EXAMPLE 6 Group Theory in Your Room, or More Than Once Upon a Mattress
Have you ever turned a mattress? The purpose of this usually seasonal practice is to even out lumps and sags and make the mattress last longer and be more comfortable.
There are various ways to “turn” a mattress, but they all require that the mattress fit back on the bed. What are all those ways?
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For this example, make yourself a “mattress” from a (nonsquare) rectangular sheet of paper. (It’s a lot easier to flip a piece of paper than a real mattress!) Label the initial position of the “mattress": Write “UP” in the middle of the top surface of the paper and label the edges in clockwise order “TOP,” “RIGHT,” “BOTTOM,” and “LEFT.” Then turn the sheet over so that the top edge is again at the top. Label this surface of the paper “BOTTOM” in the middle and label the edges in clockwise order “TOP,” “RIGHT,” “BOTTOM,” and “LEFT.” Now turn the sheet over again to the initial position.
We’ll designate the turning over of the paper that you just did as follows:
- Flip (turn it over, left and right switch), denoted by , amounts to a 180° rotation around an axis running from bottom to top of the mattress.
- Rotation (up side stays up, top and bottom switch), denoted by , amounts to a 180° rotation of the mattress as seen from above, that is, around a vertical axis through the center of the mattress.
- Toss (turn it over and switch top and bottom), denoted by , corresponds to a 180° rotation of the mattress around a horizontal axis across the middle of the short side of the mattress.
How many different positions are there for the mattress? Satisfy yourself that there are really only four positions for the mattress that fit it back on the bed: the results of the turns , , and —together with the “un-turn,” which we denote by . These four turns together form a group {, , , }. The group operation is to perform one turn followed by another.
Performing one of the turns, followed by a subsequent one in the next season, actually gives the same result as a single turn; for example, doing followed by puts the mattress in the same position as doing just . This fact is the closure property of the group.
The identity element of the group is the “un-turn” . Each turn is its own inverse: Performing it twice in a row puts the mattress back in the initial position .
Associativity of the turns is true, though it is tedious to check, but this necessary property of a group indeed holds.
The same structure of operations holds for turning your pillow. Any group with four elements, each of which is its own inverse, has basically the same abstract structure, called the Klein 4-group [after Felix Klein (1849–1925), who classified geometries by their symmetry groups].