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Monsters and Moonshine lieven le bruyn 2012 universiteit antwerpen neverendingbooks.org CONTENTS 1. Monsters :::::::::::::::::::::::::::::::::::: 4 1.1 The Scottish solids..............................4 1.2 The Scottish solids hoax...........................4 1.3 Conway’s M13 game.............................8 1.4 The 15-puzzle groupoid (1/2).........................9 1.5 The 15-puzzle groupoid (2/2)......................... 11 1.6 Conway’s M13-groupoid (1/2)........................ 14 1.7 Mathieu’s blackjack (1/3)........................... 16 1.8 Mathieu’s blackjack (2/3)........................... 18 1.9 Mathieu’s blackjack (3/3)........................... 20 1.10 Conway’s M13 groupoid (2/2)........................ 22 1.11 Olivier Messiaen and Mathieu 12 ....................... 24 1.12 Galois’ last letter............................... 26 1.13 Arnold’s trinities (1/2)............................ 28 1.14 The buckyball symmetries.......................... 32 1.15 Arnold’s trinities (2/2)............................ 37 1.16 Klein’s dessins d’enfants and the buckyball................. 39 1.17 The buckyball curve.............................. 42 1.18 The ”uninteresting” case p = 5 ........................ 45 1.19 Tetra lattices.................................. 46 1.20 Who discovered the Leech lattice (1/2).................... 47 1.21 Who discovered the Leech lattice (2/2).................... 50 1.22 Sporadic simple games............................ 53 1.23 Monstrous frustrations............................ 57 1.24 What does the monster see?.......................... 57 2. Moonshine ::::::::::::::::::::::::::::::::::: 63 2.1 The secret of 163 ............................... 63 2.2 The Dedekind tessellation........................... 65 Contents 3 2.3 Dedekind or Klein?.............................. 67 2.4 Monsieur Mathieu............................... 69 2.5 The best rejected research proposal, ever................... 72 2.6 The cartographer’s groups (1/2)........................ 75 2.7 The cartographer’s groups (2/2)........................ 79 2.8 Permutation representations of monodromy groups............. 81 2.9 Modular quilts and cuboid tree diagrams................... 84 2.10 Hyperbolic Mathieu polygons........................ 86 2.11 Farey codes.................................. 88 2.12 Generators of modular subgroups....................... 90 2.13 Iguanodon series of simple groups...................... 92 2.14 The iguanodon disected............................ 93 2.15 More iguanodons via kfarey.sage....................... 95 2.16 Farey symbols of sporadic groups...................... 97 2.17 The McKay-Thompson series......................... 99 2.18 Monstrous Easter Egg Race.......................... 101 2.19 The secret revealed.............................. 102 2.20 The monster graph and McKay’s observation................ 103 2.21 Conway’s big picture............................. 106 2.22 Looking for the moonshine picture...................... 108 2.23 E(8) from moonshine groups......................... 111 2.24 Hexagonal moonshine............................ 113 series 1 MONSTERS 1.1 The Scottish solids John McKay pointed me to a few interesting links on ’Platonic’ solids and monstrous moon- shine. If you thought that the ancient Greek discovered the five Platonic solids, think again! They may have been the first to give a correct proof of the classification but the regular solids were already known in 2000BC as some neolithic stone artifacts discovered in Scot- land show. These Scottish solids can be visited at the Ashmolean Museum in Oxford. McKay also points to the paper Polyhedra in physics, chemistry and geometry by Michael Atiyah and Paul Sutcliffe. He also found my posts on a talk I gave on monstrous moonshine for 2nd year students earlier this year and mentionted a few errors and updates. As these posts are on my old weblog I’ll repost and update them here soon. For now you can already hear and see a talk given by John McKay himself 196884=1+196883, a monstrous tale at the Fields Institute. 