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

published on August 1, 2020

welcome to another mythology video today's video is about one of my own mathematical adventures my quest to pin down the mathematically best ways to lace shoes yes you heard right the guy who only ever wears Pickens dog and back

On a mathematical shoelace expedition this goes back to the time when my own kids were little teaching them to tie their shoelaces suggested to me to look at various mathematical aspects of lacing and tying shoes because of course

That's the crazy way mathematicians think we're always on the lookout for the mathematical soul of things no matter how big or small the whole thing started out as a bit of innocent fun but then i obsessed a bit happens didn't

Quit quite a bit and things got out of control first i ended up publishing an article about the best ways to lace shoes a bit of mathematical fun in the heavy duty journal nature effect which made the

Evening news on TV here in australia and resulted in a couple thousands of emails overnight and requests for interviews from all over the planet believe it or not two years later i published a whole book about shoeless mats for the

American mathematical society that's my daughter Lara on the cover well her feet anyway Lara's tying her laces and that's me up there in the corner with hair anyway time to get into it so what's the best way to lace your

Shoes let's begin by looking at some familiar lacings there that's a mathematical shoe a rectangular array of evenly spaced eyelet pairs looks comfy right a lacing consists of the same number of straight

Line segments as they are eyelets and these segments form a closed path that visits every eyelet exactly once just like in this first example now that's the crisscross lacing that most shoes come with here's a second most popular

Lacing this is zigzag placing and here are a couple of other examples we era that's a lacing that a friend of mine actually found in a French shoe shop a couple of years ago he is something else you've probably not seen before and

Here's one that it's safe to say has never appeared in a shoe shop pretty insane right well don't worry I didn't send my kids off to school with shoes laced like that how about this one here well seems to

Fit the bill right the lace forms a closed path visiting every aisle at once but do we really want to call this a lacing something's clearly not right can you pinpoint it well the problem is there are eyelets that don't help us

Pulling the two sides of the shoe together like this one here right doesn't do anything so in a real lacing we want at least one of the two segments of the path ending in an eyelet those ones there to connect to the other side

Of the shoe right that lacing over there works there are also extra special lacings like the two most popular lacing the criss cross and a zig zag lacings in these two lay since every eyelet contributes twice to pull in the two

Sides of the shoe together I call such lacings tat in other words in a tight lacing every segment of the closed path connects the two sides of the shoe and there are no vertical segments at all this has the effect that as you travel

Along the lacing you constantly zigzag back and forth between the two sides of the shoe like this ok there's eggs eggs eggs eggs eggs eggs the exact zigzag zigzag see like okay down to work I've shown you

Five different lacings for a shoe with seven eyelet pairs so let's list them all no let's not it turns out there are more than 38 million such lacings almost 2 million of these are tight lacings the number of

Lacings increases rapidly with the number of eyelets for example for God shoes with 100 eyelet pairs we get in the order of 10 to the power of three hundred fifty four different lacings now the formula for the exact number of

Lacings is this scary-looking monster here and my first challenge for the Keen among you to turn the number of tight lacings of a shoe with five eyelet pairs remember tight lacings are the ones that go back and forth and the challenge for

The super keen what's the general formula for the number of tight lacings as always post your answers and ponderings in the comments okay so we have our lacings sort of that means we can now ask which of all the gazillions

Of lacings of a particular shoe is the very best of course there are many types of best people often say french things are best so maybe the french shoe shop lacing is best by default anyway ignoring the French I thought there were

Two natural interpretations of the word best to consider first the shortest lacing and second the strongest lacing make sense anyway it was these two interpretations of best that I went for okay so first what is the shortest

Lacing of a given shoe easy right just of computer list all possible lacings of this shoe and figure out which one is the shortest with 38 million lacings at a modern computer that's definitely not a problem at all

However even with seven eyelet pairs there are infinitely many different shoes to consider depending on the spacing of the eyelets right and of course it's true mathematicians we are honorbound to consider all infinitely

Many different spacing of all the infinitely many different mathematical shoes shoes with just two allit pairs three four pairs etc all right for now best means shortest so

What are the shortest ways to laser shoes well with all these infinitely many possibilities you'd expect many structurally different shortest lacings not true this came as a real surprise to me but no matter the islet spacing the

Shortest solution is essentially the same first the answer for the tight lacings remember those are relations that zip back and forth between the two sides of the shoe it turns out the shortest tight lacing is always the

