Evidence of the earliest known knotted neckcloth lies in the tomb of China’s first emperor, Qin Shih-huang-di (259 – 221 BC). In an elaborate underground mausoleum, 7,500 sculptured terracotta soldiers each sport knotted neckerchiefs.

Two Cambridge University mathematicians, Thomas Fink and Yong Mao, have researched the physics and equations of tying a necktie. Their research paper was published in the respected science journals Nature and Physica A.

The authors prove that there are exactly 85 ways of tying a necktie using the conventional method of wrapping the wide end of the tie around the narrow end. They describe each one, and highlight those that they determine to be historically notable or aesthetically pleasing.


The mathematics


The discovery of all possible ways to tie a tie depends on a mathematical formulation of knots being equivalent to persistent random walks on a triangular lattice, with some constraints on how the walks begin and end. Thus enumerating tie knots of n moves is equivalent to enumerating walks of n steps.

The basic idea is that tie knots can be described as a sequence of five different possible moves: left (L), centre (C) right (R), into (i), out (o), and through the loop (T). With this shorthand, traditional and new knots can be compactly expressed. Any knot that begins with an o move must start with the tie turned inside out around the neck.

The authors turned their academic paper into a popular book which is a delight to read. Fink and Mao provide an informative history of tie-knot evolution. They also provide much more – a guide to taste in knot tying. The book shows which knot works best with a given tie and collar, and shows that tie knots that can be enjoyed as things of beauty in themselves.

Fink and Mao have performed a great service for civilization, doing for tie-knot tying what Isaac Newton did for the motion of the heavens: lifting it from the darkness of secrecy, ritual and superstition to the light of rational, scientific good taste. The scientific force of the work is that Fink and Mao have created a formal model that captures the salient characteristics of tie-knot tying in the real world, and have then analysed the formal model, guided by the scientific lights of simplicity and symmetry. Their model predicts the knots most commonly used, and provides several new possibilities.

Fink and Mao have obeyed the imperative of the scientific entrepreneur: create a niche, and then fill it. Their book is the definitive work on tie knots, and on one of the most common applications of knot theory.

The applications of knot theory are infinite: a test tube of DNA may contain billions of knots, but sometimes they are hard to see. Polymers in general may gain many of their characteristics from tangling, knotting and linking, but this may not be apparent when you are holding the material in your hand. Magnetic field lines are often knotted, linked or otherwise entangled. But now imagine the morning dressing routines around the world – imagine how many tie knots are tied in a day.

Fink and Mao have shown that it is possible to be both smart and smart – in brains and in style.  Symmetry in tie knots refers to two qualities: visual symmetry (the knot appears to be shaped identically on the left and right), and mathematical symmetry (the number of L and R moves being as close to equal as possible).


(Source: The 85 Ways to Tie a Tie, by Thomas Fink and Yong Mao, published in 1999)

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