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Pdf of the Talk I Construction of highly symmetric Riemann surfaces, related manifolds and some exceptional objects, I Iain Aitchison 6 8 12 5 7 11 2 3 5 4 6 10 Hiroshima University March 2018 (IRA: Hiroshima 03-2018) Highly symmetric objects I 1 / 80 Highly symmetric objects I Motivation and context evolution Immediate: Talk given last year at Hiroshima (originally Caltech 2010) 20 years ago: Thurston's highly symmetric 8-component link Klein's quartic: sculpture at MSRI Berkeley Today: (p = 5) Historical context { Antiquity, 19th-20th Centuries Exceptional objects { Last letter of Galois, p = 5; 7; 11 Arnold Trinities Transcription complexes Construction of some special objects (p = 5) Tomorrow: (p = 7; 11) Construction of some special objects (p = 7; 11) WTC = Weeks-Thurston-Christie 8-component link New picture for Mathieu group M24, Steiner S(5; 8; 24) (Golay code, octads) New Arnold Trinity involving Thurston's 8-component link (IRA: Hiroshima 03-2018) Highly symmetric objects I 2 / 80 The Eightfold Way: Klein's Quartic at MSRI, Berkeley Eightfold Way sculpture at MSRI, Berkeley, commissioned by Thurston (IRA: Hiroshima 03-2018) Highly symmetric objects I 3 / 80 Excerpt of Thurston: `How to see 3-manifolds' View of link in S3 View from cusp in universal cover H3 Hyperbolic link complement: extraordinary 336-fold symmetry Weeks - Thurston -Joe Christie, seminar talk MSRI 1998 Immersed 24-punctured totally geodesic Klein quartic Complementary regions 28 regular ideal tetrahedra (IRA: Hiroshima 03-2018) Highly symmetric objects I 4 / 80 Old Babylonian Antiquity { 3,4,5 3 + 4 + 5 = 12 32 + 42 = 52 3:4:5 = 60 Tables of integers, inverses Right-triangles: 1 1 x2 − 1 1 1 x2 + 1 (x − ) = (x + ) = 2 x 2x 2 x 2x Rescale, give Pythagorean triples when x rational: x2 − 1; 2x; x2 + 1 (IRA: Hiroshima 03-2018) Highly symmetric objects I 5 / 80 Plimpton 322 circa 1800 BC Old Babylonian, Larsa 564 ** Translated about 70 yearsJ. P. Britton agoet al. 4 Appendix**B: ListPhoto of t ofhe tablet Pythagorean triples? Plimpton Cuneiform 322, Rare Book and Manuscript Library, Columbia University, Courtesy(egJane Siegel, (3,4,5),photo C. Proust (5,12,13),(see Figs.14, 15). ... Integers forming sides of a right-angled triangle.) Fig. 14 Plimpton 322, obverse (IRA: Hiroshima 03-2018) Highly symmetric objects I 6 / 80 Fig. 15 Plimpton 322, reverse 123 Ancient Greece Xenophanes c. 570-475 BC { all is opinion, no one can know truth Pythagoras c. 570-495 BC Heraclitus c. 535-475 BC { all existence is change Parmenides c. 515BC, pupil of Parmenides, { all change is illusion ZENO, pupil of Xenophanes, c. 490-430 BC { paradoxes, space, time, motion Leucippus, contemporary of Zeno { atomism, mentor of DEMOCRITUS c. 460-370 BC { atomic theory of the universe Socrates c. 470-399 BC PLATO c. 427-347 BC { unchanging mathematical reality, space, ... Fire:Tetrahedron Air:Octahedron Earth:Cube Water:Icosahedron Universe:Dodecahedron (IRA: Hiroshima 03-2018) Highly symmetric objects I 7 / 80 Consequence 2: Democritus and atomism (Aristotle) `... the differences in atoms are the causes of other things. They (Democritus et al) hold that these differences are three { shape, arrangement