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Group Theory in the Bedroom

An insomniac's guide to the curious mathematics of mattress flipping

Brian Hayes

Mattress Multiplication

If you are inclined to undertake a search for such a magic combination, by all means flip away; but it's less strenuous to bring some mathematics to bear on the problem. Particularly helpful is the branch of mathematics known as group theory, which is the traditional tool for studies of symmetry.

In general, a group is a set of objects together with an operation for combining them. For example, one can have a group in which the objects are numbers and the combining operation is addition or multiplication. In this case, however, the "objects"—the elements of the group—are themselves operations. They are the various ways of flipping a mattress. The rule for combining the elements is simply to perform one operation after another.

Not just any set of operations can qualify as a group; they have to meet four criteria. First, among the operations there must be an identity element—an operation that leaves the system unchanged. For mattress flipping, the identity operation is obvious: Just do nothing.

Second, every element of the group must have an inverse, an "undo" action that returns the system to its former state. Again this requirement is easily met for mattress flipping: Each of the three basic rotations is its own inverse. If you flip the mattress a half-turn around the roll axis, and then do exactly the same thing again, you come back to where you started. The same is true of half turns around the pitch and yaw axes. (Even more obviously, the do-nothing identity element is also its own inverse.)

The third criterion for grouphood is that the operations obey an associative law, so that (f g) h means the same as f (g h), where f, g and h are any operations in the group. For mattress flipping this is true but not very informative, so I'll say nothing more about it.

The final requirement is closure, which says the set of operations is in some sense complete. More formally, if f and g are any elements of the group, then the combination of f followed by g must also be an operation in the group. The implications of closure are made clear by drawing up a "multiplication table" for the group, as in figure 2. The table gives the result of all possible pairwise combinations of the four operations I, R, P and Y (which stand for the identity operation and for 180-degree rotations around the roll, pitch and yaw axes). The crucial point is that every such combination is equivalent to one of the fundamental operations. For example, a roll turn followed by a yaw turn has exactly the same effect as a pitch turn. This fact is what dooms the search for a golden rule. The table shows that any combination of two basic operations can be replaced by a single operation. Since we already know that none of the single operations yields a cycle through all four states of the mattress, it's clear that no pair of operations can be composed to make a golden rule.

What about longer sequences of symmetry moves? They too are ruled out. Someone might come forward to claim that a complicated, n-step sequence of roll, pitch and yaw turns has the desired effect. But by consulting the multiplication table, we can replace the first two of these actions with a single turn, creating an equivalent procedure with n-1 steps. Continuing in the same way, we eventually reduce the entire sequence to a single symmetry operation, which cannot be a golden rule.

But wait! Maybe there's some tricky maneuver that involves motions other than the symmetry operations, as with the quarter turns in the diagram in figure 1. The trouble is, any move that qualifies as a mattress flip has to begin and end with the mattress in one of the four canonical positions on the bed frame. In between, you are welcome to twirl it over your head on one finger while riding a unicycle, but when you put it down again, only the net effect of your gyrations can be observed. And the multiplication table for the group says that all your manipulations, no matter how acrobatic, can be replaced by a single symmetry operation—-either I, R, P or Y.

There is no golden rule hidden under the mattress.

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