721:Torelli theorem
3691:Dessin d'enfant
3671:Belyi's theorem
3657:
3643:PlĂŒcker formula
3574:
3565:Hurwitz surface
3534:
3513:
3447:
3421:Analytic theory
3413:Elliptic curves
3407:
3388:Projective line
3375:Rational curves
3369:
3363:
3313:
3308:
3291:
3275:
3244:
3213:
3206:
3204:
3201:
3194:
3187:
3185:
3170:
3161:
3143:
3142:. Reviewed by:
3115:
3096:
3075:
3056:math.NT/0005160
3041:
3000:
2996:
2991:
2990:
2983:
2976:
2975:
2971:
2962:
2958:
2941:
2940:
2936:
2927:(5 June 2017).
2923:
2922:
2918:
2909:
2902:
2887:
2872:
2867:
2866:
2857:
2847:
2846:
2842:
2832:
2831:
2824:
2811:(23 May 2013).
2807:
2806:
2802:
2794:
2790:
2781:
2772:
2763:
2750:
2741:
2737:
2732:
2704:Hurwitz surface
2690:
2662:Vladimir Arnold
2628:Bring's surface
2612:Hurwitz surface
2605:
2596:
2592:
2584:
2580:
2576:dessin d'enfant
2572:
2565:
2552:Wayback Machine
2450:â sculpture by
2433:
2429:
2425:
2421:
2402:
2368:
2363:
2362:
2332:
2327:
2326:
2296:
2291:
2290:
2260:
2255:
2254:
2224:
2219:
2218:
2188:
2183:
2182:
2152:
2147:
2146:
2116:
2111:
2110:
2080:
2075:
2074:
2044:
2039:
2038:
2008:
2003:
2002:
1972:
1967:
1966:
1936:
1931:
1930:
1900:
1895:
1894:
1864:
1859:
1858:
1828:
1823:
1822:
1795:spectral theory
1779:
1777:Spectral theory
1724:
1683:
1642:
1601:
1560:
1519:
1500:
1496:
1491:
1490:
1459:
1458:
1430:
1429:
1373:
1369:
1353:
1345:
1344:
1295:
1282:
1263:
1259:
1243:
1235:
1234:
1186:
1147:
1134:
1119:
1115:
1114:
1110:
1094:
1086:
1085:
1044:
1043:
1036:
1019:
1011:
1004:
990:
979:
968:
945:
895:universal cover
862:
817:
812:
811:
803:
770:
756:
755:
747:
714:
709:
708:
650:
637:
632:
631:
624:
616:
600:
590:
563:
535:
518:
517:
503:
492:
481:
444:
443:
431:
421:
405:
372:
356:
340:
335:
334:
320:
310:
308:complex numbers
304:algebraic curve
297:
269:
260:
252:
244:
240:
234:
230:
224:
216:
206:
187:
168:
139:homology theory
125:
109:
102:Hurwitz surface
97:
93:
85:
79:Riemann surface
43:
42:
38:
28:
17:
12:
11:
5:
3991:
3989:
3981:
3980:
3975:
3970:
3965:
3963:Quartic curves
3955:
3954:
3948:
3947:
3944:
3943:
3941:
3940:
3935:
3930:
3924:
3922:
3920:Vector bundles
3916:
3915:
3913:
3912:
3907:
3902:
3897:
3892:
3886:
3884:
3878:
3877:
3875:
3874:
3869:
3864:
3859:
3853:
3851:
3847:
3846:
3844:
3843:
3838:
3833:
3828:
3823:
3817:
3815:
3811:
3810:
3808:
3807:
3802:
3797:
3792:
3787:
3782:
3777:
3772:
3766:
3764:
3757:
3753:
3752:
3750:
3749:
3744:
3739:
3733:
3731:
3727:
3726:
3724:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3678:
3673:
3667:
3665:
3659:
3658:
3656:
3655:
3650:
3645:
3640:
3635:
3630:
3625:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3584:
3582:
3576:
3575:
3573:
3572:
3567:
3562:
3557:
3552:
3546:
3544:
3540:
3539:
3536:
3535:
3533:
3532:
3527:
3521:
3519:
3515:
3514:
3512:
3511:
3506:
3501:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3455:
3453:
3449:
3448:
3446:
3445:
3440:
3435:
3430:
3424:
3422:
3415:
3409:
3408:
3406:
3405:
3400:
3398:Riemann sphere
3395:
3390:
3385:
3379:
3377:
3371:
3370:
3364:
3362:
3361:
3354:
3347:
3339:
3333:
3332:
3326:
3320:
3312:
3311:External links
3309:
3307:
3306:
3289:
3242:
3224:(6): 564â631.
