With a single, double or queen size mattress, there are four ways of changing the mattress: don't change it; rotate it halfway; flip it from left to right, flip it from head to toe. These look like:





Null  Rotate  Flip LR  Flip HT 
What are some properties of changing mattresses? Changing the mattress leaves it looking like it did before. So if we change it again, again it looks the same as before. So the operation of doing both changes, one after another, is itself a way of changing mattresses. In other words, we can "add" changes. Using Method X to change a mattress and then following that by the combination of doing Method Y to it followed by Method Z is the same as doing first the combination of X followed by Y and then doing Z to that. Both are really longwinded ways of saying "Do X, Y, and Z in that order." Of course, we really don't have to change the mattress. If we feel comfortable with it as it is, we may just leave it alone. That is the null operation, which I shall call e. Finally, no matter how we change the mattress, if we want to, we can change it back again and have it back where it was. This means for every operation, there is a counteroperation that undoes the operation. Let us formulate the rules of mattress operations, then, where G is the group of all operations:
A set of operations which have these properties is called a group. The study of groups forms an important area of research in mathematics. Groups cover a wide range of phenomena. For example, the integers, or the real numbers, where one combines numbers with addition, or with multiplication, form groups. The symmetries of a polygon or polyhedron form a group. Interactions between elementary particles (such as protons, mesons, and so forth) form a group. Groups appear whenever symmetry is involved. Groups also appear when one can perform operations on an object that leave it looking the "same" as before. Mattress changing is such a group. So also is moving or rotating a cube so that it looks the same as before, like throwing a die. The rotations of a cube form a group, the octahedral group. The cube can be split so that each face looks like a 3x3 square, and the 3x3x1 planes can be rotated. A game inventor named Ernö Rubik created such a cube. The operations on this split cube are complicated enough so that he was able to market it as a puzzle, which we know as Rubik's cube.
The group of the mattress is called the fourgroup. It has the null element and three elements of order two: if you do any of them twice in a row, you get back to where you started. If you flip head to toe and then flip head to toe again, the mattress is back to where it was.
A more complicated group arises for a superking size bed. A kingsize bed is nearly square; hence one can lie on it in a number of directions, a fact which is useful for a number of purposes. Suppose such a bed is elongated slightly more so that it forms a square. In that case there would be more ways of flipping the mattress; for example, turning it right 90 degrees. Let us call that rotation R. Suppose one does R to a mattress and then flips it from head to toe (H). One gets this result:


We can describe a group by a multiplication table, similar to those we learned addition and multiplication from in our grade school days. Unlike those tables, this one gives all possible multiplications. We can do this since there are only a finite number of operations we can do to a square bed, namely 8. The table looks like:
*  N  R  T  L  V  H  D  U 

N  N  R  T  L  V  U  H  D 
R  R  T  L  N  U  H  D  V 
T  T  L  N  R  H  D  V  U 
L  L  N  R  T  D  V  U  H 
V  V  D  H  U  N  L  T  R 
U  U  V  D  H  R  N  L  T 
H  H  U  V  D  T  R  N  L 
D  D  H  U  V  L  T  R  N 
What other things in our life can be described by groups? Every year I go to the Southeast Unitarian Universalist Summer Institute (SUUSI). One of the things I do there is contra dance. Contra dancing is a traditional form of dance in which a number of men and women line up in two rows, and do various dance steps with each other. After a series of such steps, the formation looks like it did before but the people making it all up are switched around (permuted) in some way. Contra dance steps satisfy all the requirements for a group: doing one step followed by another is itself a step; three steps are always do step 1 followed by step 2 followed by step 3 no matter how you conceive it; the dancers don't have to do any dancing, and if one does some steps, by doing them backwards, the line gets back to where it started. So contra dancing is an example of a group. What kind of group is it?
This is the formation in contra dancing:
BAND  M  W  M  W  M  W  M  W  M  W 
W  M  W  M  W  M  W  M  W  M 
BAND  A 
B 
C 
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Alice  Bob 
Carl  Diane 
Bob  Diane 
Alice  Carl 
Bob  Alice 
Diane  Carl 
Diane  Bob 
Carl  Alice 
But what is the goal of this switching around? With a mattress the goal is to produce all the possible group elements in turn so that the bed will get even wear, considering that not all activities on a square bed will be of the inactive sort! In contra dancing the goal is to interchange the two opposing partners. This means that
Alice  Bob 
Carl  Diane 
Bob  Alice 
Diane  Carl 
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So contra dancing involves two groups: one to dance the foursomes, and one to do the progression. A lot of fancy steps are performed in between, such as promenading, balancing, gypsying, and doing heys for 4, but the end result is that the circles of 4 switch partners, so that the overall group can progress. Both groups, by the way, are called dihedral groups, because they are the symmetry groups of twosided polygons, as a pentagon cut out of paper. Such a pentagon has two sides, a front and a back.
So if you can flip a mattress on occasion, you can contra dance, and further, you have an interesting design for a quilt, which shows how versatile group theory, and mathematics, can be.
Dr. James V. Blowers
2001 August 12