Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. r(z)=⇣ z and s(z)= . (7w 48) (3w 32) (5w +12) n z − − 2 3 3 I Elongated Pyramids (J7, J8, J9) Graphs I A4 = r, s s = r =(sr) =1 are the rigid rotations of the 4(835+872p2) w 4 (w 1)3 (11+8p2) w+1 − tetrahedron,with I φ(w)= 3 1 z (8+9p2) w+1 (8 5p2) w 1 ⌦ r(z)=⇣3 z ↵ and s(z)= − . I A (finite) graph is an ordered pair V , E consisting of vertices V and 2 z +1 −⇥ − ⇤ 2 3 4 I Diminished Trapezohedron⇥ ⇤ ⇥ ⇤ edges E. I S4 = r, s s = r =(sr) =1 are the rigid rotations of the 4(835 872p2) w 4 (w 1)3 (11 8p2) w +1 v = V =thenumberofvertices octahedron and the cube,with φ(w)= − − − | | I 3 e = E =thenumberofedges ⌦ ⇣4 + z ↵ 1 z (8 9p2) w +1 (8 + 5p2) w 1 r(z)= and s(z)= − . − ⇥ − ⇤ | | ⇣4 z 1+z f = F =thenumberoffaces − ⇥ ⇤ ⇥ ⇤ | | 2 3 5 I A5 = r, s s = r =(sr) =1 are the rigid rotations of the Hypermaps: Rotation Group Dn I A connected graph is a graph where, given any pair of vertices z1 and icosahedron and the dodecahedron,with 3 z ,onecantraverseapathofedgesfromonetotheother. 64 z3 z3 1 2 ⌦ 4 ↵ 4 ⇣5 + ⇣5 ⇣5 z ⇣5 + ⇣5 z β(z)= − : v =4+6, e =2 6, f =4 r(z)= − and s(z)= − . 3 I Two vertices z1 and z2 are adjacent if there is an edge connecting them. 4 4 − 3 · I Elongated Bipyramid ⇣5 + ⇣5 z + ⇣5 ⇣5 + ⇣5 z +1 8 z +1 A bipartite graph is a graph where the vertices V can be partitioned (J , J , J ) 14 15 16 4 into two disjoint sets B and W such that no two edges z1, z2 B w (32 w 5) 2 Platonic, Archimedean, Catalan, and Johnson Solids Rotation Group S4: Rhombicuboctahedron / Deltoidal I φ(w)= − (respectively, z1, z2 W )areadjacent. (80 w +1)3 2 Icositetrahedron I A planar graph is a graph that can be drawn such that the edges only I A Platonic solid is a regular, convex polyhedron. They are named after intersect at the vertices. Plato (424 BC – 348 BC). Aside from the regular polygons, there are five I Orthobicupola such solids. (J27, J28, J30) The Sphere as The Extended Complex Plane I An Archimedean solid is a convex polyhedron that has a similar I φ(w)= arrangement of nonintersecting regular convex polygons of two or more (w 2.411)4 (w +0.138)4 − di↵erent types arranged in the same way about each vertex with all sides w (w +3.086)3 (w 0.441)4 − the same length. Discovered by Johannes Kepler (1571 – 1630) in 1620, Through stereographic projection, we they are named after Archimedes (287 BC – 212 BC). Aside from the I Elongated Gyrobicupola can establish a bijection between the prisms and antiprisms, there are thirteen such solids. 