Saturday October 28, 2006 6:45 am Lethbridge Sunrise 8:13 Sunset 18:17 Hours of daylight: 10:04

A. Morning Musings

6:45 am It is +10 C at the moment with a forecast high of +17 C.

7:53 am

Sunrise at 7:53 am

From rear window	South patio
Both images taken at 12:35 PM

B. Plan

Immediate
Health	Walk & exercise	1 hr
Model Trains	Begin laying track for Black Diamond	2 hr
Mathematics	Read "Fearless Symmetry" chap 7: quadratic reciprocity	1 hr
	Continue reading "The Equation that Couldn't be Solved"	1 hr
Literature	Complete reading "The Mystery of the Yellow Room" by Gaston Leroux	1 hr
Later
Chores	Investigate water softeners for home
Technology	Read manual for cell phone
	Make notes for chap. 4 of "Switching to the Mac"
	Begin reading "iPhoto"
	digital photography - learn about using the various manual settings
Literature	Read "The Art of Living" by Epictetus
	Read "The Song of Roland"
Mathematics	Larson "Calculus"
	Read "The Computational Beauty of Nature" Chap 3
	Gardner "The Colossal Book of Short Puzzles"
History	Continue reading "Citizens"
	Watson "Ideas"
Model Trains	Build oil refinery diorama: add ground cover
	Assemble second oil platform kit
	Assembly of CN 5930, an SD40-2 with a NAFTA logo
Puzzles	The Orange Puzzle Cube: puzzle #9

C. Actual/Notes

	Mathematics 15 October 28
Mathematics Chronology
9:20 am I skim-read chapter 7 Quadratic Reciprocity from "Fearless Symmetry" this morning and then switched to "The Equation that Couldn't be Solved" where I read the best biography of Galois that I have seen (chapter 5 The Romantic Mathematician [p. 112 - 157]. Meticulous detail. I followed this with a brief glance at the next chapter, called simply "Groups". It appears to be a clear description of the essential nature of group theory, showing the connections to permutations and to symmetry. Both this chapter and the Quadratic Reciprocity chapter involve some actual mathematics and thus require (me) to slow down and make sure that I understand what is being said. The "Fearless Symmetry" book uses the mathematical concept of Representations as its cornerstone idea. "The Equation that Couldn't be Solved" focuses on the general problem of the solution of algebraic equations as the historical genesis of group theory. Both books emphasize the importance of the young French mathematician Everiste Galois, of group theory, and of symmetry. A "representation" in mathematics is simply a function or mapping of a set of elements to another standard set that reveals some essential feature of the original set. I think I now have that idea.   Fearless Symmetry Chapter 6 Equations and Varieties [p. 49 - 66] (Review notes) Z is the set of integers. A Z-equation is an equality of polynomials with integer coefficients. A variety S defined by a Z-equation is the function that assigns to the number system A the set of solutions S(A). It is important to realize that a variety refers to a particular set of Z-equations and then restricts the solution to a set of elements in a "number system" such as Z, Q, R, C, . Fearless Symmetry Chapter 7 Quadratic Reciprocity [p. 67 - 83] "Quadratic reciprocity was the first of all reciprocity laws, and it is closely connected with the theory of Galois representations." [p. 67] "Quadratic reciprocity is part of 'classical' number theory going back to the eighteenth century. It inspired the many other reciprocity laws that eventually developed into a large part of number theory." [p. 67] At this point I do not know what a reciprocity law is, let alone quadratic reciprocity. Nor Galois representations. These are all new terms to me. I am continuing to learn... "Since the time of ... Descartes, mathematicians have realized that the degree of a polynomial is a good indication of how complicated it is." [p. 67] This chapter focuses on quadratic equations (i.e. of degree 2) and on only one aspect of them - quadratic reciprocity. Consider the quadratic equation (or equivalently, ) and let a = 1. Then S(Z) = {1, -1} Note the notation. S(Z) may be read as "the solution set over the field Z of integers for the equation and let a = 1, is {1, -1}. Now suppose a = -1. Then S(Z) is the empty set, so is S(Q) and S(R). But S(C) is {i, -i}. The real fun, and interest, begins when we consider the solution sets using various groups with a prime generator. (i.e. ).

 

D. Reflection