### Contra Dancing and Matrices

Bernie Scanlon, a mathematics instructor at Bakersfield College in California, has been dancing nearly every weekend since 1990, even traveling to distant parts of the country to join in the fun. His passion is contra dancing—a dance form unknown to most people yet practiced with great devotion and abandon throughout the United States, from New England to California.

The origins of contra dancing go back to colonial days, and its roots can be traced to English country dance. It's really a group rather than a couples effort, and it has elements that might remind you of traditional square dancing. Rhythm and pattern are the keys.

What's striking, says Scanlon, is that a remarkably high percentage of its practitioners are highly educated, often involved in mathematics, computers, or engineering. "The appeal seems to lie in its being a kind of 'set dancing,' where one's position relative to others while tracing patterns on the dance floor is paramount," he says. "Timing is also crucial, as is the ability to rapidly carry out called instructions and do fraction math on the fly." Scanlon introduced both the mathematical and performance sides of contra dancing to attendees earlier this year at the 2nd Annual Recreational Mathematics Conference (see "Fun and Games in Nevada").

The music for contra dancing is highly structured. Everything occurs in units of four. The band plays a tune for 16 beats, repeats the tune, then plays a new tune for 16 beats and repeats that. An eight-beat section is known as a call, during which each block of four dancers executes a called-out instruction. An entire dance is precisely 64 beats long.

When the dancers line up in their groups of four to produce a long column down the floor extending away from the band, each square block consisting of two couples can be thought of as a matrix. Each dancer (element of the matrix) is in a specific position within the block. The called instructions correspond to rearrangements of the elements of the matrix. After 64 beats, however, the first and second rows of the matrix must be interchanged. Of course, that can be done in one step, but the fun comes in all the different ways in which groups of four can get to that inevitable end result.

 Matrix representing initial configuration of two couples in a contra dance (so-called improper formation).

There are all sorts of called instructions, which range from simply circling once around to the left or right within each group of four (the matrix doesn't change after this operation) to sequences of moves that exchange partners or involve stepping one-quarter, one-half, or three-quarters of the way around the ring. With each call, the matrix representing four dancers changes, though it can end up the way it started.

So the emphasis is not so much on the specific footwork needed to get somewhere as on being in the right place at the right time. The physical movements can get quite complicated, and you always need to keep in mind where you started and where you ought to be. And it all happens at a lively, breathtaking, whirling pace.

"The secret is the fractions and the patterns," Scanlon says. In effect, "you're dancing like a fool, but it's totally controlled!"

Here's the "CMC3 Reel" that Scanlon created for the session participants—who turned out to be quick learners.

 No. of Beats Call 8 Circle left once around 8 Star left once back 8 Allemande neighbor 1 1/2 8 Circle left 3/4 way around 8 Men allemande right twice around 8 Women allemande left 1 1/2 times 4 Circle left 1/2 time around 8 Star right once around 4 Allemande right your partner 1/2 time around

In terms of matrices, the final configuration has the two rows of the original 2 x 2 matrix interchanged. If you know what the calls mean, you can work out the sequence of matrices along the way.

Who says matrices (in motion) can't be fun? This calls for some research in the field—or local dance hall!

For an update and more on mathematics and contra dancing, see "Contra Dances, Matrices, and Groups" (March 8, 2003).

#### References:

Peterson, I. 2003. Contra dances, matrices, and groups. Science News Online (March 8). Available at http://www.sciencenews.org/articles/20030308/mathtrek.asp.

______. 1997. Mailbox: Magic dice, Monopoly, and contra dancing. Science News Online (Dec. 20). Available at http://www.sciencenews.org/pages/sn_arc97/12_20_97/mathland.htm.