About a week ago, I realized that I didn't know how to tie a tie. I mean, I did something to the tie when I put it on, but I'm not sure what. It ended up looking alright though. Anyhow, I decided enough was enough, so I went online to learn for myself the full range of tie knots, expecting maybe three or four. I soon discovered that two physicists, Yong Mao and Thomas Fink, had set out in 2001 to unravel a mathematical model for understanding tie knots. Based on the supposition that there are three different moves one can make with a tie (cross to the left, right or go down through the top) and the two possible directions a tie can face, they figured out that there are exactly 85 different ways to tie a tie, about a dozen of which are really aesthetically pleasing.
I take a weird delight in seeing an element of fashion reduced down to mathematical principles to be dissected. By denoting each move by a pair of letters (Li, for instance, means go to the Left, facing In) Mao and Fink are able to set out the instructions for a tie knot simply and concisely. For instance, the Windsor knot is Li Co Ri Lo Ci Ro Li Co T, which translates to:
For a full explanation, plus how to tie all 85 knots, go to Fink's homepage, or buy his book The 85 Ways to Tie a Tie (not cheap; I recommend looking for it used). If you are further intrigued, read his summary Designing Tie Knots by Random Walks, or the full paper that inspired the book, Tie Knots, Random Walks, and Topology.
(I do not endorse the ad at top in any way, other than that I think it's funny.)