2 (J36, J37, J39) 4 unit sphere S ( )andtheextended 3 2 R I (w +(0.739+0.223 i)w (0.754+0.034 i) w+(0.002 0.020 i) A Catalan solid is a dual polyhedron to an Archimedean solid. They are − − I φ(w)= 3 4 1 w w (0.041 0.283 i) w 2 (2.004+0.189 i) w+(0.005+0.324 i) complex plane P (C)=C . named after Eug`ene Catalan (1814 – 1894) who discovered them in 1865. ⇥ − − − ⇤ [{1} ⇥ ⇤ ⇥ ⇤ Aside from the bipyramids and trapezohedra, there are thirteen such solids. Theorem (E. Baeza, L. Baeza, C. Lawrence, and C. Wang, I A Johnson solid is a convex polyhedron with regular polygons as faces Define stereographic projection as that map from the unit sphere to 2014) but which is not Platonic or Archimedean. They are named after Norman the complex plane. Johnson (1930) who discovered them in 1966. There are ninety-two There are explicit Bely˘ımaps β for S 2(R) ⇠ P1(C) Johnson solids. ! I Wheel/Pyramids (J1, J2) I Elongated Pyramids (J7, J8, J9) 2x 2y x2+y 2 1 u + iv − u, v, w = x2+y 2+1, x2+y 2+1, x2+y 2+1 x + iy= I Cupola (J3, J4, J5) I Diminished Trapezohedron 7! 1 w Proposition (Wushi Goldring, 2012) Rotation Group A5: Truncated Icosahedron / Pentakis − which have rotation group Aut(β) Zn;and ⇣ ⌘ Dodecahedron ' Let %(w)bearationalfunction.Thecomposition% β is also a Bely˘ı Bely˘ımaps ◦ map for every Bely˘ımap β : X P1(C)ifandonlyif% is a Bely˘ımap I Elongated Orthobicupola (J35, J38) ! I Bipyramid (J12, J13) Let β : X 1( )beameromorphicfunctiononaRiemannsurfaceX . which sends the set 0, 1, to itself. I Elongated Gyrobicupola (J36, J37, P C I Elongated Bipyramid (J , J , J ) ! 1 1 14 15 16 J ) We say z X is a critical point if β0(z)=0,andw P (C)isa 39 2 2 I Gyroelongated Bipyramid (J17) critical value if w = β(z)forsomecriticalpointz X .Arational Proposition (Nicolas Magot and Alexander Zvonkin, 2000) I Dipole/Hosohedron 1 2 I Orthobicupola (J27, J28, J30) function β : X P (C)whichhasatmostthreecriticalvalues I Truncated Trapezohedron ! The following % are Bely˘ımaps which send the set 0, 1, to itself. I Gyrobicupola (J29, J31) 0, 1, is called a Bely˘ımap. I Bifrustum/Truncated Bipyramid 1 1 (w 1)2/(4 w)isarectification, which have rotation group Aut(β) D . n Dessin d’Enfant − − 3 ' 8(4 w 1) /(27 w)isatruncation, − Acknowledgements Fix a Bely˘ımap β : X P1(C). Denote the preimages >1/w is a birectification, > ! > 3 2 1 1 1 β 1 >(4 w) /(27 w )isabitruncation, We would like to thank the following people for their support: B = β− 0 W = β− 1 E = β− [0, 1] X P (C) %(w)=> { } { } ! > −2 3 2 2 >4(w w +1)/ 27 w (w 1) is a rhombitruncation, I The National Science Foundation (NSF) and Professor Steve Bell, <> − − 4 2 I Dr. Joel Spira, (w +1)/ 16 w (w 1) is a rhombification, black? white? ? ? − 5 3 > I Andris “Andy” Zoltners, ? ? edges? R? > 7496192 (w + ✓) verticesy verticesy y y > 3 is a snubification, > I Professor Edray Goins, and ⇢ ⇢⇢ >25 (3 + 8 ✓) w 88 w (57 ✓ +64) The bipartite graph ∆β = V , E with vertices V = B W and edges E > − I Dr. Kevin Mugo. [ where ✓2> (7/64) ✓ +1=0. is called Dessin d’Enfant.WeembedthegraphonX in 3-dimensions. :> − Department of Mathematics, 150 North University Street, West Lafayette, IN 47907-2067 http://www.math.purdue.edu/.