The Windsor or the Half Windsor are the two most-used necktie knots, pretty much the standard for day-to-day school and business attire. Chances are, you are taught how to do it once, then forever continue to use the exact same method for the remainder of your life -- even though prior knowledge suggests you could mix it up a bit of you wanted.
In 1999, two mathematicians, Thomas Yink and Yong Mao, examined the actions involved in tying a necktie and calculated there were 85 different ways to do so. However, a new team of mathematicians has trumped their research. Mikael Vejdemo-Johansson of the KTH Royal Institute of Technology in Stockholm, Sweden, along with a small team of mathematicians, found that Fink and Mao had left out some possibilities.
"We extend the existing enumeration of necktie knots to include tie knots with a textured front, tied with the narrow end of a tie," Vejdemo-Johansson wrote in the abstract of the team's paper, titled "More ties than we thought." "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)."
With this discovery, the team realized something wasn't quite right, so they had a look at Fink and Mao's research. They realized that Fink and Mao had restricted the number of tucks that occur at the end of knotting the tie to just one. They had also made the assumption that any knot work would be covered by a flat section of fabric, and restricted the number of windings to just eight.
Armed with this information, Vejdemo-Johansson's team adjusted the parameters of Fink and Mao's language and calculated that the number of possible knots is much, much higher than the previous calculations: 177,147, to be precise.
"We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free," the team wrote. "It has a sub-language that covers all the knots that inspired the work, and that is regular. From this regular sub-language we enumerate 177,147 distinct tie knots that seem tieable with a normal necktie. These are found through an enumeration of 2,046 winding patterns that can be varied by tucking the tie under itself at various points along the winding."
The average US lifespan for men is 77.4 years. That's 28,271 days, roughly correcting for leap years. You would have to tie more than six different knots a day for your entire life to get through them all.
If you care to decipher the team's language, you can give it a shot by taking a look at the research paper on the site arXiv.org here (PDF).