6:40 am

A Note to the Reader

• Goodman begins with an exerpt from "A River Runs Through It" by Norman Maclean, a book that I have, and have enjoyed reading. They made a movie based on the book starring Robert Redford, but I have yet to see it. The exerpt describes how one of the men had reasoned through why the fish were biting in one location but not in another just a few yards further downstream.
• Goodman then comments, "In mathematical practice the typical experience is to be faced by a problem whose solution is a mystery: The fish won't bite. Even if you have a toolbox full of methods and rules, the problem doesn't come labeled with the applicable method, and the rules don't seem to fit. There is no other way but to think things through for yourself." [p. xii]
  

• Goodman continues, "It's not at all easy to work things out for yourself, and it's not at all easy to explain clearly what you have worked out. ... You must have patience, or learn patience, and you must have time. You can't learn these things without getting frustrated, and you can't learn them in a hurry. If you can get someone else to explain how to do the problems, you will learn something, but not patience, an not persistence, and not vision. So rely on yourself as far as possible." [p. xii]

Chapter 1 Algebraic Themes

1.1 What is Symmetry?

• One way to conceptualize the idea of symmetry is to view a symmetry as "an undetectable motion". An object is then said to be symmetric if it has some symmetries.

1.2 Symmetries of the Rectangle and the Square

• By taking a rectangular card one can show that there are 3 rotational symmetries as well as 1 symmetry that corresponds to doing nothing (often called the identity symmetry).
• In order to compare various transformations of this card, it is important to adopt some notational conventions so one can clearly communicate what one is trying to do.
• The number of symmetries of a square turns out to be 8.
• Essential Observation: The result of two symmetries, one after the other, is also a symmetry.
• The idea of applying two symmetries (i.e. two undetectable motions) one after the other leads to the idea of a "multiplication" of symmetries, where xy means "first do y, then do x". Note that xy could just as easily mean "first do x, then do y". The important point is simply to be clear on the meaning, which is an arbitrary decision, but one that must be made.
• Now one can create a multiplication table of all possible symmetries of the object. Once again, it is essential to adopt a notational convention of how to represent each of the symmetries. One approach might be to use the letter s with a subscript going from 1 to n, to represent each symmetry. However Goodman uses a more sophisticated lettering system that has additional information embedded in the labels. Thus r, r-squared and r-cubed represent successive rotations in the plane and letters like a, b, c, ... represent rotations in a third dimension. One can then see additional patterns when one develops the complete multiplication table.

1.3 Multiplication Tables

• Once one has created a multiplication table for an object, one can then examine this table for patterns and regularities.
• One observation to make is whether the operation is commutative (i.e. is ab = ba ?).
• Another observation is to note whether every operation has an inverse (i.e. for each a, is there a b such that ab = e ? where e represents the identity or "do nothing" operation).