"Does not commute"

I sometimes talk about science and models of reality. The math may be as precise as you like, but if the application of it isn't right it doesn't help much.

One thing that surprises people from time to time is that the math that best describes the real world doesn't always have A+B=B+A. I like to remind them that putting on your socks and shoes isn't the same as putting on your shoes and socks--the order makes a little difference.

Typically they are unconvinced. So if they are rich enough to wave about two dollar bills, I suggest the standard example.

Lay two dollars bill in front of you face up, each with George's head oriented up as he sadly contemplates the state of the nation. You will apply the same kinds of rotations to both. On the left, rotate the bill 90 degrees clockwise. Then flip it over far end to near and near end to far. Take the other bill and flip it far end to near and near end to far. Then rotate it 90 degrees clockwise. Notice the difference.

What happens if you rotate by 180 degrees instead of 90?

In a closely related example, suppose you start at Indianapolis and travel 100 miles due south, ignoring roads and construction. Then travel 100 miles due east, then 100 miles due north, then 100 miles due west. As we all know, you are not back where you started from; you overshoot because the Earth is round.

In fact the effect is fairly common. Do A, then do B, then do the opposite of A, then do the opposite of B. It isn't always the same as doing nothing. Sometimes you are not back where you started. The difference between where you started and where you wind up tells you something about the kind of space you are in (which needn't be a space with the usual distance dimensions--it can be a space of momenta or something else).

If you have points in a plane represented by (x,y) you can rotate clockwise by θ about the origin with a simple matrix (quick reminder about matrix multiplication)

cos(θ)	-sin(θ)
sin(θ)	cos(θ)

Suppose θ is vanishingly small. Then cos(θ) ≈ 1 and sin(θ) ≈ θ . So you can write the tiny rotation as the identity plus the first order rotation plus higher order terms:

1	0
0	1

+

0	-θ
θ	0

+ higher order terms

So far this is nothing particularly startling. A small rotation mostly leaves the situation the same (that's the identity matrix with 1's in the diagonal), but there's a tiny set of first order changes and even tinier higher order. (If you think .001 is small, how about .001 squared: .000001?)

In 3D, with rotations about the x, y, and z axes, when you play the same game you have 3 tiny rotation matrices, which I'll call A, B, and C, with extremely tiny angles a, b, and c. You start with the identity matrix (1's down the diagnonal) and

A=

0	-a	0
a	0	0
0	0	0

B=

0	0	b
0	0	0
-b	0	0

C=

0	0	0
0	0	-c
0	c	0

Notice that AB is not BA, and AB-BA is not a zero matrix.

In fact, if we just look at the unscaled base matrices, setting a=b=c=1, we see the following:

• AB-BA=-C
  
• BC-CB=-A
  
• CA-AC=-B
  

It isn't as simple as 1+2=2+1, but the structure gives you some interesting symmetries you wouldn't have seen otherwise. Or to put it in layman's terms, that's kind of cool. And if you have a mathematical bone anywhere in your body you'll ask: "what happens if..." (if rotations are in 4 dimensions, if you have 4 similar cycling equations in ABCD, if AB-BA=-C+A, etc, etc).