Crisscross lacing so the most popular lacing also turns out to be the shortest for each of those infinitely many possible mathematical shoes nice to know isn't it the first to prove this shoelace theorem was the mathematician

John Horton in overt sync 1995 in an article in the mathematical Intelligencer a couple of years before I got interested in shoelace nets see I'm not the only one now what about general lacings here turns out the absolutely

Shortest lacing is always what I call a bowtie lacing bowtie lacings have horizontal segments at the top and at the bottom and the rest of the tickets come in pairs either making short parallel pairs like this 1 2 3 or short

Crosses like this 1 2 3 the basic bowtie lacing of a shoe starts with a parallel pair at the bottom and then the parallel pairs and the crosses alternate okay in the case of an odd number of eyelet pairs like in the shoe over there apart

From the basic type of bowtie lacing there are also these variations there there and there obviously since they are just a few segments shuffled around all these variations have the same overall lengths and so if one is of shortest

Lengths then all of them are okay so this is the situation for mathematical shoes with an odd number of eyelet pairs for an even number of eyelet pairs there is only one bowtie lacing just like for the case of six eyelet pairs this one

Here and this particular instance of this type of shorted lacing also inspired the name bowtie lacing look yes the bowtie okay so I've told you the winners of the shortest lacing competitions but how would we prove

Something like this well most mathematicians presented with this task will place it within a whole circle of such puzzles known as the Traveling Salesman problems let's say you've got a number of towns like for example all the

Major towns in the state of Victoria where I live yeah that's all of them the Traveling Salesman problem asks for a shortest round trip that visits every one of these cities once he is the solution to this problem and here's the

Solution of the same problem for more than 18,000 towns in Germany fantastic stuff isn't it and there is one very striking aspect of these solutions have a closer look can you see it yep I'll bet a lot of you got it

There's no crossings and that turns out to always be true no solution to a Traveling Salesman problem will ever intersect itself it's actually really easy to see that crossings can't happen just imagine if it did then we'd have a

Closed path that intersects itself somewhere like this color the crossing segments like this but now it's clear that if we replace both the blue and green parts by straight-line segments we get a shorter closed loop through all

The points that means any closed path with an intersection like this can always be shortened and so the shortest closed path the solution to our travelling salesman problem cannot have any self intersections back to our shoes

What is the solution to the travelling salesman problem for a shoe yep it's just the boring non lacing loop that we stumble across earlier so how can insights about the general traveling salesman problem help solve our shortest

Shoe lacing problem well have a look at this lacing can this possibly be a shortest lacing doesn't seem likely does it and we can actually prove that it's not the shortest we can create a shorter lacing just by doing a little bit of

Rewiring and straightening just like in the Traveling Salesman problem right yeah straighten and we've got something shorter okay what about this new lacing can this won't be the shortest nope we can

Shorten again with some more rewiring right just straighten out the green and blue bits yeah voila once again a shorter lacing what about this third lacing well I'm sure you can guess and here we go again okay and shorter oh

Damn no lacing but we could fix that all good so finally after three rewiring we've ended up with one of our bowtie lacings this rewiring idea was at the heart of my first proof that the bowtie lacings are the shortest very simple

Idea right but in the end completely nailed down the proof it was a bit tedious because of the ridiculous number of different cases that had to be considered there just to give you an idea that's one of the diagrams listing

Two different cases in one part of the proof of course I was pretty happy of having found my rewiring proof but somehow my brain subconsciously kept working on the problem and about six months after I finished the first proof

I woke up one morning in the middle of dreaming about another much shorter proof now pretty much all mathematicians have had dream proofs and it's wonderful you wake up really excited and then two minutes later you realize your dream

Proof is completely ridiculous just like your dream of suddenly being able to levitate or co-starring in a movie with Scarlett Johansson but amazingly my dream lacing proof really worked for the really keen ones among you I'll

Illustrate a special case of my dream proof at the end of this video proving to you that the crisscross lacing is the shortest tight lacing before that let me show you some other really neat shoelace facts remember my shoeless book have a

Look at the subtitle a mathematical guide to the best and worst ways to lace your shoes yep I also pinned down things like the longest lacing is among the different classes I know pretty strange but totally the thing to do if you are a

Mathematician just like worrying about lacings with a hundred eyelet fares what turns out to be the longest tight lacings behold there devil lacings pretty devilish huh here are the devil's for small shoes up to six eyelet pairs

And what are the longest lacing is overall well for short shoes shoes with short horizontal spacing the longest placings are still the devil lacings for long shoes shoes with long horizontal spacing those are the angel Asics I've

