Reading Ξ’s post, I was struck at first by strong memories of my youth, learning various knots in scouting, and thought about how I don’t really tie knots very often anymore (Friday’s expedition to come home with a holiday tree notwithstanding).

But then again, most adults *do *tie knots on a regular basis. Many men wear cloth neck ornaments, and most men and women are faced with tying shoes — if not their own, then their childrens’.

I remembered reading about the various ways to tie a necktie (more on this issue later this week), but didn’t know much about tying shoes. I now happily know a lot more about shoe tying and shoe lacing, thanks to Ian’s Shoelace Site, a fantastic resource written and maintained by Ian Fieggen. Much of the site consists of practical advice:

- How do you tie your shoes?
- Are there alternate methods of shoe tying?
- Are there one-handed methods?
- Why do shoe laces come undone, and how can this be avoided?
- Why are my laces crooked?
- Are there alternate methods of lacing shoes?

While much of the site is practical, some has a decidedly mathematical bent, and while Fieggen avoids technical detail, he does quote a variety of counting and combinatorial results related to shoe tying and lacing problems. These intruige me, and I want to explore some of these problems in forthcoming posts.

I’ll close this introduction of the topic with a question: How many different ways are there to lace your shoes? (Assume that you have twelve eyelets, six on each side, symmetrically placed. Assume your lace is sufficiently long, and that each eyelet will have the lace pass through it exactly once. Further, the free-ends of the laces should both be at the top of the shoes.)

Answer and further explorations to come….

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This entry was posted on November 26, 2007 at 12:01 am and is filed under Knots. You can follow any responses to this entry through the RSS 2.0 feed.
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December 5, 2007 at 6:28 pm |

So, when you are refering to lacing your shoes, are you counting going through the eyelets in different directions – the bottom up vs. the top down – as different paths? Just curious. =]

December 7, 2007 at 10:22 am |

That’s part of the allure of the applied question — it is open to so many variations!

One could start by counting the number of lacings ignoring direction at each eyelet; then counting direction (multiplies the first result by 2^12).

The next level of complication is worrying about crossings of laces. If we assume the laces cross each other in the order they’re laid down, we need to enumerate all possible sequences for a given lacing. If we are allowed to weave a lace through the laces it crosses at the time we lay it down, the enumeration becomes quite a bit more intricate.

If laces are allowed to twist around previous laid laces, then there are infinitely many lacings….

There are other variations as well….

More to come under a separate post.