1.2 The Scottish solids hoax A truly good math-story gets spread rather than scrutinized. And a good story it was : more than a milenium before Plato, the Neolithic Scottish Math Society classified the five regular solids : tetrahedron, cube, octahedron, dodecahedron and icosahedron. And, we had solid evidence to support this claim : the NSMS mass-produced stone replicas of their finds and about 400 of them were excavated, most of them in Aberdeenshire. Six years ago, Michael Atiyah and Paul Sutcliffe arXived their paper Polyhedra in physics, chemistry and geometry, in which they wrote : ”Although they are termed Platonic solids there is convincing evidence that they were known to the Neolithic people of Scotland at least a thousand years before Plato, as demon- series 1. Monsters 5 strated by the stone models pictured in g. 1 which date from this period and are kept in the Ashmolean Museum in Oxford. ” Fig. 1 is the picture below, which has been copied in numerous blog-posts (including my own, see the previous section) and virtually every talk on regular polyhedra. From left to right, stone-ball models of the cube, tetrahedron, dodecahedron, icosahedron and octahedron, in which ’knobs’ correspond to ’faces’ of the regular polyhedron, as best seen in the central dodecahedral ball. But then ... where’s the icosahedron? The fourth ball sure looks like one but only because someone added ribbons, connecting the centers of the different knobs. If this ribbon-figure is an icosahedron, the ball itself should be another dodecahedron and the ribbons illustrate the fact that icosa- and dodeca-hedron are dual polyhedra. Similarly for the last ball, if the ribbon-figure is an octahedron, the ball itself should be another cube, having exactly 6 knobs. Who did adorn these artifacts with ribbons, thereby multiplying the number of ’found’ regular solids by two (the tetrahedron is self-dual)? The picture appears on page 98 of the book Sacred Geometry (first published in 1979) by Robert Lawlor. He attributes the NSMS-idea to the book Time Stands Still: New Light on Megalithic Science (also published in 1979) by Keith Critchlow. Lawlor writes ”The five regular polyhedra or Platonic solids were known and worked with well before Plato’s time. Keith Critchlow in his book Time Stands Still presents convincing evidence that they were known to the Neolithic peoples of Britain at least 1000 years before Plato. This is founded on the existence of a number of sphericalfstones kept in the Ashmolean Museum at Oxford. Of a size one can carry in the hand, these stones were carved into the precise geometric spherical versions of the cube, tetrahedron, octahedron, icosahedron and dodecahedron, as well as some additional compound and semi-regular solids, such as the cube-octahedron and the icosidodecahedron.” Critchlow says, ’What we have are objects clearly indicative of a degree of mathematical ability so far denied to Neolithic man by any archaeologist or mathematical historian’. He speculates on the possible relationship of these objects to the building of the great astro- nomical stone circles of the same epoch in Britain: ’The study of the heavens is, after all, a spherical activity, needing an understanding of spherical coordinates. If the Neolithic inhabitants of Scotland had constructed Maes Howe before the pyramids were built by the ancient Egyptians, why could they not be studying the laws of three-dimensional coordi- nates? Is it not more than a coincidence that Plato as well as Ptolemy, Kepler and Al-Kindi attributed cosmic significance to these figures?’ ” As Lawlor and Critchlow lean towards mysticism, their claims should not be taken for granted. So, let’s have a look at these famous stones kept in the Ashmolean Museum. The Ashmolean has a page dedicated to their Stone Balls, including the following picture (the Critchlow/Lawlor picture below, for comparison) series 1. Monsters 6 The Ashmolean stone balls are from left to right the artifacts with catalogue numbers : • Stone ball with 7 knobs from Marnoch, Banff (AN1927.2728) • Stone ball with 6 knobs and isosceles triangles between, from Fyvie, Aberdeenshire (AN1927.2731) • Stone ball with 6 knobs and isosceles triangles between, from near Aberdeen (AN1927.2730) • Stone ball with 4 knobs from Auchterless, Aberdeenshire (AN1927.2729) • Stone ball with 14 knobs from Aberdeen (AN1927.2727) Ashmolean’s AN 1927.2729 may very well be the tetrahedron and AN 1927.2727 may be used to forge the ’icosahedron’ (though it has 14 rather than 12 knobs), but the other stones sure look different. In particular, none of the Ashmolean stones has exactly 12 knobs in order to be a dodecahedron. Perhaps the Ashmolean has a larger collection of Scottish balls and today’s selection is different from the one in 1979? Well, if you have the patience to check all 9 pages of the Scottish Ball Catalogue by Dorothy Marshall (the reference-text when it comes to these balls) you will see that the Ashmolean has exactly those 5 balls and no others! The sad lesson to be learned is : whether the Critchlow/Lawlor balls are falsifications or fabrications, they most certainly are NOT the Ashmolean stone balls as they claim! Clearly this does not mean that no neolithic scott could have discovered some regular poly- hedra by accident. They made an enormous amount of these stone balls, with knobs ranging from 3 up to no less than 135! All I claim is that this ball-carving thing was more an artistic endeavor, rather than a mathematical one.
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