Abstract
We extend the existing enumeration of neck tie-knots to include tie-knots with a textured front, tied with the narrow end of a tie. These tie-knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000). We show that the relaxed tie-knot description language that comprehensively describes these extended tie-knot classes is context free. It has a regular sub-language that covers all the knots that originally inspired the work. From the full language, we enumerate 266682 distinct tie-knots that seem tie-able with a normal neck-tie. Out of these 266682, we also enumerate 24882 tie-knots that belong to the regular sub-language.
Highlights
There are several different ways to tie a necktie (Fig. 1)
In a sequence of papers and a book, Fink & Mao (2001), Fink & Mao (2000) and Fink & Mao (1999) defined a formal language for describing tie-knots, encoding the topology and geometry of the knot tying process into the formal language, and used this language to enumerate all tie-knots that could reasonably be tied with a normal-sized necktie
Attempts by fans of the movie to recreate the tie-knots from the Merovingian have led to a collection of new tie-knot inventions, all of which rely on tying the tie with the thin end of the tie—the thin blade
Summary
There are several different ways to tie a necktie (Fig. 1). Classically, knots such as the four-in-hand, the half windsor and the full windsor have commonly been taught to new tie-wearers. Attempts by fans of the movie to recreate the tie-knots from the Merovingian have led to a collection of new tie-knot inventions, all of which rely on tying the tie with the thin end of the tie—the thin blade. We produce a novel enumeration of necktie-knots tied with the thin blade, and compare it to the results of Fink and Mao. Formal languages The work in this paper relies heavily on the language of formal languages, as used in theoretical computer science and in mathematical linguistics. Languages that are described by finite state automata are regular; languages that require a pushdown automaton are context free; languages that require a linear bounded automaton are context sensitive and languages that require a full Turing machine to determine are called recursively enumerable This sequence builds an increasing hierarchy of expressibility and computational complexity for syntactic rules for strings of some arbitrary sort of tokens. Each time the active blade is moved across the tie-knot—in front or in back—we call the part of the tie laid on top of the knot a bow
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