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COMPUTING SCIENCE

Group Theory in the Bedroom

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

Pillow Talk

I'm not a mathematician, but I've been hanging around with some of them long enough to know how the game is played. Once you've solved a problem, the next step is to generalize it beyond all recognition. There's a nerdy joke with the punchline, "Consider a spherical cow"; in the same spirit I ask, "Consider a cubical mattress."

A cube has a much higher order of symmetry than the generic orthotope of an ordinary mattress. Any of six faces can be turned uppermost for sleeping on, and each face has four orientations, so there are 24 states in all. (The group is S 4, the same group we encountered at the breakfast table and in the garage.) Is there a golden rule for flipping an unlabeled cubical mattress—a single maneuver that can be repeated 24 times to cycle through all the configurations? The answer is no, but I'll leave the proof of that fact as an exercise.

There is a silver rule for cubical mattress flipping: As with an ordinary mattress, if we label the configurations, we can step through them one by one by counting. But there is a subtle difference. On the ordinary mattress, there are six essentially different ways to label the configurations—six ways to arrange the numbers 0 through 3— but it doesn't matter which one you choose. With any arrangement of the numbers, you can always go from one state to the next in a single flip. For the cube, there are many more labelings (the exact number is 23!, or 25,852,016,738,884,976,640,000), and they are not all equal. When you tour all 24 states in numerical order, some labelings allow every transition to be accomplished by a simple quarter turn around the roll, the pitch or the yaw axis. Other labelings require more-complicated maneuvers—rotations around multiple axes (or, equivalently, around a diagonal). How many of the labelings fall into each category? Is there any simple rule that distinguishes the two classes?

If you have answered those questions and you still can't get to sleep, you are welcome to go on and consider a hyper-cubical mattress—one that takes the form of a cube in n-dimensional space, for some n greater than 3. The 4-D mattress has 24 square faces and 96 configurations.

As far as I know, neither Sealy nor Simmons nor anyone else is yet selling a four-dimensional mattress. As a matter of fact, the trend seems to be in the opposite direction. Mattress makers now promote the one-sided, "no-flip," mattress, whose only symmetry operation is a 180-degree turn around the yaw axis. This innovation finally gives us a golden rule for mattress flipping, but it solves the problem in a trivial, dull and unsatisfying way.

Still, there remain ways to increase the number of permutations, to provide opportunities for creative problem-solving, to make life more interesting. One could get married.