Mathematics Chronology

7:10 am Day 2 of my resolution to begin the day with some maths. I want to continue with the Goodman book on Algebra, which I enjoyed yesterday.

Algebra: Abstract and Concrete. Edition 2.5 (online) Frederick Goodman

Chapter 1. Algebraic Themes

There are 12 sections to this chapter. The main themes appear to be symmetry (first four sections), permutations, divisibility, integers, group, rings, and fields.

1.1 What is Symmetry?

See Mathematics 12 notebook.

1.2 Symmetries of the Rectangle and the Square

Goodman points out that one could also include the motion of "no motion" (i.e. leaving the card unchanged). He claims that while this is an arbitrary choice, there are good mathematical reasons for including it.

I assume these good reasons have something to do with the idea of zero (i.e. do nothing).

Goodman also points out that one might consider the notion that there are an infinite number of symmetries as one could continue endlessly rotating the object by multiples of the original symmetry.

I had not thought of that! But I agree that it makes sense to exclude them and focus on just those symmetries that uniquely move the object.

Goodman then adds that he is excluding potential reflection symmetries for the moment.

I had not thought of these either.

• "Here is an essential observation: If I leave the room and you perform two undetectable motions one after the other, I will not be able to detect the result. The result of two symmetries one after the other is also a symmetry." [p. 4]

Of course. But I had not realised this until Goodman mentioned it. It helps to have a good teacher, particularly at the early stages.

One can consider a sequence of symmetries as a form of "multiplication". Then the product of two symmetries x and y would be written as xy where this would mean to first do y and then x.

One might also consider the sequence to a form of "addition", where x + y would mean first do y and then x. The only reason I see for prefering multiplication as the metaphor rather than addition is that it is easier to write. I understand that it is simply a convention to say the xy means do y first. We could just as easily say xy means do x first. The issue is one of both reader and writer understanding which convention is being used. It also helps if there is a form of universal agreement as it makes the interpretation that much easier and automatic.

• "Let's label the three nontrivial rotations of the rectangular card by and let's call the nonmotion e. If you perform first , and then , the result must be one of or e (because these are all of the symmetries of the card)." [p. 4 - 5]

The next task is to complete a "multiplication table" of all possible pairs of motions:

	e
e	e
		e
			e
				e

Done.

Now do the same for the square card.

	e				a	b	c	d
e	e				a	b	c	d
				e	d	c	a	b
			e		b	a	d	c
		e			c	d	b	a
a	a	c	b	d	e
b	b	d	a	c		e
c	c	b	d	a			e
d	d	a	c	b				e