# Posts Tagged: symmetry

• stories

## What’s Pippa got to do with the Bourbaki wedding?

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Last time we’ve seen that on June 3rd 1939, the very day of the Bourbaki wedding, Malraux’ movie ‘L’espoir’ had its first (private) viewing, and we mused whether Weil’s wedding card was a coded invitation to that event. But, there’s another plausible explanation why the Bourbaki wedding might have been scheduled for June 3rd :… Read more »

• stories

## So, who did discover the Leech lattice?

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For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen. In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of… Read more »

• groups

## the monster graph and McKay’s observation

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While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry group $A_5$. The alternating group $A_5$ has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say $C$ and… Read more »

• stories

## ceci n’est pas un corps

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To Gavin Wraiht a mathematical phantom is a “nonexistent entity which ought to be there but apparently is not; but nevertheless obtrudes its effects so convincingly that one is forced to concede a broader notion of existence”. Mathematics’ history is filled with phantoms getting the kiss of life. Nobody will deny the ancient Greek were… Read more »

• groups

## the “uninteresting” case p=5

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I was hoping you would write a post on the ‘uninteresting case’ of p=5 in this context. Note that the truncated tetrahedron has (V,E,F)=(12,18,8) which is a triple that appears in the ternary (cyclic) geometry for the cube. This triple can be 4 hexagons and 4 triangles (the truncated tetrahedron) OR 4 pentagons and 4… Read more »

• geometry, groups, number theory

## the buckyball curve

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We are after the geometric trinity corresponding to the trinity of exceptional Galois groups The surfaces on the right have the corresponding group on the left as their group of automorphisms. But, there is a lot more group-theoretic info hidden in the geometry. Before we sketch the $L_2(11)$ case, let us recall the simpler… Read more »

• geometry, groups

## the buckyball symmetries

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The buckyball is without doubt the hottest mahematical object at the moment (at least in Europe). Recall that the buckyball (middle) is a mixed form of two Platonic solids the Icosahedron on the left and the Dodecahedron on the right. For those of you who don’t know anything about football, it is that other ball-game,… Read more »

• groups, math

## Arnold’s trinities version 2.0

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Arnold has written a follow-up to the paper mentioned last time called “Polymathematics : is mathematics a single science or a set of arts?” (or here for a (huge) PDF-conversion). On page 8 of that paper is a nice summary of his 25 trinities : I learned of this newer paper from a comment by… Read more »

• geometry, groups, math, number theory

## Arnold’s trinities

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Referring to the triple of exceptional Galois groups $L_2(5),L_2(7),L_2(11)$ and its connection to the Platonic solids I wrote : “It sure seems that surprises often come in triples…”. Briefly I considered replacing triples by trinities, but then, I didnt want to sound too mystic… David Corfield of the n-category cafe and a dialogue on… Read more »

• stories

## Finding Moonshine

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On friday, I did spot in my regular Antwerp-bookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me… To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more,… Read more »

• stories

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A first year-first semester course on group theory has its hilarious moments. Whereas they can relate the two other pure math courses (linear algebra and analysis) _somewhat_ to what they’ve learned before, with group theory they appear to enter an entirely new and strange world. So, it is best to give them concrete examples :… Read more »

• stories

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Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal. These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic… Read more »

• featured

## Mathieu’s blackjack (3)

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If you only tune in now, you might want to have a look at the definition of Mathieu’s blackjack and the first part of the proof of the Conway-Ryba winning strategy involving the Steiner system S(5,6,12) and the Mathieu sporadic group $M_{12}$. We’re trying to disprove the existence of misfits, that is, of non-hexad… Read more »

• featured

## The 15-puzzle groupoid (1)

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Before we go deeper into Conway’s M(13) puzzle, let us consider a more commonly known sliding puzzle: the 15-puzzle. A heated discussion went on a couple of years ago at sci-physics-research, starting with this message. Lubos Motl argued that group-theory is sufficient to analyze the problem and that there is no reason to resort to… Read more »

• stories

## The miracle of 163

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On page 227 of Symmetry and the Monster, Mark Ronan tells the story of Conway and Norton computing the number of independent _mini j-functions_ (McKay-Thompson series) arising from the Moonshine module. There are 194 distinct characters of the monster (btw. see the background picture for the first page of the character table as given in… Read more »

• featured

## NeverEndingBooks-groups

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Here a collection of pdf-files of NeverEndingBooks-posts on groups, in reverse chronological order.

• featured

## the taxicab curve

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(After-math of last week’s second year lecture on elliptic curves.) We all know the story of Ramanujan and the taxicab, immortalized by Hardy I remember once going to see him when he was lying ill at Putney. I had ridden in taxicab no. 1729 and remarked that the number seemed to me rather a dull… Read more »

• featured

## The cartographers’ groups

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Just as cartographers like Mercator drew maps of the then known world, we draw dessins d ‘enfants to depict the associated algebraic curve defined over $\overline{\mathbb{Q}}$. In order to see that such a dessin d’enfant determines a permutation representation of one of Grothendieck’s cartographic groups, $SL_2(\mathbb{Z}), \Gamma_0(2)$ or $\Gamma(2)$ we need to… Read more »

• featured

## Monsieur Mathieu

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Even a virtual course needs an opening line, so here it is : Take your favourite $SL_2(\mathbb{Z})$-representation Here is mine : the permutation presentation of the Mathieu group(s). Emile Leonard Mathieu is remembered especially for his discovery (in 1861 and 1873) of five sporadic simple groups named after him, the Mathieu groups $M_{11},M_{12},M_{22},M_{23}$… Read more »

• stories

## way too ambitious

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Student-evaluation sneak preview : I am friendly and extremely helpful but have a somewhat chaotic teaching style and am way too ambitious as regards content… I was about to deny vehemently all assertions (except for the chaotic bit) but may have to change my mind after reading this report on Mark Rowan’s book ‘Symmetry and… Read more »

• featured

## football representation theory

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Unless you never touched a football in your life (that’s a _soccer-ball_ for those of you with an edu account) you will know that the world championship in Germany starts tonight. In the wake of it, the field of ‘football-science’ is booming. The BBC runs its The Science of Football-site and did you know the… Read more »

• stories

## symmetry and the monster

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Mark Ronan has written a beautiful book intended for the general public on Symmetry and the Monster. The book’s main theme is the classification of the finite simple groups. It starts off with the introduction of groups by Galois, gives the classifivcation of the finite Lie groups, the Feit-Thompson theorem and the construction of several… Read more »

• stories

## Alain Connes on everything

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A few days ago, Ars Mathematica wrote : Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes‚Äô flavor is the most successful. Be that as it may, do not print this 106 page… Read more »

• featured

## a Da Vinci chess problem

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2005 was the year that the DaVinci code craze hit Belgium. (I started reading Dan Brown’s Digital Fortress and Angels and Demons a year before on the way back from a Warwick conference and when I read DVC a few months later it was an anti-climax…). Anyway, what better way to end 2005 than with… Read more »

• featured

## micro-sudoku

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One cannot fight fashion… Following ones own research interest is a pretty frustrating activity. Not only does it take forever to get a paper refereed but then you have to motivate why you do these things and what their relevance is to other subjects. On the other hand, following fashion seems to be motivation enough… Read more »