# Mattresses, Contra Dancing and Quilts

One of the tasks that many of us have to do at home at times is to change the mattress. This is to even out the wear on the mattresses. If one keeps the mattress the same all the time, all the wear is on the heavy parts of the body, primarily the torso, and these parts get worn down, and the bed becomes unorthopedic and unhealthy. So every once in a while, we flip or turn the mattress around so that the wear is evened out. What are the ways we can change the mattress? After changing, it should look the same as before.

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:

 A B C D
 C D B A
 B A D C
 C D A B
Null     Rotate    Flip LR   Flip HT
Where I designate by A, B, C, and D the four corners of the mattress. We generally want to go around to each way in turn to even out the wear on the mattress.

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 long-winded 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:

1. If x and y are in G, then x followed by y (x*y or xy) is in G
2. For all x, y, z in G, (x*y)*z = x*(y*z) (associativity; it makes sense to say x*y*z)
3. There is a null operation e such that x*e = e*x = x for all x in G
4. For every x there is a u (un-x, so to speak) such that x*u = u*x = e

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 four-group. 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 super-king size bed. A king-size 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:

 D B C A
Suppose one does it the other way. Suppose we H flip it first, then do R to it. Here is the result:
 A C B D
They are not the same. This group is useful for describing the flipping of a square bed, so we don't want to rule it out. So we don't require that the group operation be commutative; i.e., for all x and y in G, that x*y = y*x, so that the order does not matter. For square mattresses, it does matter. Some groups have commutative operations; we call them Abelian (after the mathematician Henrik Abel). The four-group is Abelian but the square-bed group is not.

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 = Null; do nothing
• R = turn Right 90 degrees
• T = Turn it halfway
• L = turn Left 90 degrees
• V = flip along Vertical (left to right) axis
• U = diagonal flip so Up diagonal changes
• H = flip Head to toe
• D = diagonal flip so Down diagonal changes
* 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
For example, a 90-degree right rotation followed by a head-to-toe flip is R*H = D, which is a flip that interchanges the down diagonal and leaves the up one fixed.

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
The men and women alternate, sort of like a checkerboard. This is called an improper position (which is about as improper as an improper fraction, but that is the term that contra dancers use). There are an even number of men and the same number of women. The men and women across from each other are partners, and the partners pair further into foursomes across the arrangement. If we designate the partners by letters, we get an arrangement something like:
 BAND A B C D E F G H I J
A series of dance steps are now performed. These operate mostly on the foursomes. A typical foursome looks like:
 Alice Bob Carl Diane
Let's take a typical contra dance step: circle around. The four people hold hands and go around in a circle until they are back where they started. They have performed the null element. They could have just as well stood in place and did the same thing but going around in a circle is more fun! If they went only 3/4 around to the right, then they would wind up like this:
 Bob Diane Alice Carl
Another movement is to allemande your neighbor. To do this, each person steps up to his or her neighbor and goes around in a tight circle around them. If this is done an odd number of times, it switches the man and woman so that this results:
 Bob Alice Diane Carl
Yet another is "ladies chain". In this one the two women go to the center of the square, swing each other half way around, then go to the man on the opposite side. The man then swings her around from left to right; the "courtesy turn". This switches the two women:
 Diane Bob Carl Alice
In contra dancing the couples perform a number of actions like these. This is the same as combining or multiplying the corresponding group elements. What kind of group is it? By now you may have seen the similarity to changing the square mattress. In fact, the two groups are essentially the same! In group theory terminology, we say they are isomorphic. Doing a circle 3/4 to the right corresponds to turning the mattress around to the right 3/4 of the way. Doing a 1 1/2 allemande neighbor is the same as flipping the bed from left to right. Ladies chain is like flipping the bed diagonally, holding two opposite corners and flipping the other corners over. To see that they are the same group, imagine that the contra dancers have to dance holding a mattress by the corners. A 3/4 turn of the dancers effects a 3/4 turn of the mattress. It would be some trick, but one could allemande neighbor in such a way that the mattress corners interchange. And so forth. The idea is this: to change the mattress, four contra dancers do a dance step. They notice where they are, and then they perform the same operation on the mattress corners. Lo and behold. The mattress is changed. So contra dancers ought to have no trouble changing a mattress!

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
becomes
 Bob Alice Diane Carl
The reason why is so that the dance can progress. With the partners switched, the men turn to the left and the women to the right and find a new couple to dance four with. This means that the contra dancing setup:
 BAND A B C D E F G H I J
becomes
 BAND B A D C F E H G J I
We will put these together, so that going down means progressing in time from one step to another:
 BAND A B C D E F G H I J BAND B A D C F E H G J I
Note then that couples A and B switch, C and D switch and so forth. The result is BADC... Call this x. Couple A was with couple B and now is with couple D. C in turn is with F and so forth. What about B? Sorry. B can't dance this round. Neither can couple I. This is a regular part of contra dancing. In every other set of steps, there are two couples, one at each end, who don't dance with the others. We say these couples are "out". The dance is performed, and the result is the third line:
 BAND A B C D E F G H I J BAND B A D C F E H G J I BAND B D A F C H E J G I
Now A dances with F and so forth. Call this sequence y. This time all the dancers get in the act. Why don't the dancers simply repeat the first set of steps? Instead of A grouping with D, why not A grouping with B, this time with sides reversed? Because this group element is an involution. Doing it twice in a row, like flipping a mattress the same way twice in a row, gets you back to where you started. That would not be interesting. Having A dance with D gives couple A a new couple to dance with, and that makes the dance fun. A full set of steps in a contra dance will allow everyone to dance with everyone else of the opposite sex! Note that A, C, and so forth move away from the band. These couples are called "active" in contra dance terminology. The others move towards the band and are called "inactive". Some steps are performed only by actives. For example, it is common for only actives to swing their partners in a step. When a couple goes out, it changes from active to inactive or vice versa and starts moving the opposite direction. If a full set of steps are danced out the result could make a good design for a quilt:
 A B C D E F G H I J B A D C F E H G J I B D A F C H E J G I D B F A H C J E I G D F B H A J C I E G F D H B J A I C G E F H D J B I A G C E H F J D I B G A E C H J F I D G B E A C J H I F G D E B C A J I H G F E D C B A I J G H E F C D A B I G J E H C F A D B G I E J C H A F B D G E I C J A H B F D E G C I A J B H D F E C G A I B J D H F C E A G B I D J F H C A E B G D I F J H A C B E D G F I H J A B C D E F G H I J B A D C F E H G J I B D A F C H E J G I
Note that a different set of permutations would have resulted if A and J had been out in the first round. This corresponds to performing y first and then x. This shows that the contra dance group is not Abelian.

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 two-sided 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