Tuesday April 22, 2008 6:00 am Sydney NSW

I want to make a few notes with a historical framework. I need a table that gives me a few important touchstones for both mathematics as well as other historical events. 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.   early 1500s del Ferro, Tartaglia, Cardano, Ferrari solve cubic & quartic equations   1777 - 1855 Carl Friedrich Gauss   1789 -   French Revolution 1802 - 1829 Niels Henrik Abel proves that no formula exists for equations of degree 5.   1832 Evariste Galois dies at age 20.       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

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

Prologue

• "Most of the basic building blocks for symmetry come in one of several infinite sub-families. These sub-families combine together in larger families, but then there are the exceptions: 26 of them that don't fit into any of these families, the largest being the Monster." [p. 3]
  
  
• "Finding these infinite sub-families and then finding the exceptions is a story that takes us from France in 1830 to the 30 years following the Second World War. Showing that they form a complete list takes us right up to the present day. And finding the underlying connections between the Monster and other branches of mathematics and physics takes us into the future." [p. 3]

Chapter 1 Thaeatetus's Icosahedron

• "A cube has a great many different symmetries - how many? The total number is 48, and they form what we call the symmetry group of the cube.Those that can be done by rotations - there are 24 of them (a cube has 6 faces, any one of which can be placed on the bottom. This face can then be rotated into 4 different positions, and 6x4=24) - form a sub-group. I shall call it the rotation group of the cube." [p. 8]
  
  
• "Consider the cube and the octahedron. They are deeply interconnected. Where the cube has 6 faces, the octahedron has 6 vertices, and where the cube has 8 vertices the octahedron has 8 faces.They both have 12 edges. ... Each one can be inscribed in the other ... We say the cube and the octahedron are dual to one another. This duality between the cube and the octahedron means that a symmetry of one is also a symmetry of the other - they have the same group of symmetries. As concrete objects they are different, but at the abstract level of symmetry they are the same." [p. 8 - 9]
  
  
• "In a similar way the dodecahedron and the icosahedron are dual to one another." [p. 9]
  
  
• "Galois was the first mathematician to use symmetry in solving a deep problem, and framing a new branch of mathematics." [p. 10]
  

Chapter 2 Galois: Death of a Genius

• "It is important that students bring a certain ragamuffin, barefoot, irreverence to their studies; they are not here to worship what is known, but to question it." (J. Brownoski, The Ascent of Man) [p. 11]

A gorgeous quote!

• "Galois's ideas for using symmetry were profound and far-reaching, but none of this was fully understood at the time ..." [p. 22]

I have enjoyed this morning's session. I like the idea of creating a historical table to provide a framework for understanding the history of mathematics. I am now working through 2 books and am at about the same historical time (the death of Galois). Both books now appear to focus on the basic ideas of group theory.

Chapter 3 Irrational Solutions

• Equations that cannot be reduced into a series of factors are called irreducible. That is, the roots of the equation are not rational numbers. [p. 27]
  
  
• "Galois knew that the solutions to an irreducible equation must be irrational, and the number of solutions equals the degree of the equation." [p. 32]
  
  
• "Irrational solutions only appear in multiples - and in each multiple there is a symmetry between them. Galois's genius was to analyse this symmetry ... " [p. 32]
  
  
• "Interchanging several things at once is called a permutation." [p. 32]
  
  
• "What Galois examined was a system of permutations having the property that one followed by another always yields a third permutation in the same system. He called such a system a group. Groups arise naturally when you permute several parts of a fixed pattern. If two permutations both preserve the pattern, then so does the permutation obtained from one followed by the other." [p. 33]
  
  
• "Galois's group of permutations for a given equation allowed him to ignore the technical details of how the solutions could be expressed." [p. 33]
  
  
• "For a given equation, Galois grouped all the allowable permutations together. This is now called the Galois group of the equation. It has become a central part of mathematics, going beyond the solution of algebraic equations to play a vital role in modern number theory." [p. 34]
  
  
• "In Galois's work a vital component was the idea of deconstructing a group into simpler groups." [p. 34]
  
  
• "Take an operation - such as a rotation or a permutation - and keep doing it until everything is back where it started. The number of times you must do the operation to achieve this is called its order, so, for example, a mirror symmetry has order 2, and a rotation by 90 degrees has order 4. A group generated by a single operation is called cyclic, and its size is the order of the operation you started with. A cyclic group of size 2, for instance, is generated by an operation of order 2. The nature of the operation is not important, and an operation of order 2 could take many forms: a mirror symmetry, a rotation by 180 degrees, a permutation switching a pair of objects and so on." [p. 34]
  
  
• "Cyclic groups are basic, but the prime cyclic ones are the most basic of all. For each prime number there is exactly one cyclic group of that size. Many groups can be deconstructed into prime cyclic groups - but not all. The distinction between those that can and those that cannot is critical in Galois's work." [p. 35]
  
  
• "... if you can do a permutation using an even number of transpositions, then you cannot do it using an odd number, and vice versa." [p. 37]

I am missing an understanding of the significance of even permutations and am not sure why such a decomposition that contains such a sub-group cannot be decomposed into prime cyclic sub-groups. I am not even sure I have expressed this properly! I need to find another source for a description of the significance of even permutations. Meanwhile I will continue reading ...

Tags: mathematics, symmetry