3211:
3199:
3192:
3168:
3159:
3157:. MAA reviews.
3113:
3094:
3073:
3039:
3034:Translated in
3016:(3): 428â471.
2997:
2995:
2992:
2989:
2988:
2969:
2956:
2934:
2916:
2900:
2885:
2855:
2840:
2822:
2800:
2788:
2770:
2748:
2734:
2733:
2731:
2728:
2727:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2689:
2686:
2604:
2601:
2589:Belyi function
2571:
2568:
2563:
2529:truncated cube
2401:
2398:
2395:
2394:
2391:
2388:
2375:
2371:
2359:
2358:
2355:
2352:
2339:
2335:
2323:
2322:
2319:
2316:
2303:
2299:
2287:
2286:
2283:
2280:
2267:
2263:
2251:
2250:
2247:
2244:
2231:
2227:
2215:
2214:
2211:
2208:
2195:
2191:
2179:
2178:
2175:
2172:
2159:
2155:
2143:
2142:
2139:
2136:
2123:
2119:
2107:
2106:
2103:
2100:
2087:
2083:
2071:
2070:
2067:
2064:
2051:
2047:
2035:
2034:
2031:
2028:
2015:
2011:
1999:
1998:
1995:
1992:
1979:
1975:
1963:
1962:
1959:
1956:
1943:
1939:
1927:
1926:
1923:
1920:
1907:
1903:
1891:
1890:
1887:
1884:
1871:
1867:
1855:
1854:
1851:
1848:
1835:
1831:
1819:
1818:
1815:
1812:
1778:
1775:
1763:
1762:
1751:
1747:
1740:
1736:
1733:
1730:
1727:
1720:
1717:
1714:
1711:
1708:
1705:
1699:
1695:
1692:
1689:
1686:
1679:
1676:
1673:
1670:
1667:
1664:
1658:
1654:
1651:
1648:
1645:
1638:
1635:
1632:
1629:
1626:
1623:
1617:
1613:
1610:
1607:
1604:
1597:
1594:
1591:
1588:
1585:
1582:
1576:
1572:
1569:
1566:
1563:
1556:
1553:
1550:
1547:
1544:
1541:
1535:
1531:
1528:
1525:
1522:
1515:
1512:
1509:
1506:
1503:
1499:
1475:
1472:
1469:
1466:
1443:
1440:
1422:pairs of pants
1409:
1408:
1397:
1394:
1390:
1384:
1379:
1376:
1372:
1368:
1363:
1360:
1356:
1352:
1334:
1333:
1322:
1318:
1313:
1307:
1304:
1298:
1294:
1289:
1285:
1281:
1278:
1272:
1269:
1262:
1258:
1253:
1250:
1246:
1242:
1228:
1227:
1216:
1215:3.93594624883.
1213:
1209:
1204:
1198:
1195:
1189:
1185:
1182:
1178:
1172:
1169:
1165:
1159:
1156:
1150:
1146:
1141:
1137:
1128:
1125:
1118:
1113:
1109:
1104:
1101:
1097:
1093:
1054:
1051:
1035:
1032:
967:
966:Affine quartic
964:
943:
925:
924:
921:
906:(2,3,7) family
861:
858:
841:
838:
835:
832:
829:
824:
820:
788:
785:
782:
775:
769:
766:
763:
732:
729:
726:
719:
702:
701:
690:
687:
684:
681:
678:
675:
672:
668:
665:
662:
657:
653:
649:
644:
640:
627:and relations
613:triangle group
587:
586:
575:
570:
566:
562:
559:
556:
553:
550:
547:
542:
538:
534:
531:
528:
525:
469:
466:
463:
460:
457:
454:
451:
429:Fuchsian group
420:
417:
402:
401:
390:
387:
384:
379:
375:
371:
368:
363:
359:
355:
352:
347:
343:
296:
293:
268:
265:
249:covering space
123:
70:, named after
19:For the Klein
15:
13:
10:
9:
6:
4:
3:
2:
3990:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3960:
3958:
3939:
3936:
3934:
3931:
3929:
3926:
3925:
3923:
3921:
3917:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3887:
3885:
3883:
3882:Singularities
3879:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3854:
3852:
3848:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3818:
3816:
3812:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3767:
3765:
3761:
3758:
3754:
3748:
3745:
3743:
3740:
3738:
3735:
3734:
3732:
3730:Constructions
3728:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3701:Klein quartic
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3681:Bolza surface
3679:
3677:
