Sunday April 20, 2008 6:30 am Lismore NSW

I have now completing reading chapter 4 of "Symmetry". 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                                                                           Now to make a few notes on chapter 4 of "Symmetry".

M. du Sautoy. (2008). Symmetry.

Chapter 4 November: Tribal Gathering

• "Science is about discovery, but it is also about communication. An idea can hardly be said to exist if you do not awaken that same idea in someone else. That is why conferences are an important part of giving life to an idea." [p. 88]

Yes. And this also suggests that notation is critically important as one tries to explore the implications of various ideas.

• "One of the key issues is determining whether one has built a truly new group of symmetries or has merely found a previously known group in a new guise. ... This is one of the difficulties facing anyone who tackles this subject: two objects can look very different yet have the same underlying symmetries." [p. 93]
  
  
• "... the nature of the underlying symmetry of an object starts to reveal itself only when you begin to explore what happens when you combine symmetrical moves." [p. 94]

This is a critical point. One way of viewing symmtry is to consider symmetry as a result of some form of transformation of an object or figure. That is, it is not so much a property of an object as a stationary object but a property that describes what happens when one moves the object according to some well defined procedure.

• "As I add more and more triangles, the number of possible symmetrical objects goes up. There are 15 objects made from the symmetries of four triangles, and 67 made from the symmetries of five triangles. But it is a complete mystery how many symmetrical objects you can make from the symmetries of ten triangles. I am trying to find a way to predict how the numbers of objects grow as I add more triangles. ... The challenge of trying to find patterns in the way these numbers evolve as I add more triangles goes to the heart of what it means to me to be a mathematician." [p. 95]

Perfect. A mathematician is one who explores unknown territory, looking for patterns and attempting to find formulas and rules that apply to new situations. The study of calculus is primarily about refining the ideas of continuity, limit and the nature of number. The study of symmetry is about looking for regularities in geometric figures and quickly also becomes an exploration of the nature of integers - particularly prime numbers.

• "The human mind is desparate to find patterns. ... But sometimes things can be random and without patterns." [p. 97]
  
  
• "There are always several different ways to make sense of any finite sequence orf numbers which can then be used to generate the next numbers in the sequence." [p. 97]
  
  
• "I'd been exploring things called zeta functions. The German mathematician Bernhard Riemann first introduced this object into mathematics in 1859 as a powerful way to try to find order in the chaos of prime numbers." [p. 99 - 100]

This is another critical point. Zeta functions are an important topic when studying both prime numbers as well as symmetry. I must at least get on top of the general idea of this function.

• "Riemann discovered in his thirties that the zeta function gave him a powerful new way of looking at the primes. It acted like a bilingual dictionary, enabling properties of numbers to be translated into geometry. Ever since riemann's discovery, variations on the zeta function have been exploited as a way of revealing patterns and structure in mathematical settings where at first there just seemed to be mess and disorder." [p. 105]

Time for a quick summary of the key ideas so far. symmetry is the study of properties of objects under transformations. prime numbers are important in the study of symmetry the zeta function is important as well I need to set up an index page for accessing my various "notes".

Tags: mathematics, symmetry