So let’s talk about that post with the diagram that I said had something to do with a contradance.
In the last week, caller and fiddler Amy Cann and I co-taught a seven-day-ish course called Patterns and Mathematics in Traditional Folk Dancing. It was basically a short, intensive workshop on analyzing contra and English country dances as mathematicians, dancers and choreographers. The goal was to use mathematics to help us become better dancers and choreographers. As part of the course, Amy solicited from a dance caller’s mailing list various caller’s lists of favorite and least favorite transitions. The diagram I had in the previous post is a still from an animation that illustrates a geometry problem that came from one of these transitions.
First, let’s look at the problem mathematically.
In the diagram above, we have segment a formed by points A and B and segment b formed by points C and D. The goal is to rotate segment a as one rigid object on to segment b so that point A rotates to point C and point B rotates to point D. In order to do that, we need the center of rotation. This is not a hard geometric problem; it takes some thinking, especially if your geometry is rusty, but it’s perfectly solvable using straightedge and compass in a few steps.(Side note: I proclaimed when giving this problem that I did not expect any of my students educated in the United States to be able to solve this problem. I was half right; two of our students solved this problem: one learned geometry in Germany and the other in a Waldorf school in the United States. Double side note: I’m considering sending my kids to Waldorf schools. Triple side note: I don’t have any kids yet and don’t really plan to have any.)
The fun part is where this problem came from!
While reading through the list of responses from the surveyed callers, Amy noticed that there was a huge division on the transition “lines of four up the hall, bend into a circle and circle left/right”. Some people loved it; some people hated it. There was no in-between. So she started thinking about it physically.
Before I go into where the math comes in, let’s talk about this sequence of movements.
At the beginning of the sequence, four people, side by side and holding hands, walk up the dance floor. They then “bend” the line, still holding on to each other, so that the two people on the end can grab each other’s free hand to form a circle of four. They then rotate this circle.
Mathematically, if we represent each dancer as a point, we have a line of four looking like this:
Now, they need to form a “circle”, which translates to a square when you are only thinking of them as four points. One way of doing it is by having the middle dancers stop and the dancers at the end of the line rotating around them to get into a square shape. So from the figure above, we get this:
Another way of doing this involves the dancers in the middle moving back a step or two while swinging the dancers in the end forwards. From the original diagram of four people, we get this as a result of having the middle dancers move back:
Amy’s conjecture is that the second method of doing this is superior in terms of smoothness and flow. Her argument involves the “center” of the minor set; physically, she is concerned with the center of mass of these four equally massive point masses. Notice that as the line of four is about to become a circle, the center of mass between all four dancers is the midpoint between the middle dancers. So, in our first diagram, it is the origin, or (0, 0). Where are the centers of mass in the second and third diagrams? (0, 1) and (0, 0), respectively. Amy’s argument is that since the center of mass is preserved from line to circle when the middle dancers back up, it feels better.I think it’s more than that, but let’s finish talking about the problem first.Let’s concentrate on dancers A and B for the moment; as the other dancers are mirror images of them we only need to worry about A and B. The problem now becomes the problem I proposed originally: if you want to rotate segment a to segment b (or A to C and B to D) where is the center of rotation?The answer is point E, which is (-1.5, -0.5).
The standard geometric answer to the question which Abijah Reed, my geometry-teaching colleague, produced on a napkin as soon as the problem was proposed: take the perpendicular bisectors of the segments AC and BD and find their point of intersection.Here’s another way of doing the problem; the interesting thing about this solution is that it takes the fact that it’s a dance into account. Notice that when dancer B begins his rotation, he’s facing up. Once he’s done, he’s facing right. This means that he made a right angle turn. The same applies to A. This means that the angle of rotation must be 90 degrees. From there, it’s easy to construct something like the following and find E:
If dancer A stayed put, B would have to travel from (-3, 0) to (-1, 2) on a 90 degree arc; the radius of the turn would be 2. If A had moved back according to this method of rotation, B would only need to travel to (-1, 1) on a 90 degree arc with radius sqrt(10)/2, or about 1.162. This is a significant decrease in the amount that B has to travel, which could be important in a fast-paced dance as there are really only two beats to bend the line with! Since dancer A shared some of the weight and work for B, this move can be completed more quickly if A took a step back. Ideally, A would actually arc back to maintain a perfect rotation.Tighter turns also feel more comfortable, at least to me.Of course, there are always exceptions to a mathematical explanation. Sometimes you do in fact want to have dancer A swing B as fast as possible into the circle in a wide arc. For example, if dancers B and D were to pull by, swing, or do anything that could use a boost of momentum into the set, it may feel better for the outer dancers to be swung in as a large arc to build up some momentum.
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