What is the best way to lace your shoes? Dream proof.

A blast from the past. A video about my fun quest to pin down the best ways of lacing mathematical shoes from almost 20 years ago. Lots of pretty and accessible math. Includes a proof that came to me in a dream (and that actually worked)!

0:00 Intro
1:31 What's a mathematical lacing?
4:42 What does "best" mean?
5:15 What is the shortest lacing? Crisscross and bowtie lacings.
8:42 How to prove that the shortest are the shortest? Travelling salesman problem
12:36 What are the longest lacings? Devil and angel lacings.
13:48 What about real lacings?
15:16 What are the strongest lacings?
17:17 Can proofs hatched in dreams be true?

Some links:
Ian's shoelace site
and his explanations of what's wrong with the way a lot of people tie their shoelaces

John Halton's proof that the crisscross lacing is always the shortest tight lacings
Halton, J.H. The shoelace problem. The Mathematical Intelligencer 17 (1995), 37–41

My shoelace article in Nature

A preview of my shoelace book at Google books

Here is a page on the German travelling salesman problem that I mention in the video
I actually got the number of cities a bit wrong. It's 15,112 cites and not 18000.

My article on shoelaces was inspired by this fun article by Thomas Fink and Yong Mao about Designing tie knots by random walks (also in Nature)
The extended version
They also wrote a really nice book about tie knots

Enjoy!

Burkard

[Mathologer]

What a crazy first half of the year! Pretty much did not have any time for anything but trying to survive the first semester here in Australia :( Now, hopefully, with most of my teaching out of the way, I'll have a bit more time for Mathologer in the second half :)

A blast from the past. This is a video about my fun quest to pin down the best ways of lacing mathematical shoes from almost 20 years ago. Lots of pretty and accessible math(s). Includes a proof that came to me in a dream (and that actually worked)!

Enjoy!

[jeffborders]

When dealing with actual shoes what goes commonly overlooked is the order in which the laces overlap at the intersection. It matters immensely. For experiment tie your shoe with standard zigzag but make sure every overlap lays right over left. Walk around for a day and then try again with overlaps left over right now. You'll notice they pull vastly differently on the foot. The lace overlapping the other lace is free to adjust slightly while pinning down other down restricting its movement. I have a bony knuckle top cent of my feet so I make sure the overlaps at this spot are laced a certain way. It's so much more comfortable than overlapped the other way. All this while never changing the actual layout of the lacing.

[MrLordZenki]

As a programmer, I sometimes wake up with "dream functions" lol

[jiggy]

Am i really watching a 29 min video at 3am about the best way to tie your shoe laces ?
YES.

[genelong]

When I was a kid, I created my own lacing, where each eyelet pair had one string going across, out the first eyelet and in the corresponding one, and that was all that was visible - no diagonal lacings. Everything else was under the eyelets and did not cross, and were thus invisible. I was very proud of it. I still lace my shoes with this style.

[dangnabbit]

This video has what most ML videos lack, that bit of enthusiasm. Glad you can't fake it but this feels like the best ML video I have seen in a long time.

[MrMutebe]

I haven't seen a comment on this yet but the intro with the "o" lacing connecting with the word Mathologer right at the beginning of the video put a smile on my face.

[Frownlandia]

I think I independently discovered the devil lacing when I was weaving my shoelaces so as to use up extra lace length and turn lace-up shoes into slip-ons. With the right goal in mind, it can be the best way to lace up your shoes!

[sylsummery]

I’d like to add to your two criteria for the “best” lacings: shortest and strongest, please. As I am a sneakerhead and lacing is an intrinsic aspect of one’s “stylishness”, that is a top criterion, though quantifying it might be a matter of subjectivity! Thank you for your studies and presentation 😄👏🏼👌🏼

[theprogrammer]

I've been tying the "Ian's knot" ever since 5th grade (roughly 11 years ago) and only today I just found out where the name came from.

[richardbloemenkamp]

Some elements to consider when defining the best lacing:
- Equally tight at the top in the middle and at the bottom. The X-cross lacing tends to tie tight at the top and not at all at the bottom, when you pull the laces because of the friction at the holes.
- Minimal vertical tightness
especially asymmetrical vertical tension. Here the zag-zag lacing pulls the shoe out of shape.
- Strength at the bottom, in the middle and at the top
- Simplicity to remember
- Aesthetics, symmetry, originality
It seems the best lacing would somehow hold the middle between a x-cross and a zigzag lacing with all the 1 elements facing outwards.

[nemesisurvivorleon]

I have trouble focusing on any particular youtube video but a full 30-minute nerdout about shoelace math is what grips me to the end. dude, what.

[piscopopasco]

0:06 Just started and I already pressed like for the t-shirt

[gabor]

When I saw Burkard with hair, my head turned into a black hole.

[Adityarm.]

Apart from all the extremely beautiful ideas, Your style always makes me happy. Wholesome content.

[benjaminbrady]

I was hoping for a crazy and unexpected method that I could start using today but it seems the criss-cross method reigns supreme (probably should have seen that coming). At least I can check out Ian's site and see if there's some other weird tricks to use!

[LaMirah]

That "weird" french lacing at 2:09 is the one I've always used and for many years thought everyone else did, too. It is quite surreal to me that the "criss-cross" lacing may be the most common, _especially_ with dress shoes in formal settings.

[ProfShoelace]

G'day, my old mate and fellow Melburnian!
Fascinating video that's natually right up my alley. A couple of comments:
1. I was surprised that your proof didn't use signed integers to refer to a positive or negative slope. That way, it would be trivial to add up all the signed integers and ensure that the result should equal zero.
2. In the real world, I also have to consider the eyelets as being more than just a node in that it has a distinct entry and exit, one of which is on the inside and the other of which is outside.
3. I liked your optimizations based on the "Travelling Salesman" solution. When I work on lacings with multiple passes through eyelets, I use a similar optimization pass to simplify lacings.
Thanks also for referring folks to my website for the more practical aspects of real-world shoe lacing.

[hooya]

Whenever I feel smart, I watch Mathologer to knock my ego down a few pegs.

[avoirdupois]

This is the best video about lacings ever. I love the production, the editing, and the segments. Thanks for the great content!