Drawn the wings curved to make the pictures less ambiguous and more angelic fun no well I don't care I think it's fun but what about real shoes I hear you asked like those on the cover of the book real shoes aren't flat eyelids on

Points laces are made up of line segments and so on well it turns out that the shortest laces for ideal mathematical shoes are surprisingly robust and are also the shortest lacings for most real shoes this is particularly

True for the shortest tight lacings the crisscross lacings at least for any shoe I've ever owned the crisscross lacing has always been the shortest tight lacing but if you're really tired of my pure mathematical weirdness and you want

To find out everything conceivable and inconceivable about real shoelaces I've got just the site for you you absolutely must visit Ian's shoelace site which my friend Ian Fagan has been obsessing over for ages insane lacings ways to tie

Laces shoelace box shoelace apps etc yep I am the pure shoelace knot Ian is the applied shoelace knot the weirdest thing is that Ian lives just a couple of kilometres away from me here in Melbourne that definitely makes

Melbourne the shoelace capital of the world doesn't it now your next easy challenge for the day head over to Ian's website and find out whether you belong to the half of the people on earth who are tying their shoelaces incorrectly

Intrigued okay after meeting with Ian we're definitely in good shape with the shortest and longest lacings so what about the strongest lacings also strongest in what sense

I'll postponed in what sense and just start by telling you the surprising answer to the first question what are the strongest lacings it turns out that the two most popular lacings the criss cross and a zig zag are also the

Strongest lacings for short shoes like the one over there the strongest lacing is the criss cross lacing as you stretch the shoe the strongest lacing stays criss-cross up to a certain changeover point at this point the criss cross

Lacing is as strong as the zig zag lacing stretching beyond a changeover point the zig zag lacing is uniquely strongest this basic behavior is the same no matter the number of eyelet pairs just the change of a spacing

Changes with that number they're more eyelets there are the quicker we reached a changeover spacing okay so you know they're strongest lacings you just don't know what strongest means so I'll tell you have a

Look at this picture see the pulley on the right ideally and lacing is a pulley like this turn sideways when a shoelace is tied we assume ideally that the tension everywhere along the shoelace is the same this tension then translates

Into the tension of the pulley in the horizontal direction that is the direction in which the two sides of the shoe are being pulled together so for a given tension throughout the lace then the larger the horizontal pulling

Tension the stronger we say the lacing is proving that the crisscross and zigzag laces are strongest is super nitty-gritty and also only really become feasible after I had my dream so let's talk about that now

To finish off I want to show you my dream proof really quite special a proof hatched in a dream that isn't completely crazy and actually works that had only happened to me once before in what follows I'll focus on proving to you

That the crisscross lacing is the shortest tight lacing of the shoe over there this proof is completely general and works for any number of eyelets and any spacing anyway onto proving that the crisscross lacing is the shortest tight

Lacing what I'll do first is to just give you an outline of the proof while I go over this outline don't get hung up on any details just run with it and try to understand gist of what's going on I'll flesh out the details afterwards

And things should come together nicely then okay let's say over there that's a list of all our tight lacings let's explore all of them into the different segments they consist of they explored explore explored and so on we'll see

That these segments collections resulting from the explosion of tied lacings share for easy to see properties first each collection consists of ten segments second each collection has at most five horizontals I'll tell you the

Remaining two properties in a minute now what we can do is to study these collections in their own right to have a close look at all collection of segments that satisfies these four special properties why do that wait and see I've

Named these special collections exploded lacings apart from the exploited blessings arising from real shoes we see that there are lots of others like for example this one here the lengths of an exploded lacing is

Just the sum of the lengths of all its segments this means that the length of a real lacing is the same as the length of its exploded counterpart pretty obvious right now comes the nifty part of the proof and the whole point of this

Exploded stuff although there are a lot more exploited lacings then the real lacings we start with it will be extremely easy to figure out what the shortest exploded lacing is why is that well it comes from the overall lack of

Structure of exploded lacings which makes them very easy to manipulate the shortest exploded lacing turns out to be the explosion of the crisscross lacing consisting of two horizontals and eight short diagonals it's also easy to see

That the crisscross lacing is the only tight lacing consisting of two horizontal and H or diagonals consequently since the exploded crisscross lacing is shortest among all explored lacings the crisscross lacing

Must be the shortest real tight lacing really cooler so my dream proof dodges all the complicated structure of lacings by taking a shortcut through some strange world of phantom laces how neat is that only in a dream

