Tuesday April 1, 2008 7:30 am Lethbridge, Alberta

This page last updated on: Sunday, June 1, 2008 9:00 AM

It is -8 C with a high forecast of +2 C. Sunrise 7:07 Sunset 20:02 Hours of daylight: 12:55

A. Morning Musings

There is a fog this morning. The first sip of coffee goes well with the fog.

We have a coffee set up with a fellow at the uni this morning. Other than that the day is free.

Learning Category	Planned Activities for Today	Time
Literature	Begin morning with a Rumi reading
Literature	Begin reading "Time Regained" by Marcel Proust	1 hr
Mathematics	Begin reading "Fearless Symmetry" by Ash & Gross	1 hr
Mathematics	Begin reading "Symmetry and the Monster by Ronan	1 hr

B. Actual Learning Activities

1:00 PM

I am on an emotional high. The recent reading the quantum book together with the books that I have on symmetry are simply exciting.

As a result I bought 3 more books this morning:

Symmetry and the Monster (2006) by Mark Ronan.
- I would not have bought this book but know that I know what the Monster is (from the du Sautoy book "Symmetry") I am interested in another description of the same topic.
Science 101 Physics (2007) by Barry Parker.
- This is a nice background book that gives me a better sense of the context for the quantum book.
The Book of Games (2008) by Jack Botermans
- This is a return to an interest I have always had in games of strategy that have been popular in different times and cultures.

Notes on Symmetry - 2

Dale Burnett

From my notes of Symmetry - 1:

du Sautoy also talks about always having a yellow notepad at his disposal. This reinforces my idea of a lined notepad of high quality paper and a good fountain pen. My notepads have a bright orange cover. My fountain pen is a Pelikan M250 Tradition with a fine nib.

1. Use my camera to take photos of objects that appear to possess symmetry. Begin to create a catalogue of such images.

2. Begin making notes using my fountain pen and Rhodia notepads.

a. Notes on Herstein textbook.

b. Notes on Goodwin textbook.

c. Notes on Ranucci & Teeters.

d. Notes on Ash & Gross.

3. Familiarize myself with the notational conventions of group theory and symmetry operations.

4. Read and make notes on du Sautoy's book on symmetry.

I have completed rereading the first chapter of "Fearless Symmetry" by Ash & Gross. Now for a few notes. Once again, this is a descriptive chapter and my notes will be on this web page rather than by using a pen & paper.

A. Ash & R. Gross. (2006). Fearless Symmetry.

Chapter 1 Representations

The book contains 23 short chapters.

"... the basic concept of representation ... is the key concept underlying the number-theoretic methods of Galois representations that are our goal." [p. 3]
"The distinction between truth and appearance, the thing-in-itself and its representation, is a keynote of philosophy." [p. 3]
"...representations ... explain one thing by means of another. The object we want to understand is the 'thing: the thing-in-itself, the source. The object that we know quite a bit about already, to which we compare the source via a representation, we call the standard object." [p. 5]
Definition: A set is a collection of things, which are called the elements of the set. [p. 6]
Definition: A one-to-one correspondence from a set A to a set B is a rule that associates to each element in A exactly one element in B, in such a way that each element of B gets used exactly once, and for exactly one element in A. [p. 6]
Definition: A function from a set A to a set B is a rule that assigns to each element in A an element of B. [p. 9]
Definition: A morphism is a function from A to B that captures at least part of the essential nature of the set A in its image in B. [p. 9]
"There are many kinds of functions, but the mose useful ones for us are the morphisms from a source to a well-understood standard target. We will call this a representation.
Definition: A representation is a morphism from a source object to a standard target object. [p. 10]

This chapter conveys a very important idea - that of representation as a particular kind of function that maps a set that we want to understand onto a set that we already understand fairly well.

I am going to see if I can create a chart to monitor my reading of books on symmetry. Each cell will correspond 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 that I have read and made notes for the first chapter from two of these books, I need to identify my next chapter. I also want to begin taking photos of symmetric patterns and I want to begin playing with cardboard cutouts. The online book "Algebra" by Goodman contains activities involving cardboard cutouts. I will not have easy access to the next three books in my table for awhile. The last book, like the first book in the table, is intended for the general reader and may provide a nice break from the more serious books.

I will try the first chapter of "Symmetry and the Monster".

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

Chapter 1. Theaetetus's Icosahedron

There are some connections between the Monster symmetry and both number theory and string theory, but these connections are not well understood at the moment.
The 5 Platonic solids: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons) , icosahedron (20 triangles).
The cube and the octahedron are called dual objects. Each can be inscribed within the other, implying that they have the same symmetries.
The dodecahedron and the icosahedron are also dual objects.

I am enjoying this approach of reading from a number of different books that all deal with the same topic. Each description adds something to the whole. This is so much more interesting that restricting oneself to one resource (e.g. a textbook).

What I am trying to do is learn more about symmetry. I am NOT trying to say that I finished a particular book. There is a hint of a suggestion here for education.

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. Bronowski, The Ascent of Man]" [p. 11]
"Galois's ideas for using symmetry were profound and far-reaching but none of this was fully understood at the time ..." [p. 22]

There is no math in this chapter, but the quote by Bronowski made the chapter worthwhile.

The chapter describes the historical context for Galois's death in a duel at age 20. Sometimes it is fun to simply read a math book for relaxation.

Tags: mathematics, symmetry