**Dancing
with Mathematics:**

**Mathematics
related to contra dancing**

**What's
contra dancing?**

New England or
Southern contra dancing apparently is a decendent of English country dancing
("Jane Austen" dancing), via France (where the name was changed to
*contra*). Many of the figures are similar to those of square dancing,
but a contra dance is preformed in long lines instead of squares, and the caller
doesn't have so much freedom to improvise. It's the kind of dancing that anyone
who forms and discerns simple patterns can learn to do and enjoy.

Each contra dance begins with two lines, everyone facing a partner. Most of the action takes place in pairs of couples. These four dancers execute a squence of 7-9 figures for 64 beats of music. By the end of the sequence, the couples will have exchanged places. Then each couple performs the same sequence with the next couple in line, working its way up or down the lines. After arriving at one end, the couple turns around and works it way back in the other direction.

More information and details can be found at many sites. There's also web-based program for writing contra dances.

**What mathematics is in contra
dancing?**

You can find mathematics at a number of levels.

At an elementary level, there are shapes— lines and squares and circles. Some use is also made of fractions, as well, such as for the figure in which a group circles left 3/4 or a pair of dancers allemandes right 1-1/2 times. And there are regular patterns to the music, which is in AABB form, with each of the four parts consisting of two eight-beat phrases,

At a somewhat higher level, matrices can describe the possible arrangements of dancers in a group of four after various figures.

More abstractly, changes to these arrangements can be thought of as geometric transformations. Most figures executed by each group of four can be seen as reflections, rotations, and translations. In fact, each couple is translated up or down the set from one round to the next.

From the point of view of abstract algebra, these transformations can be thought of as forming the group of symmetries of a square. The transformations accomplished by the various figures can be represented in a variety of ways. Cosets of a subgroup of this group can be used to help count the number of dances that illustrate all eight symmetries of a square.

Equivalently, the transformations form a subgroup of eight of the group of all 24 permutations on four elements, as articulated by James V. Blowers. He also points out that a different group of permutations describes the interchanging of all the dancers in the set.

In various MathTrek columns, Ivars Peterson has given a layperson's view of the mathematics of contradance. In one, he describes a matrix approach. In a response, computer systems analyst and contra dance caller Ted Crane describes how to model dances with 24 x 24 transition matries.

**More abstractly**

Besides actual mathematical content, the orderliness and repeated patterns are attractive to mathematicians. The mental activity of reflecting one's motions after switching directions at the end of the lines is good practice at visualization. And the interplay between mind and body (how often the mind forgets what it's just done 20 times as the body tires!) is analogous to the interplay between logic and intuition in doing high-level thinking.

Last update 07-Mar-2010