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Group Theory in the Bedroom

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

Brian Hayes

Group Theory in the Garage

Not all household chores are as mathematically intractable as mattress flipping. In rotating the tires of a car, for example, it's easy to find a golden rule. One simple strategy says: Always rotate a quarter turn clockwise. In other words, move the right-front tire to the right-rear, move the right-rear tire to the left-rear, and so on. The analogous counterclockwise rule works just as well. In either case, when you repeat the process four times, the tires return to their original positions, each one having visited all four corners.

The cyclic 4-group...Click to Enlarge Image

Why is tire rotation so different from mattress flipping? It is governed by a different group. The quarter-turn operations are elements of a group known as the cyclic 4-group, which describes the symmetries of a square that rotates in a plane (but cannot be lifted out of the plane and flipped over). The fundamental symmetry operations are turns of 0 degrees (the identity element), 90 degrees, 180 degrees and 270 degrees. (Alternatively, the 270-degree turn could be described as a 90-degree rotation in the opposite direction.) The multiplication table for the cyclic 4-group is shown at left. The quarter-turn and three-quarter turn operations are golden-rule moves within this group: Repeatedly applying either of them cycles through all the orientations of the square.

Transformations induced by the two four-element groups...Click to Enlarge Image

The other group we have been discussing, the one associated with mattress flipping, is known as the Klein 4-group, after the German mathematician Felix Klein. This group describes the symmetries of a rectangular object rather than a square; moreover, the rectangle resides in three-dimensional space, so that it can be flipped over (or, equivalently, reflected in a mirror).

Comparing the multiplication tables for the two groups reveals an important similarity. Both tables are symmetrical with respect to the main diagonal (running from upper left to lower right). In other words, the symbol at row j and column k is invariably the same as the symbol at row k and column j. This implies that any two actions can be performed in either sequence with the same effect. For the mattress, a roll followed by a yaw is the same as a yaw followed by a roll. Groups with this property are said to be commutative, or Abelian, after the Norwegian mathematician Neils Henrik Abel.

It turns out that the cyclic 4-group and the Klein 4-group are the only groups with exactly four elements; there is just no other way to combine four operations and satisfy all the requirements of grouphood. But both of the four--element groups are embedded within a larger group, called S 4, which describes all possible permutations of four things. Taking group theory into the kitchen now, S 4is the group enumerating the ways to arrange a family of four at the breakfast table. The earliest riser can choose any of the four places; then the next person to get out of bed has three chairs available; the third person has two choices; and the last to arrive must take whatever's left. Thus the number of arrangements is 4 factorial (denoted 4!), equal to 4 x 3 x 2 x 1, or 24. The 24 elements of the group S 4are all the ways of reshuffling the seat assignments.

Patterns for rotating the tires of a car...Click to Enlarge Image

Among the 24 permutations, nine are derangements—they leave no person in the same chair. Except for the identity, all the elements of both the Klein 4-group and the cyclic 4-group are derangements. There are four more derangements in S 4, not members of either four-element group. These additional derangements provide further useful patterns for rotating tires—as shown in the illustration at left—but they still offer no help for the mattress problem.

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