and position; being, they say, differs only in `rhythm, touching and turning', of which `rhythm' is shape, `touching' is arrangement and `turning' is position' rhythm touching turning shape arrangement position A − N AN − NA Z − N (Aristotle) `..these atoms move in the infinite void,separated one from the other and differing in shapes, sizes, position and arrangement; overtaking each other they collide, and some are shaken away in any chance direction, while others, becoming intertwined one with another according to the congruity of their shapes, sizes, positions and arrangements, stay together and so effect the coming into being of compound bodies.' (IRA: Hiroshima 03-2018) Highly symmetric objects I 8 / 80 Consequence 1: Plato's concept of space Fundamental polyhedra : PlatonicTHE PLATONIC solids (dodecahedron SOLIDS problematic) 3 SPACE: What is enclosed by a surface configuration of special 3-4-5-gons Fundamental triangles 3 4 5 60-60-60 30-60-90 45-45-90 36-54-90 54-54-72 (IRA: Hiroshima 03-2018) Highly symmetric objects I 9 / 80 Thales {3,4,5, Pythagorean triples, rational points z = p + iq; z2 = (p2 − q2) + i:2pq zz p2−q2 2pq (p+iq)2 Q = p2+q2 + i p2+q2 = (p+iq)(p−iq) Q 1+iq=p 1+iσ = (1−iq=p) = 1−iσ ; σ = q=p R z a P R = iσ ! 1+iσ = Q S aa 1−iσ Stereographic projection from S from Cayley transform, Half plane to Unit Disc Pythagorean Triples (p2 − q2; 2pq; p2 + q2) $ Rational points on circle (IRA: Hiroshima 03-2018) Highly symmetric objects I 10 / 80 Cayley transform= Octahedron rotational symmetry StereographicCayley transform projection is and conformal, the octahedron/cube sends circles on S2 to circles, lines Rotated octahedron Projection to C Cayley transform: order-3 rotation of an octahedron around an axis through opposite triangle faces. (IRA: Hiroshima 03-2018) Highly symmetric objects I 11 / 80 (IRA-VAC-31-29-11-13) Star of Ishtar, Daughter of Sin 14 / 62 Modular Diagram for SL2(Z) Modular diagram in upper-half-space and unit disc Rational numbers on the real line (as boundary of the upper-half-plane) and as rational points on the circle under the Cayley transform. (IRA-VAC-31-29-11-13) Star of Ishtar, Daughter of Sin 10 / 62 (IRA: Hiroshima 03-2018) Highly symmetric objects I 12 / 80 Elliptic curves as Euclidean triangles Torus from parallelogram; Parallelogram gives two Euclidean triangles Always possible: any Euclidean torus from triangle, angles ≤ π=2 A A D B C B C Choice of two triangles Unit base, apex in fundamental region (IRA: Hiroshima 03-2018) Highly symmetric objects I 13 / 80 4, 6 : Moduli space as Euclidean triangles Parallelograms give Euclidean triangles defining moduli space Rectangle 1/h I _ 1 1 I S Equilateral 1 Square i i a E a h a Rectangle 1 a Isosceles Singular points S, E correspond to Euclidean tori arising from the tessellation of the plane by SQUARES or regular HEXAGONS (honeycomb lattice). (IRA: Hiroshima 03-2018) Highly symmetric objects I 14 / 80 4, 6: Escher's Heaven and Hell, Circle Limit IV Hyperbolic plane tessellated by hexagons 4 at each vertex; dually, by squares 6 at each vertex. Remark. (A, 1988): All possible decompositions of all 3-manifolds as a union of generic 3-polyhedra arise from quotient surfaces of this pattern of