Representations of contra dance moves

Permutation representation

Suppose that the positions of the members of a contra dance group of four are numbered like this

Then you could indicate that the dancers in positions 1 and 3 switch places (somehow) by the notation (1 3). Aloud, you'd read "the person in position 1 goes to position 3, and the person in position 3 goes to position 1." Similarly, if everyone circled left 3/4 of the way around, you could write (1 4 3 2) and read, "The dancer in position 1 goes to position 4, the dancer in position 4 goes to position 3, the dancer in position 3 goes to position 2, and the dancer in position 2 goes to position 1." Of course, you wouldn't say all that. You'd probably mutter to yourself something like "1 to 4, 4 to 3, 3 to 2, 2 to 1." But you'd need to remember that the numbers refer to places rather than people.

For a complete sequence that ends with everyone progressing, you need to determine a sequence that's equivalent to (1 2)(3 4)—interchanging the dancers in positions 1 and 2, and also those in positions 3 and 4.

An advantage of this permutation notation is that you can see where a person moves over a sequence of figures. For example, suppose you have this sequence:

Then you can trace the path of the person who begins in location 1:

from position 1 to position 2 in the neighbor swing

on to position 4 in the right and left through (because it says the person in position 2 moves to position 4)

no net change in the next two figures

to position 3 after circling left 3/4

So you'd start describing the composite by writing

(1 3

and then go on to consider what happens to the person who begins in position 3. If you trace this all the way though, you see that this dancer ends up in position 1, so you close off your sequence:

(1 3)

This doesn't tell yet what happens to the person starting in position 2, so you trace that person through the dance sequence tosee that they end in position 4: :

(1 3)(2 4

To be sure you've got it, check that the person starting in position 4 ends up in position 2, so that the composite is

(1 3)(2 4).

If you've started to write a dance with these figures, you might ask what else is needed to finish it. The result has to be (1 2)(3 4) to interchange the two couples.

Group element representation

Rather than writing out all of these numbers, you might find it easier to represent them by letters that show how they are part of the group of symmetries of a square. One particular way of using letters helps us see relationships among the permutations.

First, there are figures, like lines forward and back, that don't change the locations of the dancers in the square. Let's call the resulting permutation I, for "identity."

Then there is the permutation that rotates the square counterclockwise by 90 degrees (circle left 3/4). Let's call that R. It would be (1 4 3 2) in the previous notation. A rotation that rotates the square by 180 degrees is the same as two of these first rotations, so let's call it R2. In the permutation notation above, R2 is (1 3 )(2 4). R3, then, is a rotation through 270 degrees, or (1 2 3 4).

The result of each sequence of steps in a dance is to interchange the dancers in positions 1 and 2 and also those in positions 3 and 4. This is a reflection of the square about a vertical line through the center of the horizontal sides. Let's call that permutation, which is (1 2)(3 4), by the name F, for "flip." Then each of the other possible permutations can be written as a product of F with one of the R's:

FR is (2 4)

FR2 is (1 4)(2 3)

FR3 is (1 3)

To see what happens when one group element is followed by another, we write them next to each other and then simplify using the realization that RF = FR2 and RF2 = FR. Also, we need to keep in mind that FF and R4 are the same as I. Then, for example, we can see that FR2 followed by F is

FR2F = FFR = IR = R

and FR followed by FR2 is

FRFR2 = FFR3R2=FFR4R = IIR = R.

Note that FR2 followed by FR is different:

FR2FR = FFR2R = IR3 = R3.

So we have to worry about the order in which elements are combined as we make a table:

The element listed in the left column comes first, and is followed by the element in the top row.

The table, once worked out, allows us to continue the example we began with the permutation notation. There we had gone through a sequence of figures to get to (1 3)(2 4), and we wanted to know what else to do to get to (1 2)(3 4). In the notation of group elements, we had gotten to R2 and we want to get to F. Looking at the table, we find R2 in the left column and look to see what element across the top will give us F. It's FR2. So to finish the dance we must choose a figure or sequence of figures that result in FR2.

The properties of this table show that, in the terminology of abstract algebra, these elements form a group. It's called the dihedral group of order eight, symbolized by D4.

We must add that some advanced contra dance figures result in rearrangements that are not among the eight symmetries of a square. For example, a half-figure-8 by the number 1 couples results in either (2 4) or (1 3), depending on their starting position. Since square are no longer square if two adjacent vertices are interchanged, these rearrangements are not among the symmetries in D4.

Matrix representation

Because the original square

looks like a matrix, we might think that the the transformations can be represented by matrices as well, and that the result might be found by multiplying matrices in the standard way.

Indeed, matrices do represent linear transformations of the plane; but as such they operate on the coordinate plane, in which points (or vectors) are described by a pair of coordinates. Toward that end, we can rename our four positions using coordinates:

and then represent the entire square as a 4 x 2 matrix:

This 4 x 2 matrix can be multiplied by a 2 x 2 matrix to get another 4 x 2 matrix, representing the new arrangement of dancers. For example, if the transformation is represented by the matrix

then the product is

This means that

the dancer in position (-1 1) moves to position (1, 1);

the dancer in position ((1, 1) moves to position (1, -1);

the dancer in position (1, -1) moves to position (-1, -1); and

the dancer in position (-1, -1) moves to position (-1, 1).

Using previous notations, this is the transformation R or (1 2 3 4).

There are eight elementary matrices that contain 0's on on diagonal and 1 or -1 on the other. These eight matrices represent the eight permutations or group elements under consideration. You can find these matrices in the table below.

Another table

Here's a list of the equivalent notations for transformations and some common standard contra dance figures that result in each.

permutation	group element	transformation matrix	typical figures
(1)(2)(3)(4)	I		lines forward and back, stars, hay, swing on side (if lady is already to right of gent), dos-á-dos or gipsy or allemande once around
(1 2 3 4)	R3		circle right 3/4 (or left 1/4)
(1 3)(2 4)	R2		right and left through, circle 1/2
(1 4 3 2)	R		circle left 3/4 (or right 1/4)
(1 2)(3 4)	F		swing on side (if lady is to left of gent)
(1 3)	FR3		ladies' chain (if ladies are in positions 1 and 3)
(1 4)(2 3)	FR2		California twirl, box the gnat
(2 4)	FR		gents allemande 1-1/2 (if gents are in positions 2 and 4)

contra dancing

Larry Copes

Last update 29-Jan-2003