Woops! - Ladies Chain

Today I looked at the mathematical structure of contra dancing. That link to "mathematical structure" refers to an article I wrote for my mathematics page. It shows how the mathematical structure of contra dancing is like the mathematical structure of changing a mattress, and it ends with a quilt based on contra dancing patterns.

One of the moves of contra dancing is "ladies chain". This is a step where the two ladies in a foursome come to the center and allemande around, going to the opposite side, whereupon the opposite man twirls them around 360 degrees. The result is that the two ladies have switched position, while the two men stay the same. I found an error, or at least a questionable point, in my paper, which makes for an interesting conundrum or paradox.

The part of mathematics that I relate contra dancing to is group theory. A group is a set with an operation (as plus or times) such that the group is closed (adding or multiplying group elements always results in group elements, is associative ( (x*y)*z=x*(y*z) ), has an identity element, as 1, where 1 times anything is that same thing again, and inverses, so that for each a there is an a' such that a*a' = 1. Some operations are easily seen to be associative. For example, let x, y, and z be moves on a Rubik's Cube. then (x*y)*z and x*(y*z) both mean do x, then do y, and then do z, with the only difference being a trivial linguistic difference. So therefore the Rubik's cube group is associative.

So I thought surely contra dancing moves are associative. After all, (x*y)*z and x*(y*z) both mean dance x, then dance y, then dance z. So the two should be the same. No. They are not. It turns out that if y is ladies chain, and x switches the men and women around, then the two are different!

I will explain this in a sequel to my contra dance article, but the idea is this. Imagine a foursome in a square, and from left to right top, then from left to right on the bottom, label the dancers A, B, C, and D, in positions 1, 2, 3, and 4. Then A and D are ladies, and B and C are men. If they do a ladies chain, then A and D trade places. These are in places 1 and 4. But suppose they quarter turn left first. Then the women are in 2 and 3, and ladies chain trades 2 and 3 instead. This means that in (x*y)*z, then y trades 1 and 4, while in x*(y*z), y trades 2 and 3. The definition of y depends on the grouping of the terms. This results in the non-associativity.

I emailed a colleague about this, Larry Copes. He told me that a ladies chain when the ladies are in positions 2 and 3 is not often seen. I thought about it. It is a mirror image of the usual one! It violates the principle of "lady on the right". So most contra dances seen on dance floors do not have such ladies chains. So as far as movements on the dance floor are concerned, they are associative. But that is not good enough for a mathematical group. There one should be able to take any two elements and multiply them. I suppose the way to handle this is to regard ladies chain and gypsy once-and-a-half (which switches ladies 2 and 3) as the same mathematical group element. Then the group of contra dancing is D4, as I say in my article.