Monday April 21, 2008 7:00 am Lismore NSW

I have read chapters 5, 6 & 7 over yesterday and today. Now to make a few notes. Here is my chart of symmetry readings: Each cell will corresponds to a chapter. Yellow indicates the number of chapters in the book, green indicates that I have read and made notes on the chapter.   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

M. du Sautoy. (2008). Symmetry.

Chapter 5 December: Connections

• Another topic that is relevant to the study of symmetry is that of elliptical curves.

At the moment I have no idea what these are. This book is an excellent catalyst for generating an interest in the topic of symmetry but, at least so far, it is short on providing details on the exact mathematics that is required.

• The remainder of the chapter is a historical description of the mathematical battles between two Italian mathematicians (Tartaglia and Cardona) in the mid-sixteenth century. The competition involved trying to solve cubic equations. I have read other accounts of this and was able to skim through it fairly quickly.
  

The important point is simply to realize that the origin of the development of ideas for exploring symmetry was in an area that appears at first sight to have little to do with symmetry.

Chapter 6 January: Impossibilities

• "Each group of symmetries has a zeta function attached to it. This zeta function can be described by a formula built from polynomial expressions. If the group of symmetries is not too complicated I can calculate the formula for the zeta function of that particular group." [p. 141]

I still don't know what a zeta function looks like, but it seems to have something to do with a group of symmetries. I do have a sense of what a group of symmetries looks like but am not yet familiar with some of the important properties of such a group.

• Much of the remainder of this chapter is a historical description of the Norwegian mathematician, Neils Abel, who was the first to show that an equation of degree 5 did not always have a formula for determining its roots.
  
  
• "The history of mathematics shows how different cultures wrestled with the discovery of new types of number." [p. 149]
  
  
• Here is a list of such numbers:
  • Whole numbers (the positive integers)
  • Negative numbers
  • Rational numbers (numbers that can be expressed as a fraction)
  • Irrational numbers (numbers that cannot be expressed as a fraction. e.g. the square root of 2)
  • Transendental numbers (numbers that are not rational nor irrational. e.g. p)
  • Imaginary numbers (numbers of the form a + bi where i is the square root of -1)

Chapter 7 February: Revolution

• Much of this chapter is a historical description of the young French mathematician Everiste Galois, considered the father of group theory.
  
  
• "The French capital is blessed with some fantastic examples of symmetry in architecture ... ". [p. 175]

This gives me an idea for tomorrow. I could take the train to the Sydney CBD and see how many pictures I can take of buildings that exhibit some form of symmetry. This makes much more sense that reading the next few chapters of the Du Sautoy book as I can do this at any time whereas I rarely have an opportunity to look for examples of architectural symmetry in Sydney.

Tags: mathematics, symmetry