The 85 Ways to Tie a Tie

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The 85 Ways to Tie a Tie

Author	Thomas Fink and Yong Mao
Publisher	Fourth Estate
Publication date	November 4, 1999
ISBN	1-84115-249-8

The 85 Ways to Tie a Tie (ISBN 1-84115-249-8), by Thomas Fink and Yong Mao, was published by Fourth Estate on Nov 4, 1999, and subsequently published in nine other languages.

1 The Book
2 The Science
3 Knot Representation
4 Knots
5 Reviews
6 References
7 External links

[edit] The Book

The 85 Ways to Tie a Tie is about the history of the knotted neckcloth, the modern necktie, and how to tie both. It is based on two mathematics papers published by the same authors in the journal Nature^[1] and Physica D.^{[citation needed]} The authors prove there are exactly 85 ways of tying a necktie and enumerate them. Of the 85, 13 stand out: the four traditional knots (the four-in-hand, Pratt, half-Windsor and Windsor) and nine others, which the authors name.

The cover painting contains an amusing error: the tie knot shown is an impossible knot. The stripes on the knot and the wide blade of the tie cannot be oriented in the same direction - they must differ by a 90 degree rotation. (This assumes that the pattern repeats along the length of the tie, and does not reverse direction at some point.)

[edit] The Science

The discovery of all possible ways to tie a tie depends on a mathematical formulation of the act of tying a tie. In their papers (which are technical) and book (which is for the layman, apart from an appendix), the authors show that necktie knots are 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. Imposing the conditions of symmetry and balance reduces the 85 knots to 13 aesthetic ones.

[edit] Knot Representation

The basic idea is that tie knots can be described as a sequence of six different possible moves, although not all moves can follow each other. Moreover, there are two ways of beginning a tie knot, and two ways of ending a tie knot. These are summarized as follows. All diagrams are drawn as the tie would appear were you wearing it and looking in a mirror.

L = left; C = centre; R = right.

i = into the page; o = out of the page.

T = through the loop just made.

Above: The two ways of starting a tie knot. For knots beginning with Lo, the tie must begin inside out. For knots beginning with Li, the seam of the tie should lie against the collar. In both cases the wide end of the tie should be hung from the right-hand-side of your neck.

Above: The six tie knot moves. L, C and R represent moving the wide end of the tie to your left, center and right respectively. i represents moving the wide end down and in towards the shirt, while o represents moving the wide end up and away out from the shirt.

Above: The two ways of finishing a tie knot.

With this shorthand, traditional and new knots can be compactly expressed as below.

[edit] Knots

The thirteen useful knots (out of 85 total) described in the book, in order of size, are as follows. The knots are sometimes designated by their number alone, e.g., FM2 for the four-in-hand (FM stands for Fink-Mao). A knot is self-releasing if, when the thin end is pulled out through the knot, no knot is left.

Number	Sequence	Name	Self-releasing
1.	Lo Ri Co T	Small knot	No
2.	Li Ro Li Co T	Four-in-hand	Yes
3.	Lo Ri Lo Ri Co T	Kelvin	No
4.	Lo Ci Ro Li Co T	Nicky (self-releasing Pratt)	Yes
5.	Lo Ci Lo Ri Co T	Pratt	No
6.	Li Ro Li Ro Li Co T	Victoria	Yes
7.	Li Ro Ci Lo Ri Co T	Half-Windsor	No
8.	Li Ro Ci Ro Li Co T	Half-Windsor variant	Yes
12.	Lo Ri Lo Ci Ro Li Co T	St Andrew	Yes
18.	Lo Ci Ro Ci Lo Ri Co T	Plattsburgh	No
23.	Li Ro Li Co Ri Lo Ri Co T	Cavendish	No
31.	Li Co Ri Lo Ci Ro Li Co T	Windsor	Yes
44.	Lo Ri Lo Ri Co Li Ro Li Co T	Grantchester	Yes
54.	Lo Ri Co Li Ro Ci Lo Ri Co T	Hanover	No

[edit] Reviews

The book was reviewed in Nature,^[2] The Daily Telegraph, The Guardian, GQ, Physics World, and others.

[edit] References

^ Fink, Thomas M.; and Yong Mao (1999). "Designing tie knots by random walks". Nature 398: 31–32. doi:10.1038/17938.
^ Buck, Gregory (2000). "Why not knot right?". Nature 403: 362. doi:10.1038/35000270.