More ties than we thought

More ties than we thought Dan Hirsch1 , Ingemar Markström2 , Meredith L. Patterson1 , Anders Sandberg3 and Mikael Vejdemo-Johansson2,4,5 1 Upstanding Hackers Inc. 2 KTH Royal Institute of Technology, Stockholm, Sweden 3 Oxford University, UK 4 Jožef Štefan Institute, Ljubljana, Slovenia 5 Institute for Mathematics and its Applications, Minneapolis, USA 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 & 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 266,682 distinct tie-knots that seem tie-able with a normal neck-tie. Out of these 266,682, we also enumerate 24,882 tie-knots that belong to the regular sub-language. Subjects Algorithms and Analysis of Algorithms, Computational Linguistics, Theory and Formal Methods Keywords Necktie knots, Formal language, Automata, Chomsky hierarchy, Generating functions INTRODUCTION Submitted 13 February 2015 Accepted 6 May 2015 Published 27 May 2015 Corresponding author Mikael Vejdemo-Johansson, Academic editor Anne Bergeron Additional Information and Declarations can be found on page 13 DOI 10.7717/peerj-cs.2 Copyright 2015 Hirsch et al. Distributed under Creative Commons CC-BY 4.0 OPEN ACCESS 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. 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 then used this language to enumerate all tie-knots that could reasonably be tied with a normal-sized necktie. The enumeration of Fink and Mao crucially depends on dictating a particular finishing sequence for tie-knots: a finishing sequence that forces the front of the knot—the façade—to be a flat stretch of fabric. With this assumption in place, Fink and Mao produce a list of 85 distinct tie-knots, and determine several novel knots that extend the previously commonly known list of tie-knots. In recent years, however, interest has been growing for a new approach to tie-knots. In The matrix reloaded (Wachowski et al., 2003), the character of “The Merovingian” has a sequence of particularly fancy tie-knots. 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. Doing this allows for How to cite this article Hirsch et al. (2015), More ties than we thought. PeerJ Comput. Sci. 1:e2; DOI 10.7717/peerj-cs.2 Figure 1 Some specific tie-knot examples. Top row from left: the Trinity (L-110.4), the Eldredge (L-373.2) and the Balthus (C-63.0, the largest knot listed by Fink and Mao). Bottom row randomly drawn knots. From left: L-81.0, L-625.0, R-353.0. a knot with textures or stylings of the front of the knot, producing symmetric and pleasing patterns. Knorr (2010) gives the history of the development of novel tie-knots. It starts out in 2003 when the edeity knot is published as a PDF tutorial. Over the subsequent 7 years, more and more enthusiasts involve themselves, publish new approximations of the Merovingian tie-knot as PDF files or YouTube videos. By 2009, the new tie-knots are featured on the website Lifehacker and go viral. In this paper, we present a radical simplification of the formal language proposed by Fink and Mao, together with an analysis of the asymptotic complexity class of the tie-knots language. 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. For a comprehensive reference, we recommend the textbook by Sipser (2006). Recall that given a finite set L called an alphabet, the set of all sequences of any length of items drawn (with replacement) from L is denoted by L∗ . A formal language on the alphabet L is some subset A of L∗ . The complexity of the automaton required to determine whether a sequence is an element of A places A in one of several complexity classes. 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. Hirsch et al. (2015), PeerJ Comput. Sci., DOI 10.7717/peerj-cs.2 2/15 T Broad blade L Thin blade A C R W B Figure 2 Left/Center/Right. The parts of a necktie, and the division of the wearer’s torso with the regions (Left, Center Right) and the winding directions (Turnwise, Widdershins) marked out for reference. One way to describe a language is to give a grammar—a set of production rules that decompose some form of abstract tokens into sequences of abstract or concrete tokens, ending with a sequence of elements in some alphabet. The standard notation for such grammars is the Backus–Naur form, which uses ::= to denote the production rules and ⟨some name⟩ to denote the abstract tokens. Further common symbols are ∗, the Kleene star, that denotes an arbitrary number of repetitions of the previous token (or group in brackets), and |, denoting a choice of one of the adjoining options. THE ANATOMY OF A NECKTIE 1 There are neckties without a width difference between the ends. We ignore this distinction for this paper. In the following, we will often refer to various parts and constructions with a necktie. We call the ends of a necktie blades, and distinguish between the broad blade and the thin blade1 —see Fig. 2 for these names. The tie-knot can be divided up into a body, consisting of all the twists and turns that are not directly visible in the final knot, and a façade, consisting of the parts of the tie actually visible in the end. In Fig. 3 we demonstrate this distinction. The body builds up the overall shape of the tie-knot, while the façade gives texture to the front of the knot. The enumeration of Fink and Mao only considers knots with trivial façades, while these later inventions all consider more interest (...truncated)