Okay ready for the details where you're not here we go first let's give number labels to the different segment types that can occur we'll call horizontal segment zeros next we call segments that dries exactly one

Vertical step once that's a one there and that's another one and you can guess the rest the segments that rise to vertical steps are the tools and there are threes and finally there force forth are the longest possible segments for

Our particular shoe so our crisscross lacing consists of two zeros the horizontals and the top and the bottom and eight ones the zigzag lacing consists of five zeros for ones and one for now here are four simple

Properties shared by all the sets of numbers that correspond to lacings first as we've already mentioned all these lacings consist of ten segments so there are 10 numbers 10 non-negative integers second the fourth are the longest

Possible segments and so four is the largest possible number third and again we already noted this there will be at most five horizontal segments that means we'll have at most five zeros fourth and finally the distance between the top or

On the bottom of our shoe is four spaces and traveling around the lacing you must go at least once from top to bottom and back from bottom to top this means that the sum of our lacy numbers is at least two times 4 that's 8 okay so let's call

Any set of non-negative integers that satisfies these four properties and explode it lacing so for example this is an explorer lacing let's check 10 non-negative integers tick nothing bigger than 4 tick at most 5 zeros tick

And finally the sum of the numbers is 23 which is greater than 8 tick obviously in a real lacing there at most two of the longed-for segments so since our exploded lacing contains four fours it cannot correspond to real lacing in

Fact as I already mentioned in the intro many exploded lacings do not come real lacings okay getting there now remember the lengths of an exploded lacing is simply the total sum of the lengths of the segments corresponding to

The numbers for example for our exploded lacing here there are three zeros contributing the length of three horizontals add to that the length of a one segment plus two times the length of a three segment and finally add four

Times the length of a four segment all under control easy enough so far right okay and now for the easy proof that exploring the crisscross lacing gives the shortest exploded lacing overall

Three easy peasy steps first a shortest exploded lacing has some exactly equal to it why because any exploded lacing that has a sum greater than eight can be made into a shorter exploded lacing by

Replacing some of its numbers by smaller numbers for example in our exploded lacing here we can make these replacements two plus three plus one plus one plus one that's eight now second easy step because there are ten

Non-negative integers that add to eight they cannot all be one or greater this means some of these numbers must be zeros right but to sink in all okay good so that means a shortest exploded lacing contains zeros third and

Final easy-peasy step okay so maybe not so easy peasy but it's not too bad you'll see think again about our exploded crisscross lacing it contains only zeros and ones now can a shortest exploded lacing contain a number greater

Than one let's see in our example there is a three let's grab this three and one of the zeros and picture them together can you see how to shorten what we're looking at here yep just a little traveling salesman rewiring and

Straightening here we go there the green is a two and the pink is a one this means that if we replace the 0 and the 3 in our exported lacing by a 1 and a 2 we get a shorter explore it lacing right also notice that the sum of the numbers

In this new lacing is still eight one plus two plus two plus one plus one plus one is eight can be short now and you explore it lacing again absolutely we can use the same trick to replace a 0 and a 2 with two ones like this

Yeah that's a 1 and a 1 replace ok repeating this step once more we can get rid of the other two the last number greater than 1 go for it exactly same step right to ones replace the so now we're totally down to zeros and

Ones and this always works we can always shorten until there are only zeros and ones and so the shortest exploded lacing overall must contain only zeros and watts but since the sum is 8

There must be exactly 8 ones and two zeros this means that the shortest exploded lacing is the exploded crisscross lacing Tara pretty cool dream prover well actually there's one more T to cross last thing we do we've proved

That every lacing consisting of two zeros and eight ones is a shortest tight lacing and the crisscross lacing is definitely one such lazy but maybe there are others well let's see okay time to play with our building

Blocks what can we make with our two zeros and eight watts let's start at one corner eyelid we must have exactly two segments meeting at that eyelid and the segment's must be different so for that corner eyelid it's clear that one of the

Segments is a zero and the other is a one so one of the zeros has to be at the bottom similarly looking at a top corner the other zero must be located at the top but now our hands are forced all we have left are ones and we have no choice

In how to place them there and there and there and there and that really finished the proof very nice isn't it and as I said it's very easy to adapt arguments are just presented to also prove that the bowtie lacings are the shortest

Lacings overall and my proofs of the strongest lacing theorems are also based on explored lacings so mathematical dreams can become mathematical reality who would have thought and that's all for today pleasant dreams


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