3676:Bring's curve
3674:
3672:
3669:
3668:
3666:
3664:
3660:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3608:Conic section
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3588:AF+BG theorem
3586:
3585:
3583:
3581:
3577:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3545:
3541:
3531:
3528:
3526:
3523:
3522:
3520:
3516:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3456:
3454:
3450:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3425:
3423:
3419:
3416:
3414:
3410:
3404:
3403:Twisted cubic
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3380:
3378:
3376:
3372:
3368:
3360:
3355:
3353:
3348:
3346:
3341:
3340:
3337:
3330:
3327:
3324:
3321:
3318:
3315:
3314:
3310:
3303:
3299:
3295:
3290:
3286:
3280:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3218:
3212:
3203:
3202:
3193:
3184:
3180:
3176:
3172:
3164:
3163:Schulte, Egon
3160:
3156:
3155:
3150:
3146:
3141:
3137:
3133:
3129:
3124:
3120:
3116:
3110:
3106:
3102:
3101:
3095:
3092:
3088:
3084:
3080:
3076:
3070:
3066:
3062:
3057:
3052:
3048:
3044:
3040:
3037:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2998:
2993:
2982:
2981:
2973:
2970:
2966:
2960:
2957:
2951:
2947:
2946:
2938:
2935:
2930:
2926:
2920:
2917:
2913:
2907:
2905:
2901:
2896:
2892:
2888:
2886:0-9665201-7-3
2882:
2878:
2871:
2864:
2862:
2860:
2856:
2851:
2844:
2841:
2836:
2829:
2827:
2823:
2818:
2814:
2810:
2809:Baez, John C.
2804:
2801:
2798:
2792:
2789:
2785:
2779:
2777:
2775:
2771:
2767:
2761:
2759:
2757:
2755:
2753:
2749:
2745:
2739:
2736:
2729:
2725:
2722:
2720:
2717:
2715:
2714:Bring's curve
2712:
2710:
2709:Bolza surface
2707:
2705:
2702:
2700:
2699:Shimura curve
2697:
2695:
2692:
2691:
2687:
2685:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2654:
2652:
2648:
2647:Shimura curve
2643:
2640:
2639:modular curve
2635:
2633:
2629:
2625:
2624:Bolza surface
2621:
2617:
2613:
2608:
2602:
2600:
2591:(ramified at
2590:
2577:
2567:
2561:
2560:Mathieu group
2557:
2553:
2549:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2513:
2508:
2504:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2453:
2449:
2445:
2441:
2437:
2419:
2411:
2406:
2399:
2392:
2389:
2373:
2369:
2361:
2360:
2356:
2353:
2337:
2333:
2325:
2324:
2320:
2317:
2301:
2297:
2289:
2288:
2284:
2281:
2265:
2261:
2253:
2252:
2248:
2245:
2229:
2225:
2217:
2216:
2212:
2209:
2193:
2189:
2181:
2180:
2176:
2173:
2157:
2153:
2145:
2144:
2140:
2137:
2121:
2117:
2109:
2108:
2104:
2101:
2085:
2081:
2073:
2072:
2068:
2065:
2049:
2045:
2037:
2036:
2032:
2029:
2013:
2009:
2001:
2000:
1996:
1993:
1977:
1973:
1965:
1964:
1960:
1957:
1941:
1937:
1929:
1928:
1924:
1921:
1905:
1901:
1893:
1892:
1888:
1885:
1869:
1865:
1857:
1856:
1852:
1849:
1833:
1829:
1821:
1820:
1817:Multiplicity
1816:
1813:
1810:
1809:
1803:
1800:
1799:Bolza surface
1796:
1788:
1783:
1776:
1774:
1772:
1768:
1749:
1745:
1738:
1731:
1725:
1718:
1712:
1706:
1703:
1697:
1690:
1684:
1677:
1671:
1665:
1662:
1656:
1649:
1643:
1636:
1630:
1624:
1621:
1615:
1608:
1602:
1595:
1589:
1583:
1580:
1574:
1567:
1561:
1554:
1548:
1542:
1539:
1533:
1526:
1520:
1513:
1507:
1501:
1497:
1489:
1488:
1487:
1470:
1464:
1441:
1438:
1427:
1423:
1413:
1395:
1392:
1388:
1382:
1377:
1374:
1370:
1366:
1361:
1358:
1354:
1350:
1343:
