Saturday May 24, 2008 6:30 am Lethbridge, Alberta

Date Mathematics History 500 BCE Pythagorus   399 BCE Thaetetus classifies the 5 regular Platonic solids in 3 dimensions: tetrahedron, cube, octahedron, dodecahedron, icosahedron.   1048 - 1131 Omar Khayyam finds geometric method for solving cubic equations.   1200 Leonardo Fibonacci wtote the first original book on mathematics published in Europe. It introduced Hindu-Arabic numerals and place-value notation.   1439   Gutenberg invents the printing press 1452 - 1519   Leonardo da Vinci 1492   Columbus discovers America early 1500s del Ferro, Tartaglia, Cardano, Ferrari solve cubic & quartic equations   1564 - 1642 Galileo   1642 - 1727 Isaac Newton   1775 - 1783   American War of Independence 1777 - 1855 Carl Friedrich Gauss   1789 - 1799   French Revolution 1802 - 1829 Niels Henrik Abel proves that no formula exists for equations of degree 5.   1832 Evariste Galois dies at age 20.   1842 - 1899 Sophus Lie: Norwegian group theorist       The goal today will be to reread the first 6 chapters of Ronan's book "Symmetry and the Monster" and then continue the book.   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23                             Symmetry                                                                           Fearless Symmetry                                                                           Algebra                                                                           Abstract Algebra                                                                           Creating Escher-type Drawings                                                                           Handbook of Regular Patterns                                                                           Symmetry & the Monster                                                                           The Celtic Design Book                                                                           Groups & Symmetry                                                                           Groups: A Path to Geometry                                                                           A Transition to Advanced Mathematics                                                                           Modern Abstract Algebra

Mark Ronan. (2006). Symmetry and the Monster.

Here are links to my previous notes for this book: Chapters 1 - 3 ( April 22 ) Chapters 4 - 6 ( May 6 )

Chapter 7 Going Finite

• There is a description of modular arithmetic in the first part of this chapter.
  
  
• "Burnside continued turning out excellent results, and in 1904 he published an important theorem about 'simple' groups - the atoms of symmetry. This theorem showed that if a 'simple' group is not prime cyclic, then its size must be divisible by at least three different prime numbers. ... If a group is not prime cyclic, and its size is divisible by only two different prime numbers, then it cannot be 'simple'; it must be composite." [p. 85 - 86]
  
  
• "Mathematics makes progress by proving theorems, but it also advances by developing new methods to prove them." [p. 86]

Chapter 8 After the War

• This chapter describes the formation of the Bourbaki group of French mathematicians.
  
  
• "He [Michio Suzuki] had been working on symmetry atoms having a certain type of cross-section ... and discovered a whole new family of them." [p. 95]
  
  
• "After these new discoveries some experts guessed there were no more families of finite symmetry atoms to be found, at least not with infinitely many members, but it was not clear why this should be the case. If there were other families, mathematicians had to find them - and if there were not then they needed a proof of that fact. Such a proof did gradually emerge, but only in a long series of very technical papers by many different authors. In the process it turned out that although there were no other families, there were some unexpected exceptions. Finding all these exceptions and showing there were no more was a challenge, and it is from this great work that the Monster finally emerged." [p. 95 -96]

I keep bouncing between two levels. At one moment I think I understand the idea of symmetry atom and the next moment I am not sure. Worse, I have no real idea what is meant by a family of atoms. Googling "symmetry atoms" yielded this URL for the first site: http://www.nature.com/nature/journal/v451/n7179/full/451629a.html The web site is a book review in Nature magazine of Marcus du Sautoy's book "Symmetry" (the other book I just finished reading). This time the words work. "If the group of symmetries could be deconstructed into cyclic groups, then the solutions could be expressed in terms of roots. Some groups do not admit any deconstruction — they are 'atoms of symmetry' ... Atoms of symmetry are the basic building-blocks for all finite groups of symmetry ... Most symmetry atoms fit into a 'periodic table' where they belong in one of several families whose members enjoy similar properties. There are 26 exceptions. The largest of these is the 'Monster', a vast group of symmetries requiring at least 196,883 dimensions in which to operate. It exhibits numerical patterns similar to those obtained in an important branch of number theory, a connection dubbed 'Moonshine' by John Conway, who was one of the first to investigate it and marvel at its surprising magic. I am not yet sure what properties would be used to form the families, but this is a good beginning. Seeing this reference to du Sautoy is giving me the little nudge I need to send an email to du Sautoy asking about a reference to a group theory primer [p. 263 of "Symmetry"]. One can only try. du Sautoy gives his personal web URL: http://www.maths.ox.ac.uk/~dusautoy in the book and that provides me with an email address. Done.

