Mathematician Calculates 177,147 Ways to Tie a Tie

“I have tried 10 or 20 of them, and most of them to be quite honest look kind of awkward,” he says

How to knot a tie Matrix style

Just in time for New York Fashion week, mathematicians have figured out how many times you can tie a tie: 177,147. Apparently inspired by The Matrix Reloaded and the unique tie knots dreamed up by the movie's costume designer, mathematician Mikael Vejdemo-Johansson at the KTH Royal Institute of Tehcnology in Stockholm, Sweden, decided to try and calculate just how many options someone wearing a necktie has.

This apparently is not the first time math has tried to tackle this question. In 1999, two mathematicians from the University of Cambridge did the same thing. But they came up with a much smaller number than Vejdemo-Johansson. By their estimate, there were only 85 different ways to knot that tie. So, where did the difference come from? Definitions, mostly. 

Fink and Mao assumed two things that Vejdemo-Johansson did not. Jacob Aron at New Scientist explains:

It turns out Fink and Mao had made two assumptions about tie knots that drastically reduced the number available. They assumed that you would only make a tuck – folding one end of the tie under the rest to complete the knot – at the end of a given tying sequence, and that all knots would be covered by a flat stretch of fabric. Those assumptions don't hold for the new set of knots, which can involve making multiple tucks midway through a sequence – and surfaces with many folds and edges.

If you remove those two assumptions, and set the number of times you could wind the tie before it became too comically short at 11 rather than 8 (which was where Fink and Mao drew the line) you wind up with 177,147 different ties. You can even generate random knots according to these rules at a website that Vejdemo-Johansson built. He doesn’t recommend that you go out to your next black tie event with one though. "I have tried 10 or 20 of them, and most of them to be quite honest look kind of awkward,” he told Aron.

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