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At Cambridge University in England, gentlemen are still required to wear neckties in the dining halls. So much brainpower is assembled daily at these meals, perhaps it's not surprising that eventually someone would ask, "Why do we tie these knots the same way every day? Isn't there something different we can do?"
The question was asked, and answered, by Thomas Fink and Yong Mao, who are mathematical physicists holding research fellowships at the University's prestigious Cavendish Laboratories. Their answer was interesting enough to earn publication in Nature, one of the world's top science journals.
Most men know of two necktie knots. The simple "four-in-hand" knot originated in the nineteenth century and is still the most common. The larger Windsor knot was popularized in the late 1930's by the Duke of Windsor (the former King Edward VIII). There is also a simpler version of the Windsor, called the half-Windsor. A fourth knot, called the Pratt knot, was discovered in 1989.
Fink and Mao's analysis of necktie knots is actually a good example of how mathematicians attack real-world problems in an organized, abstract way. First, a necktie has two ends, but only the wide end is moved in tying the knot. Fink and Mao call this the active end.
Second, the tie and the developing knot divide a person's front into three areas: the right side (R), the left side (L), and what Fink and Mao call the center (C), by which they mean the area above the knot, under the throat. Each movement in tying the knot carries the active end toward one of these three areas.
Third, each movement in tying the knot carries the active end either toward the shirt (t) or away from it (f). In fact, these two directions must alternate: a move toward the shirt is always followed by a move away from the shirt. At the end of the knot-tying, there is a special move Fink and Mao call the pull-through (T), in which the active end is pulled downward through the knot.
In good mathematical fashion, Fink and Mao have thus created a notation for describing how to tie necktie knots:
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Here is one of Fink and Mao's illustrations, showing the tying of the familiar four-in-hand knot. You can clearly see the sequence of moves: left and toward the shirt (Lt); right and away (Rf); left and toward (Lt); center and away (Cf); pull-through (T). |
The Windsor knot has eight moves before the pull-through, and this is near the practical limit, because ties are only so long and knots can get only so big. Fink and Mao decided that practical knots should have no more than nine moves before the pull-through.
It turns out that within these restrictions there are exactly 85 possible necktie knots! Of course, many of these knots look terrible. So Fink and Mao investigated questions of style as well. For example, they found that good necktie knots have approximate symmetry: the number of left moves should be equal to, one more than, or one less than the number of left moves.
Even after allowing for these style considerations, Fink and Mao found at least 10 "good" knot designs, including the four designs already known and six others. Many of the new designs begin with a move away from the shirt; in order for this to work out it's necessary to start with the tie inside-out, that is, with the back of both ends of the tie facing out. Apparently, no one had ever thought of this before. One of their favorite designs, designated "the 7,2" is Lf Rt Lf Ct Rf Lt Cf T. It combines some of the properties of the four-in-hand and the Windsor knot.
The paper has attracted considerable interest, including a recent story in the New York Times. Because of the interest, Fink and Mao have both posted copies of the paper, with diagrams, on the Internet.
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Posted April 13, 1999.
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