Non-Commutativity and Associativity

In my last post, I introduced the ideas of commutativity (order doesn't matter) and associativity (grouping doesn't matter). In this post I want to explore a structure that's associative, but not commutative.

Messing With Triangles: Ultra-Basic Group Theory

I'm going to start with an equilateral triangle:

Triangles are pretty symmetric; rotating it 120 degrees doesn't change it:

flipping it over also leaves the triangle unchanged:

All in all, there are 5 different things I can do to a triangle and still have it appear the same. I can rotate (clockwise or counterclockwise), or I can flip it over one of three axes:

I've labeled the corners to tell the triangles apart. Also included is what mathematicians call the Identity Element (top left corner): it's what you get when you do nothing.

In order to facilitate discussion, I think each of the possible 6 actions should have names. For the purposes of this post, I'm going to call the counterclockwise rotation R1 and the clockwise rotations R2. I've labeled each of the flip actions F1, F2 and F3. In keeping with mathematical tradition, I've labeled the identity I:

At this point, I would recommend cutting your own triangle out of paper (or better yet, cardboard) and playing with it. Notice what happens when you do two actions in a row. For example, if you do F1 followed by R1, then you end up in the same place as just doing F2. If you do F2 followed by F3 you end up at R2. F2 followed by F2 again is I. To make things more readable, I'm going to use use the notation x*y to mean x followed by y. So:

"R1 followed by F1 is R2"

can be written as:

R1 * F1 = R2

and 

"F1 followed by F1 is I"

is simply

F1 * F1 = I

This will making writing and (after some practice) reading about the different triangle operations much easier. I suggest pausing here to play with your newly cut out triangle. If you haven't bothered to make your own triangle (come on, it's not that much work), or if you just want to check what you're doing, you can play with this interactagraph I made:

At this point, you might begin to notice some patterns. For example:

• I * I = I.
• You can always get back to I with just one step.
• A rotation followed by another rotation is always a rotation (or I).
• A flip followed by a rotation (or visa versa) is always a flip.
• A flip followed by the same flip is I.
• A rotation followed by a different rotation is I.

It might be helpful at this point to collect all of our observations into a sort of multiplication table.

*	I	R1	R2	F1	F2	F3
I	I	R1	R2	F1	F2	F3
R1	R1	R2	I	F2	F3	F1
R2	R2	I	R1	F3	F1	F2
F1	F1	F3	F2	I	R2	R1
F2	F2	F1	F3	R1	I	R2
F3	F3	F2	F1	R2	R1	I

Reading this is just like reading a times table. If I want to know what R2 * F2 is, I just find where the R2 row meets the F2 column:

*	I	R1	R2	F1	F2	F3
I	I	R1	R2	F1	F2	F3
R1	R1	R2	I	F2	F3	F1
R2	R2	I	R1	F3	F1	F2
F1	F1	F3	F2	I	R2	R1
F2	F2	F1	F3	R1	I	R2
F3	F3	F2	F1	R2	R1	I

So R2 * F2 = F1.

Associativity and Non-Commutativity

With our multiplication table in place, we can now begin to look for patterns. It's clear that this structure is associative. For example:

R2 * (F1 * F2) = R2 * R2 = R1

and

(R2 * F1) * F2 = F3 * F2 = R1

More abstractly, 

(x * y) * z = x * (y * z)

for all x, y and z. But the interesting thing is that this structure is non-commutative; when evaluating equations, the order in which you evaluate matters. For example:

F1 * R1 = F3

but

R1 * F1 = F2

So the structure is non-commutative. (An important observation is that some equations are commutative: R1 * R2 = R2 * R1, but you'll remember from my last post that every pair of elements has to commute in order for an operation to be considered commutative.)

Conclusion

In my last post we saw structures that were commutative and associative (like addition), and non-commutative and non-associative (like division). In this post we looked at something non-commutative and associative. Up next, a post about something commutative and non-associative.

Follow-Up Questions:

• Consider just R1, R2 and I (so, ignore flips). Is this structure commutative? Is it still associative?
• Study the rotations and flips of a square. Is this structure commutative? Associative?
• Tetrahedrons also have four corners. Is the set of possible transformations the same as the squares? (i.e. is the tetrahedron multiplication table the same as the square multiplication table?)

Further Reading, (from easiest to hardest):

• Group Theory in the Bedroom, and Other Mathematical Diversions by Brian Hayes (Chapter 12)
• Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability by Peter Pesic (Chapter 8)
• The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry by Mario Livio (Chapter 6)