Tying Down a Random Walk

The business world is gradually slipping into casual couture, not just on Fridays but throughout the work week. Even law offices in Silicon Valley and elsewhere are succumbing to the relaxed look.

One casualty of this sweeping change in style is the necktie. For many men, the ordeal of fashioning a neat tie knot is fading into the distant past.

It seems ironic that two physicists have now put necktie knots back on center stage. Thomas M.A. Fink and Yong Mao of the Cavendish Laboratory in Cambridge, England, have developed a mathematical model of tie knots. The model suggests six new "aesthetically pleasing" knots, ready for sampling by any gentleman interested in cutting-edge sartorial splendor.

The standard tie is a tapered piece of fabric. A tie knot is initiated when the tie's wide end is brought either over or under the narrow end. The knot-tying procedure then continues with a sequence of moves (half-turns) bringing the wide end to the left, center, or right (though never in the same direction two times in a row). With each half-turn, the wide end alternates between moving toward the shirt and heading away from it. A final flurry of moves wraps up the process.

Fink and Mao modeled knot-tying sequences as random walks (see Knotted Walks, Nov. 3, 1997) plotted on a triangular grid, where consecutive steps can't be made in the same direction.

"Practical considerations (namely the finite length of the tie), as well as aesthetic ones, suggest an upper bound on knot size," Fink and Mao note in the March 4 Nature. By limiting the number of moves to nine or fewer, they found that a conventional, tapered necktie can be tied in 85 ways. Those sequences include the four knots (four-in-hand, Pratt, half-Windsor, and Windsor) currently in widespread use. Six additional configurations have the appropriate symmetry (an equal number of left and right moves) and balance (mixing of moves to create a tightly bound, well-shaped knot) to merit serious consideration, the researchers remark.

The results explain why the Windsor knot is wider than the four-in-hand knot. It requires more center moves, which tend to make a knot bulkier.

Fink and Mao have submitted a lengthy paper detailing their results to the Journal of Physics A. Interestingly, similar random-walk models play an important role in studies of protein folding, which is Fink's main research interest. Meanwhile, the researchers have novel knots to show off in Cambridge University dining halls, where jacket and tie are still required.

References:

Fink, T.M., and Y. Mao. 1999. Designing tie knots by random walks. Nature 398(March 4):31.

Fountain, H. 1999. It takes a scientist to tie a necktie, 85 different ways. New York Times (March 9).

Peterson, I. 1999. Simulations nab protein-folding mistakes. Science News 155(March 6):150. (Available at http://www.sciencenews.org/sn_arc99/3_6_99/fob6.htm.)

Seife, C. 1999. It's style, but knot as you know it. New Scientist (March 6):6.

Waldman, P. 1999. Hot-button issue: Getting lawyers to ditch their suits. Wall Street Journal 232(March 19):1.

Thomas Fink's Web page can be found at http://www.tcm.phy.cam.ac.uk/~tmf20/.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.