![]() ![]() Symmetry in the case of tie knots can refer to two possible qualities: visual symmetry (the extent to which the knot appears to be shaped identically on the left and right side), and mathematical symmetry (the number of L and R moves being as close to equal as possible). (In the Nature paper, the lower bound was placed at a more restrictive 1:4, eliminating the knot classes containing the Kelvin, Victoria, and Grantchester this was likely revised specifically in order to include the Victoria/Prince Albert, which has fairly extensive historical documentation.) The most representative knot in each remaining class was then selected on the basis of symmetry and balance. ![]() There are a total of 16 classes, ranging from three moves with one center to nine moves with four centers, but only classes in which the ratio of centering moves to total moves is 1:6 or greater contain an aesthetic knot, eliminating three classes (ten knots) for a remaining 13 classes, with 75 knots. ![]() Due to the triangular nature of tie knots, the number of centering moves must necessarily be less than half the total number of moves. Knots with fewer centering moves, less than one-third of the total, appear narrower and more elongated, while knots with more centering moves appear wider and more squat. For example, the four-in-hand is a four-move, one-center knot, while the half-Windsor is a six-move, two-center knot. In Fink and Mao's classification, each of the 85 tie knots belongs to a particular "class", which is defined by its total number of moves and its number of centering moves. They made their selection based on three criteria: shape, symmetry, and balance. Of the 85 knots possible with a typical necktie, Fink and Mao selected thirteen as "aesthetic knots" suitable for use. Note that any knot that begins with an o move must start with the tie turned inside out around the neck. With this shorthand, traditional and new knots can be compactly expressed, as below.
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