Chapter 9 The Man From Uccle

• "Jacques Tits invented mathematical buildings, although he didn't call them that at first." [p. 98]
  
  
• "Buildings are mult-crystals ... A multi-crystal lives in a collection of intersecting universes ... " [p. 99]
  
  
• "A building, or multi-crystal, built out of hexagons is a network of edges satisfying two conditions: there can be no circuit of fewer than six edges; and any two edges must lie on at least one common hexagonal circuit." [p. 100]

I tried typing Jacques Tits into google and found this web site: http://en.wikipedia.org/wiki/Jacques_Tits The mathematics was too dense for me on a quick skim, but I noticed a link to Bruhat-Tits buildings http://en.wikipedia.org/wiki/Bruhat%E2%80%93Tits_building This was also too difficult, but I then tried a link to Coxeter groups http://en.wikipedia.org/wiki/Coxeter_group This looks a little more manageable, but not a lot. I seem to be in an infinite regress ... I bought a book on Coxeter about a year ago and have just pulled it from my shelf: The King of Infinite Space - Donald Coxeter, the Man Who Saved Geometry. (2006) by Siobhan Roberts. But I now have a totally new vocabulary and set of ideas: crystals, multi-crystals, buildings, ... Wow!

• There is an excellent description on pages 101 - 106 that describes and gives examples od such crystals and multicrystals and the types of symmetries that they possess. Fascinating!
  
  
• "This is how mathematics is done. People sit and talk, perhaps with a chalkboard at hand, and as they talk they clarify their own ideas." [p. 108]

This web site provides a modern example of this approach. Two (or more) individuals could make a web site similar to this where each person provides meta-comments (in brown!) as well as mathematics (in green) and they then carry on a "conversation" as they iterate their web sites. Add to this video-conferencing using iChat or Skype ... This is not to suggest that this would replace face-to-face, but it might mean that such meetings would be even more productive.

Chapter 10 The Big Theorem

• "... a theorem in the mathematics of symmetry (that branch of mathematics called group theory). It makes a deceptively simple claim: apart from the prime cyclic groups, the size of every symmetry atom must be even. A symmetry atom, remember, is a group of symmetries that cannot be doconstructed into anything simpler, and there is another way of stating this theorem: if a group of symmetries has odd size, then it can be deconstructed (and a complete deconstruction will yield a collection of prime cyclic groups)." [p. 114]
  
  
• "This theorem is vitally important ... and it set in motion a long sequence of results that led to the discovery and classification of all finite symmetry atoms." [p. 114]
  
  
• "If a symmetry atom has even size, then a theorem of Cauchy ... guarantees that it contains a symmetry of order 2.This means a symmetry that when done twice leaves everything as it was. For example, a mirror symmetry has order 2. ... Most of the symmetries will move the mirror to another mirror, but some will stabilize it. The sub-group of symmetries stabilizing the mirror I shall call a cross-section [usually called an 'involution centralizer']." [p. 118]
  
  
• "A symmetry atom cannot have a single cross-section, because it can operate as a group of symmetries on itself, moving a cross-section into many different positions. This yields lots of cross-sections having the same shape, and from these cross-sections you can reconstruct the symmetry atom." [p. 118]
  
  
• "Here is the idea. Take the symmetry atoms you already know about and look at their cross-sections. In each case show that there are no other symmetry atoms having such cross-sections: either you prove that, or you find new symmetry atoms. ... If you find any new ones, then you consider them as possible cross-sections in something larger. This is precisely how the Monster was discovered. ... When this process eventually draws to a close, you have a list of all symmetry atoms." [p. 118 - 119]

I love this paragraph that begins "Here is the idea." We need more of this! Let's see the big picture and then we can begin to see how the details fit together.

• "... meanwhile at the University of Chicago they decided to arrange a special year on finite mathematics." [p. 123]

Another great idea! Bring together a number of key people and have them interact regularly for an extended period of time.

Tags: mathematics, symmetry