Sunday May 25, 2008 5:30 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 make notes for chapters 11 - 13 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 ) Chapters 7 - 10 ( May 24 )

Chapter 11 Pandora's Box

• "In mathematics, as in other creative activities, you can get completely stuck, unable to move towards a proof of your conjecture, and unable to disprove it by finding a counter-example." [p. 127]
  
  
• "He [Janko] took an imaginary symmetry atom having the awkward cross-section, and threw all his efforts into obtaining a contradiction to show it couldn't exist. This involved figuring out numbers associated with the imaginary object, and showing they led to a contradiction. If there was no mistake in the calculation, then there was no symmetry atom having the awkward cross-section. ... But every time this happened he found an error in his work and the contradiction melted away. Could this really be because there really was something out there?" [p. 127]
  
  
• "Was it reasonable that there might be a symmetry atom not in the table? The short answer was yes, absolutely, because five exceptions were discovered in the mid-nineteenth century. But these five all had a very remarkable property, and no one seriously supposed there was anything else quite like them. ... This suggested that any further exceptions, if they existed, were probably of quite a different nature." [p. 128]

Having just reread the first few pages of this chapter, and copied out three of the important sentences, I now feel that I have a clear understanding of what was being attempted, although the actual mathematics is something I am a far way from even sensing, let alone understanding. This process of rereading , copying, and perhaps even more important, doing, is essential to finally understanding. And, as has been commented earlier, there is a difference between understanding and creating.

• "In the mid-nineteenth century, after Galois's work was published, mathematicians gradually got intrigued by groups of permutations, and developed the concept of 'transitivity': a group of permutations on a set of objects was called transitive if it could send any object to any other. ... Transitivity is a common occurrence, but there are higher levels of transitivity that are far less common." [p. 128]

Here is another new idea: n-fold transitivity. I think I now have a sense of this idea, although I am not sure what I could do with it at the moment.

• "If a group of permutations can send any pair of objects to any other pair, then it is called 2-fold transitive. ... With the square, for instance, a pair of vertices joined by an edge is quite different from a pair not joined by an edge. You can't send one pair to the other, so although the symmetry group of the square is transitive on its set of four vertices, it is not 2-fold transitive." [p. 128]
  
  
• "Of course, if you take the group of all permutations, or all even permutations, on a set of objects, then 2-fold transitivity is no problem,but those groups are too large and inclusive. They mix everything around too much and don't preserve an interesting pattern. Are there smaller groups that are 2-fold transitive? ... what about 3-fold transitivity, 4-fold transitivity, and so on?" [p. 129]
  
  
• "Mutiple transitivity is rare, and the higher the level of transitivity, the more likely it is that the group will include all permutations, or at least all even permutations. By the time you reach 6-fold transitivity nothing else is possible.This fact has been proved using the list of all symmetry atoms but as a method of proof this is not very elegant, and it only works after verifying that the list is complete. Getting everything on the list and proving that it is complete was an immense collaborative effort." [p. 130]

Both du Sautoy's book and this book attempt to give a sense of this huge collaborative effort. That is the underlying idea of both books. They do not show much of the actual mathematics as it is too difficult but they both give the reader a sense that the overall effort involved a large number of people over a period of about 100 years.

• "Emile Mathieu ... discovered two groups of permutations that were 5-fold transitive, one permuting 12 symbols, the other permuting 24. They are now called M12 and M24." [p. 131]

Now the fun begins! I tried googling "Ernst Witt 1935" to see if I could find a paper that showed his M24 symmetry atom. Unsuccessful. But I found an article that looked interesting, although not related to my original search: Geometry and Education in the Internet Age by Ulrich H. Kortenkamp and Jürgen Richter-Gebert. This mentioned a software program called Cinderella's Cafe. I googled that and found this web site: http://cinderella.de/tiki-index.php This looks like an amazing software package and the price is quite reasonable. Aaaagh! I just realized that Appendix 2 provides a description of the Witt design! On the other hand, if I hadn't tried to find this using google, I would not have discovered Cinderella's Cafe! But ... back to making notes on symmetry.

• "The way to establish the existence of a symmetry atom, known only through its cross-sections, is first to work out its character table." [p. 134]

Another new idea: character table. And I am not yet comfortable with the idea of a cross-section. I have googled "Character table" and located a number of web sites and pdf files that use this concept. I have now seen some of the actual mathematical articles related to this general topic of symmetry atoms. I have a very long way to go. ...

Chapter 12 The Leetch Lattice

• This chapter describes the general problem of how to pack spheres into boxes and then generalizes the idea to more than three dimensions.

Chapter 13 Fischer's Monsters

• This chapter describes Bernd Fischer's work and the discovery of more very complex symmetry atoms.

I am beginning to suspect that the remainder of this book can be read more as a novel than as a book that describes mathematics. The mathematics becomes too complex, but the idea of finding symmetry atoms and proving that you have eventually found ALL of them is the core message. I would still like to understand much more than I do at the moment about Group Theory and how this relates to symmetry. There are still three more chapters to read. I am not sure if I will need to make more notes. We shall see. I have completed reading this book. I have now read two books that provide a good sense of what has happened in the last century with respect to group theory and symmetry. Now to continue my personal journey along a path that has already been cleared.

Tags: mathematics, symmetry