Recently, I read a piece by David Singmaster about Rubik’s Cubes. He claims that conjugation and commutators are extremely helpful in solving the cube. He gives a wonderfully commonsense example of the power of conjugation. To clean dirty clothes in the bedroom, you carry the clothes to the laundry room, wash the clothes there, and carry them back to the bedroom. The moral of this example is: to solve a problem at an arbitrary place, you “carry it” to a place where you can solve it, solve the problem there, and then “carry” the solution back.
A more mathematical example of this heuristic is the formula for rotating about an arbitrary point $\mathbf{x}$ in the plane by an angle $\theta$. One translates the point to the origin, rotates about the origin, and undoes the translation. As a formula, we might write: $ T^{-1}_{\mathbf{x}} \circ R_\theta \circ T_{\mathbf{x}}. $
Singmaster doesn’t give a commonsense example of the commutator heuristic. I spent a fair bit of time pondering and asking around while at CMESG and don’t have a great example yet. What follows below are various calculations and ideas that came to mind while looking for one.
If we have two invertible operations $X$ and $Y$ then their commutator is \[ [X,Y] = XYX^{-1}Y^{-1}. \] If $X$ and $Y$ commute then this expression will cancel to the identity. If the operators are not commutative then you will get something much stranger. You will get a group element which expresses the degree to which the operations do not commute.
A standard example of this is rotations. Pick a non-symmetric object like a potato. Suppose that $X$ and $Y$ are the rotations by $90^\circ$ about the $X$ and $Y$ axes. If you apply the commutator of $X$ and $Y$ to the potato then it will not return to its initial position.
You don’t even need a potato to do this; you can also do this with your hand. Hold your left hand up with the palm facing away from your chest. Perform a series of $90^\circ$ rotations: down, right, up, left. If you do the sequence correctly, your fingers with be pointing away from you and your palm will be pointing towards the left.
You can, of course, do this sort of rotation calculation with quaternions: \[ ij i^{-1} j^{-1} = (-k)(-i)(-j) = (-k)(-k) = k^2 = -1. \]
A somewhat non-sensical example comes from the usual socks-and-shoes example of inverses in groups. The inverse of $XY$ is not $X^{-1}Y^{-1}$, as one might expect, but rather $Y^{-1}X^{-1}$. To illustrate this, people talk about putting on socks $X$ and shoes $Y$. To undo the operation of putting on your socks and then your shoes, you must take off your shoes and then your socks. This “explains” the reversal of the order. You undo the latter operations first to permit you to undo the earlier operation.
If we look at this example from the vantage point of commutators, we get complete gibberish. What’s the commutator of putting on socks and shoes?
If the operations don’t commute then it seems we get a strange moment in the middle. We return to our initial shoeless and sockless state but we’ve been through something. And this makes sense. The socks and shoes operations don’t commute. The operation $XY$ is a normal everyday thing. Put on your socks, put on your shoes, and go for a walk. If we try even perform $YX$ then we get nonsense. If you find yourself putting shoes on bare feet, and then trying to put on socks over top of the shoes, then you’ve done too much math and should seek professional help.
To stay with the theme of socks for a moment, consider the operation of putting a sock on your right foot $R$ and your left foot $L$. The operations $RL$ and $LR$ are both reasonable and result in the same state: you’re wearing a pair of socks like a normal person. In this case the commutator is pretty reasonable.
You’ve been through a moment of hesitation about your wardrobe but haven’t attempted anything crazy. The operations commute and so the commutator feels simple.
In summary, these examples suggest that if a commutator is “straightforward” then the operations commute. If the commutator involves some kind of strange or impossible state then the operations don’t commute.
A commutator measures how much two operations commute. The commutator is a group-element-valued measure of commutation. It is going to be a be bit non-intuitive. As a concept, or heuristic, it might be far from our ordinary commonsense view of the world.
It turns out that some common life trajectories, or romantic plots, come up as commutators. For example, consider the operation of marrying Xena and dating Yoric.
A common life trajectory involves getting married and buying a house.
In the socks-and-shoes examples, we can make up concrete groups which model the situations. It’s not clear what these latter romantic examples represent, or really mean, because the ambient group is less clear. Is it the group of invertible lifestyle changes? Is any experience in life really reversible?
Published: Jun 2, 2026 @ 15:51.
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