1342:
1341:
1339:
1320:
1316:
1311:
1305:
1302:
1296:
1292:
1287:
1283:
1279:
1276:
1270:
1267:
1260:
1256:
1251:
1248:
1244:
1240:
1233:
1232:
1231:
1214:
1211:
1207:
1202:
1196:
1193:
1187:
1183:
1180:
1176:
1170:
1167:
1163:
1157:
1154:
1148:
1144:
1139:
1135:
1126:
1123:
1116:
1111:
1107:
1102:
1099:
1095:
1091:
1084:
1083:
1082:
1074:
1070:
1068:
1052:
1049:
1041:
1033:
1031:
1029:
1023:
1017:
1008:
1002:
998:
993:
989:
983:
976:
973:
965:
963:
960:
956:
952:
947:
941:
940:Mathieu group
936:
934:
930:
922:
919:
918:
917:
915:
911:
907:
903:
898:
896:
892:
888:
883:
879:
871:
866:
859:
857:
855:
839:
836:
833:
830:
827:
822:
818:
807:
767:
764:
761:
751:
707:
688:
685:
682:
679:
676:
673:
670:
666:
663:
660:
655:
651:
647:
642:
638:
630:
629:
628:
620:
614:
610:
604:
597:
594:
573:
568:
560:
557:
554:
548:
545:
540:
532:
529:
526:
516:
515:
514:
510:
506:
499:
495:
491:of the field
488:
484:
464:
461:
458:
452:
449:
441:
435:
430:
426:
418:
416:
412:
408:
388:
385:
382:
377:
373:
369:
366:
361:
357:
353:
350:
345:
341:
333:
332:
331:
327:
323:
318:
313:
309:
305:
302:
294:
292:
290:
286:
285:
280:
279:compact space
276:
275:
266:
264:
258:
250:
237:
227:
222:
214:
211:by a certain
209:
205:
200:
194:
190:
185:
181:
175:
171:
167:
162:
160:
156:
152:
148:
144:
140:
136:
131:
129:
121:
117:
112:
107:
103:
98:168 Ă 2 = 336
91:
84:
80:
77:
73:
69:
68:Klein quartic
65:
54:
51:and its dual
47:
37:
32:
26:
25:Klein quadric
22:
3867:Prym variety
3841:Stable curve
3831:Hodge bundle
3821:ELSV formula
3700:
3623:Fermat curve
3580:Plane curves
3543:Higher genus
3518:Applications
3443:Modular form
3304:(2): 413â430
3301:
3297:
3257:(3): 37â42,
3254:
3250:
3221:
3215:
3205:, retrieved
3197:
3186:, retrieved
3174:
3152:
3099:
3046:
3013:
3009:
2979:
2972:
2959:
2950:the original
2944:
2937:
2919:
2876:
2848:Stay, Mike.
2843:
2816:
2803:
2791:
2738:
2655:
2644:
2636:
2609:
2606:
2573:
2539:, as in the
2524:
2521:hyperboloids
2517:
2464:
2460:
2457:
2447:
2438:
2417:
2415:
1792:
1764:
1419:
1338:Schmutz 1993
1335:
1229:
1079:
1037:
1021:
1006:
991:
981:
977:
971:
969:
950:
948:
937:
926:
923:56 Ă 3 = 168
920:24 Ă 7 = 168
899:
875:
805:
749:
703:
618:
602:
598:
592:
588:
508:
504:
497:
493:
486:
482:
433:
422:
410:
406:
403:
325:
321:
311:
298:
282:
272:
270:
235:
225:
207:
192:
188:
173:
169:
163:
155:class number
132:
116:simple group
67:
61:
36:Klein graphs
20:
3742:Polar curve
2485:convex hull
2459:this is an
1811:Eigenvalue
1767:cubic graph
878:regular map
615:. Namely,
611:hyperbolic
589:exhibiting
178:defined by
128:Klein 1878b
72:Felix Klein
3957:Categories
3737:Dual curve
3365:Topics in
3207:2010-04-15
3188:2010-04-17
3043:Elkies, N.
2994:Literature
2925:Egan, Greg
2730:References
1771:Fano plane
972:projective
599:The group
301:projective
219:that acts
157:one; see (
145:, and the
118:after the
3850:Morphisms
3598:Bitangent
3271:122330024
3238:120508826
3036:Levy 1999
3030:121407539
3002:Klein, F.
2965:Levy 1999
2895:1099-6702
2744:Levy 1999
2682:trinities
2535:, or the
2533:snub cube
2481:Levy 1999
2461:embedding
2410:Greg Egan
2370:λ
2334:λ
2298:λ
2262:λ
2226:λ
2190:λ
2154:λ
2118:λ
2082:λ
2046:λ
2010:λ
1974:λ
1938:λ
1902:λ
1866:λ
1830:λ
1787:FreeFEM++
1416:of pants.
