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.
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2004/09/15
Hurricanes Hurricanes and more Hurricanes
I have not talked too much about hurricanes lately, but this has been some season. Since early July, I have been tracking down some hurricane or tropical storm somewhere, even if it is only in modelspace, such as the infamous SUUSI Hurricane that ruined SUUSI this year by striking Virginia Tech with 10 inch rains and 50 mph winds on its third day, and also that did not exist at all. It was a mere figment of modelspace. Some runs of the Global Forecasting System (GFS) showed this hurricane striking Charleston, SC and hitting western Virginia, but after two or three days, it completely disappeared from the runs and we had beautiful weather for SUUSI.
But most of the other storms I have been tracking have been real storms that have caused some real trouble. Charley and Frances X-ed out Florida with their tracks, and now the Gulf Coast is being hit by Ivan the Terrible as I speak. It's hurricane after hurricane after hurricane, Alex, Bonnie, Charley, Frances, Gaston, Ivan, and Jeanne, plus Karl and Lisa in modelspace. Gaston already has given us the second worst storm that I can remember here, and it was a mere tropical storm.
So the question is, will Ivan and Jeanne strike here? A Virginia DOT meteorologist says no. Neither will affect us. The forecaster at WxRisk.com also says that both will miss. He says Jeanne will be a Florida hurricane. Certainly I don't think Ivan will strike here. He is going straight for the Great Smoky Mountains, and there he will die. He may give us some flung-out spiral arm rain, as Frances did. But that's all. But what about Jeanne? Although WxRisk thinks this is a Gulf hurricane, MillenniumWeather thinks it's headed for the Carolinas. What do the models say?
No model goes out far enough to see where it will strike, except GFS. They do show her approaching North Carolina, but no farther than that. GFS does show what happens to her, and this is really peculiar, almost comical. GFS says that Jeanne will hit Florida twice. After Charley, Francis, and Ivan, this will make for five hurricane strikes this season for the beleaguered state, out of only four storms. It shows Jeanne going out in to the Atlantic, approaching the Outer Banks, but then a huge H of a High bombs out of Canada (the season's first cold snap) and stands right in front of Jeanne and says "STOP!". The big H then pushes Jeanne back to where she came from, and then towards the southern tip of Florida. Not only that, but the 18Z run of it shows it being pushed farther than that, into the Gulf of Mexico, where it hits the Yucatan. The idea of a hurricane going backwards seems comical to me. But then when this big beautiful Autumn High finally leaves, the hurricane then turns around and heads east northeast, hitting Florida a second time and then going out to sea in the Atlantic to spend its last days spinning fish. In my opinion, I don't think GFS knows what to make of this tremendous battle between a strong high ridge and a hurricane. I think the strong high will force Jeanne to move west northwest and hit Florida and go into the Gulf, as WxRisk says. But I am still going to watch this storm.
But most of the other storms I have been tracking have been real storms that have caused some real trouble. Charley and Frances X-ed out Florida with their tracks, and now the Gulf Coast is being hit by Ivan the Terrible as I speak. It's hurricane after hurricane after hurricane, Alex, Bonnie, Charley, Frances, Gaston, Ivan, and Jeanne, plus Karl and Lisa in modelspace. Gaston already has given us the second worst storm that I can remember here, and it was a mere tropical storm.
So the question is, will Ivan and Jeanne strike here? A Virginia DOT meteorologist says no. Neither will affect us. The forecaster at WxRisk.com also says that both will miss. He says Jeanne will be a Florida hurricane. Certainly I don't think Ivan will strike here. He is going straight for the Great Smoky Mountains, and there he will die. He may give us some flung-out spiral arm rain, as Frances did. But that's all. But what about Jeanne? Although WxRisk thinks this is a Gulf hurricane, MillenniumWeather thinks it's headed for the Carolinas. What do the models say?
No model goes out far enough to see where it will strike, except GFS. They do show her approaching North Carolina, but no farther than that. GFS does show what happens to her, and this is really peculiar, almost comical. GFS says that Jeanne will hit Florida twice. After Charley, Francis, and Ivan, this will make for five hurricane strikes this season for the beleaguered state, out of only four storms. It shows Jeanne going out in to the Atlantic, approaching the Outer Banks, but then a huge H of a High bombs out of Canada (the season's first cold snap) and stands right in front of Jeanne and says "STOP!". The big H then pushes Jeanne back to where she came from, and then towards the southern tip of Florida. Not only that, but the 18Z run of it shows it being pushed farther than that, into the Gulf of Mexico, where it hits the Yucatan. The idea of a hurricane going backwards seems comical to me. But then when this big beautiful Autumn High finally leaves, the hurricane then turns around and heads east northeast, hitting Florida a second time and then going out to sea in the Atlantic to spend its last days spinning fish. In my opinion, I don't think GFS knows what to make of this tremendous battle between a strong high ridge and a hurricane. I think the strong high will force Jeanne to move west northwest and hit Florida and go into the Gulf, as WxRisk says. But I am still going to watch this storm.
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