hexagons. (Heegaard surfaces, handlebodies) (IRA: Hiroshima 03-2018) Highly symmetric objects I 15 / 80 Symmetry : Legacy of Galois Non-solvability by radicals for general quintics etc simple groups { A5 simple, series An < Sn; n > 4 simple simple groups { L2(p) := PSL2(Z=pZ) simple, p prime p > 11: L2(p) has no non-trivial permutation representation on fewer than p + 1 Exceptional: p = 5; 7; 11, related to Platonic solids (IRA: Hiroshima 03-2018) Highly symmetric objects I 16 / 80 Last letter of Galois Ainsi pour le cas de p = 5; 7; 11, l'equation modulaire s'abaisse au degre p. En toute rigueur, cette reduction n'est pas possible dans les cas plus eleves. `No subgroup F such that (set-theoretically) L2(p) = F × Cp' 1 L2(p) acts via M¨obiustransformations on PFp = Fp [ f1g p = 5; 7; 11, Galois uses involutions ∗ π5 = (0; 1)(1; 4)(2; 3) ∗ π7 = (0; 1)(1; 3)(2; 6)(4; 5) ∗ π11 = (0; 1)(1; 2)(3; 6)(9; 7)(5; 10)(4; 8) −i i L2(p) leaves invariant p involutions Π = fα πpα : 1 ≤ i ≤ pg. ∗ See later (IRA: Hiroshima 03-2018) Highly symmetric objects I 17 / 80 Arnold Trinities, Numerology, Mysticism Arnold (1976, Hilbert Problems) : find a common origin for the ubiquity of ADE occurrences in mathematics Later, TRINITIES. 3 4 5 Tetrahedronf , Cubef , Dodecahedronf : Face degrees Pythagorean Triple: right-triangle 2 4 8 Simply-laced Lie algebras: A2... D4... E8.. Division algebra dimensions: C; Q; Cay =2 1 2 4 Division algebra dimensions: R; C; Q +1 2 3 5 E8 singularity : Poincare homology sphere ×2 4 6 10 3D symmetric networks : Cubes, Diamond, K4 Laves +1 5 7 11 Galois : Exceptional actions of PSL2(p); p = 5; 7; 11 +1 6 8 12 Cubef , Cubev , Cubee Tetrahedrone , Cubev , Dodecahedronf (IRA: Hiroshima 03-2018) Highly symmetric objects I 18 / 80 Arnol'd Trinities: Cube (f ; v; e) = (6; 8; 12) ! (5; 7; 11) 5 = 6 − 1; 7 = 8 − 1; 11 = 12 = 1 Galois: exceptional finite groups for primes 5; 7; 11: ∼ ∼ L2(5) = A5 ∼ A4 × C5; L2(7) = L3(2) ∼ S4 × C7; L2(11) ∼ A5 × C11 Amazing fact: Platonic groups! TETRAHEDRON (group A4) CUBE/octahedron (group S4) { four body diagonals DODECAHEDRON/icosahedron (group A5) { five inscribed tetrahedra L2(5) ' A5 = icosahedron/dodecahedron automorphism group L2(7) = Klein quartic automorphism group Is L2(11) = nice surface??? automorphism group? A surface genus 70 (Martin-Singerman) A surface genus 26 (A-) =? a - presumably less studied! - surface studied by Klein (IRA: Hiroshima 03-2018) Highly symmetric objects I 19 / 80 Exceptional groups and symmetric surfaces Three `exceptional' Galois-groups correspond to the three Platonic groups in SO(3) Three Platonic groups in SO(3) lift to binary groups in SU(2) (spin) (Subgroups of SU(2) correspond to representations of E8 via the McKay correspondence) Three Platonic groups correspond to the three exceptional Lie algebras E6; E7; E8 Arnol'd: `I have heard from John MacKay that the 27 straight lines on a cubical surface, the 28 bitangents of a quartic plane curve and the 120 tritangent planes of a canonic sextic curve of genus 4, form a trinity parallel to E6; E7 and E8.' (IRA: Hiroshima 03-2018) Highly symmetric objects I 20 / 80 Root systems, semi simple Lie algebras A2, B2, G2, Weyl ..
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