1393:≈
1367:
1359:−
1303:π
1293:
1277:−
1257:
1249:−
1212:≈
1194:π
1184:
1168:−
1155:π
1145:
1108:
1100:−
1053:π
910:heptagons
834:η
819:η
680:−
664:η
558:−
555:η
533:η
530:−
507:= 2 cos(2
468:⟩
462:−
459:η
456:⟨
306:over the
245:PSL(3, 2)
241:PSL(2, 7)
213:cocompact
159:Levy 1999
111:PSL(2, 7)
3279:citation
3134:, 2001,
3091:18251612
3004:(1878).
2746:, p. 24)
2688:See also
2548:Archived
2525:immersed
2426:PSL(2,7)
2424:, since
2422:PSL(2,7)
2418:realized
2390:50.6283
2354:49.0429
2318:44.8884
2282:41.5131
2246:37.4246
2210:36.4555
2174:30.8039
2138:25.9276
2102:24.0811
2066:21.9705
2030:17.2486
1994:12.1844
1958:10.8691
1922:6.62251
1886:2.67793
1010:. (Here
931:and the
3910:Tacnode
3895:Crunode
3123:1722410
3083:1726059
2784:Richter
2658:trinity
2581:0, 1728
2495:) and (
2491:â see (
2471:at the
1065:by the
1014:is the
995:of the
986:on the
889:of the
854:systole
609:(2,3,7)
76:compact
74:, is a
21:quadric
3890:Acnode
3814:Moduli
3269:
3236:
3138:
3121:
3111:
3089:
3081:
3071:
3028:
2893:
2883:
2595:, and
2583:, and
2531:, the
1028:modulo
1020:SL(2,
980:SL(2,
951:tiling
860:Tiling
502:where
281:. The
274:closed
221:freely
215:group
66:, the
23:, see
3267:S2CID
3234:S2CID
3087:S2CID
3051:arXiv
3026:S2CID
2984:(PDF)
2873:(PDF)
2430:SO(3)
1005:Î(7)\
893:(the
289:cusps
247:. By
83:genus
3900:Cusp
3285:link
3217:GAFA
3136:ISBN
3109:ISBN
3069:ISBN
2891:ISSN
2881:ISBN
2593:0, 1
2574:The
2510:The
2434:O(3)
2432:(or
1765:The
1355:cosh
1245:cosh
1096:sinh
1030:7.)
1012:Î(7)
591:2 â
284:open
3259:doi
3226:doi
3179:doi
3061:doi
3018:doi
2684:".
2475:in
2467:by
1284:sin
1181:sin
1136:csc
1018:of
999:by
810:is
625:i,j
511:/7)
319:on
223:on
153:of
149:on
130:).
94:168
81:of
62:In
3959::
3302:44
3300:,
3296:,
3281:}}
3277:{{
3265:,
3255:24
3253:,
3249:,
3232:.
3220:.
3200:24
3173:,
3151:.
3130:,
3126:.
3119:MR
3117:,
3107:,
3085:,
3079:MR
3077:,
3067:,
3059:,
3024:.
3014:14
3012:.
2967:).
2903:^
2889:.
2858:^
2825:^
2815:.
2773:^
2751:^
2564:24
2487:a
2393:6
2374:15
2357:6
2338:14
2321:8
2302:13
2285:6
2266:12
2249:8
2230:11
2213:8
2194:10
2177:6
2141:6
2105:8
2069:7
2033:7
1997:8
1961:6
1925:7
1889:8
1853:1
1850:0
1092:16
946:.
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935:.
804:Î(
748:Î(
617:Î(
601:Î(
432:Î(
389:0.
330::
263:.
231:â1
141:,
137:,
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3351:t
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3261::
3240:.
3228::
3222:3
3181::
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3063::
3053::
3038:.
3032:.
3020::
2954:.
2914:)
2910:(
2897:.
2852:.
2837:.
2819:.
2786:)
2782:(
2768:)
2764:(
2742:(
2597:â
2585:â
2562:M
2158:9
2122:8
2086:7
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2014:5
1978:4
1942:3
1906:2
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1750:.
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1729:(
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1669:(
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1498:{
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1471:S
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1378:+
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1371:(
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1297:(
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1197:7
1188:(
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1171:4
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1103:1
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1024:)
1022:Z
1007:H
992:H
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