Group theory in the bedroom (OR: As the mattress turns…)

Brian Hayes‘s essay (in American Scientist) called “Group Theory in the Bedroom” applies advanced mathematics to what happens in the bedroom. The bedroom is the room with the bed in it. The essay begins:

Having run out of sheep the other night, I found myself counting the ways to flip a mattress. Earlier that day I had flipped the very mattress on which I was not sleeping, and the chore had left a residue of puzzled discontent. If you’re going to bother at all with such a fussbudget bit of housekeeping, it seems like you ought to do it right, rotating the mattress to a different position each time, so as to pound down the lumps and fill in the sags on all the various surfaces. The trouble is, in the long interval between flips I always forget which way I flipped it last time. Lying awake that night, I was turning the problem over in my head, searching for a golden rule of mattress flipping….

The search for a mattress-flipping algorithm leads to some diverting mathematics, not just in the bedroom but also in the garage and at the breakfast table. Furthermore, although I can offer no golden rule for mattress flipping, I do have some practical advice….

Particularly helpful is the branch of mathematics known as group theory, which is the traditional tool for studies of symmetry.…