Thursday June 5, 2008 5:20 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                                                                           Indra's Pearls

David Mumford et al. (2002). Indra's Pearls.

This book arrived yesterday. I have skimmed the entire book and now want to begin reading it more carefully and make some notes.

Chapter 1 The Language of Symmetry

• "This is a book about serious mathematics, but one which we hope will be enjoyed by as wide an audience as possible." [p. vii]

I like the phrase "serious mathematics". This is what I am trying to get back into.

• "Most mathematics is accessible, as it were, only by crawling through a long tunnel in which you laboriously build up your vocabulary and skills as you abstract your understanding of the world." [p. vii]

I do feel that it is a long tunnel. It is reassuring to realize that others have the same feeling. There is no shortcut.

• "How to understand and draw such limit sets is what this book is all about." [p. xix]

At the moment I dream that I will be able to understand and do this someday.

• "In a nutshell, Klein proposed viewing geometry as 'the study of the properties of a space which are invariant under a given group of transformations. To study geometry, he said, one needed not only objects (triangles, circles, icosahedra, or much wilder things like the fractal images in this book), but also movements." [p. 1]
  
  
• "In the classical Euclidean regime ... these movements had always been rigid motions ... Klein's radocal idea was that other movements, which might stretch or twist the objects quite drastically, could be thought of as gemometrical movements too." [p. 1 - 2]
  
  
• "... Galois saw that the solutions of a polynomial could be understood by defining their 'symmetries'; thus for example (+ square root of 2) and (- square root of 2) can be considered as symmetrical solutions of the equation (x squared) = 2." [p. 3]

I do not yet fully understand how Galois showed that there were no algebraic solutions for the general quintic equation. I know that it has to do with this idea of symmetries of the solutions, but I do not have the details clear - yet. This is one of my goals: to genuinely understand this idea. I must have another look at Mario Livio's "The Equation That Couldn't Be Solved" (2005).

• "Mathematicians, since Klein, have had at their disposal a more precise version [of symmetry]: symmetry is balance created by repetitions of many movements of the same kind, specifically, by all the movements in some particular group." [p. 4]
  
  
• "These two ideas, introducing the group idea into geometry and widening the class of movements to be studied, formed the background of Klein's own work." [p. 4]
  
  
• "Symmetry is created by repeating or iterating the same motion a number of times." [p. 5]
  
  
• One example of symmetry is called translational symmetry: the object is simply translated (i.e. shifted) to a new location. A second type of symmetry is called rotational symmetry: the object is rotated about a point. A third type of symmetry is called bilateral (i.e. reflection): the object is 'reflected' in a mirror.

There is no mention of a "glide-reflection". That makes sense to me since it is a combination of a reflection and a translation.

My first impression of this book is very positive. I find the text and the diagrams to be very clear.

Tags: mathematics, symmetry