# Ivars Peterson's MathTrek

September 26, 2005

## Tying Down a Random Walk

The business world is rapidly slipping into casual couture, not just on Fridays but also throughout the workweek. Even law offices 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.

This style change didn't stop retired mechanical engineer Seth Goldstein from designing and building a gangly machine that automatically ties a knot in a necktie.

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.

Goldstein's "kinetic sculpture" (or robot) uses 10 electric motors to operate an array of pulleys, levers, and other bits and pieces to tie the so-called four-in-hand knot. A computer sends a sequence of numbers to electrical circuits that convert the numbers into voltages, which in turn determine which motor moves how far and in what order. The machine even incorporates optical sensors and a self-correcting mechanism that allows it to recognize a mistake and start over.

The contraption makes about 350 distinct moves, taking a laggardly 8 or so minutes to accomplish the task. To see a video clip of the robot in action, go to http://www.asme.org/education/precollege/whyknot/. After the necktie is tied, the machine pulls the knot apart and starts all over again.

The latest version of Goldstein's machine/sculpture, called "Why Knot," is now on display at the Franklin Institute in Philadelphia.

The "Why Knot" robot ties only the four-in-hand knot. Several years ago, physicists Thomas M.A. Fink and Yong Mao of the Cavendish Laboratory in Cambridge, England, developed a mathematical model of tie knots and showed that there are actually 85 ways to tie a tie. In particular, the model suggested six new "aesthetically pleasing" knots, ready for sampling by any gentleman interested in cutting-edge sartorial splendor.

Fink and Mao modeled knot-tying sequences as random walks 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 noted in the March 4, 1999, Nature. By limiting the number of moves to nine or fewer, they found that a conventional, tapered necktie can be tied in 85 ways.

These sequences include the four knots (four-in-hand, Pratt, half-Windsor, and Windsor) currently in widespread use (pictured at http://www.tcm.phy.cam.ac.uk/~ym101/tie/aps97tie.html). 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 remarked.

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

One example of a new necktie knot (7,2) is shown at http://www.tcm.phy.cam.ac.uk/~ym101/tie/aps97tie.html. Each knot has a numerical label, which consists of a pair of numbers. The first number gives the total number of moves, and the second denotes the number of center moves. The 7,3 knot can be seen at http://www.tcm.phy.cam.ac.uk/~ym101/tie/7_3_1.gif.

To see examples of the new knots as worn, go to http://www.tcm.phy.cam.ac.uk/~tmf20/TIES/NEW/intro.html. You can judge for yourself which of the new knots, if any, are worth taking seriously.

Fink and Mao described their findings in considerable detail in a lengthy report in Physica A and in a book, The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots.

Interestingly, similar random-walk models play an important role in studies of protein folding, which is Fink's main research interest. In the meantime, the researchers had novel knots to show off in Cambridge University dining halls, where jacket and tie are still required.

Originally posted: March 31, 1999
Updated: Sept. 26, 2005

References:

2005. A robot that ties the knot. American Society of Mechanical Engineers press release. Feb. 10. Available at http://www.asme.org/pi/pr/2004/2005/021005a.html. Additional information, including a video clip, is available at http://www.asme.org/education/precollege/whyknot/.

Fink, T.M., and Y. Mao. 2000. Tie knots, random walks and topology. Physica A 276:109-121. Available at http://www.tcm.phy.cam.ac.uk/~tmf20/TIES/PAPERS/paper_physica_a.pdf.

______. 1999. The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots. New York: Broadway Books. See http://www.tcm.phy.cam.ac.uk/~ym101/preface.html.

______. 1999. Designing tie knots by random walks. Nature 398(March 4):31. Available at http://www.tcm.phy.cam.ac.uk/~tmf20/TIES/PAPERS/paper_nature.pdf.

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

______. 1997. Knotted walks. MAA Online (Nov. 3).

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

Yong Mao's Web page is at http://www.tcm.phy.cam.ac.uk/~ym101/.