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!
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.
Absolutely right, there are lots of other considerations that go into optimising something as seemingly trivial as lacing and tying your shoes. While I was obsessing about this material I did come across a lot of other interesting insights. Definitely also check out Ian's site :)
@@VaradMahashabde It's too much hassle to mirror the knots and bows, so I tie both shoes the same way. Thus, it makes more sense to lace them the same way too, rather than mirrored. If the lacing is mirrored but the knot is not, it's what we call "not elegant".
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.
Currently, it's 3:14am here and I'm procrastinating - I should be doing maths homework but instead I'm watching Mathologer videos. Arguably, the best way to procrastinate.
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.
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.
>- Equally tight at the top in the middle and at the bottom. I came here to say this. "Shortest" is not the "best" in my opinion. We have to consider the function of laces, to allow you to slip your foot in, and then tighten the shoe so it is evenly stable on your foot. None of that involves how short a lacing is.
Equal tightness is the most important to me. Second is that the force on the upper pair of holes is low. For shoes without eyelets, that is the weakest part. I use the "loop back" technique. (The lace on the right loops around the lace on the left and then returns to the right. The lace on the left returns to the left.) This forms a little pulley that halves the force on the top holes.
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.
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 😄👏🏼👌🏼
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!
I sometimes dream of the halting function that would determine for any machine if it halts or not. Ahhh, such dreams, dreams that will never come true...
how is being a programmer? i have the intention of studying to becone a game designer/programmer and im truly interested into how the programming business and worklife is.
Wow, this is a great application using math. I also like math modeling myself, and this video is in the same context of my interest so much. Great job and nice explanation!
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!
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.
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.
Great stuff! I did find myself looking at the zig-zag, and thinking - ugh, that's the one that gradually de-centres the lace each time you tighten it until it's moved so far out of centre you have to take the time tease it back into position. It can also easily look wonky, as you can pull it out of shape pulling on the long diagonal. Enter what you called the French lacing, which I use on boots, Doc Marten's, that kind of thing - horizontals are always on top, diagonals always underneath; you just see the horizontal pieces neatly perpendicular to the seam since the diagonals are hidden, but at the same time solving both the de-centring problem and the skew problem. 8-)
Yeah, it took me a while to see it from the diagram, but I started doing the French lacing since finding it in a store because it looks nice. I like to imagine that it's less stressful for the lace to be getting pulled against the edges of each side, but when has a lace ever broken there?
In recent years, when I buy shoes they tend to be laced in strange ways so that I always have to relace them (to criss-cross lacing). I wonder why this is. Are there lacings that can be done FASTER in the factory? Have you ever considered speed of lacing as a criterion?
I square lace my shoes; absolutely suboptimal for efficiency, but tops in comfort if you want control over where the shoe compresses. I could see a case for square lacing via machine as there's no need to weave the laces, they simply lay flat over each-other.
Finally a video I could actually follow the whole way through. The section on strength was particularly satisfying, because the first thing I thought was that not all tight lacings are created equal in real shoes.
Wouldn't the preferred lacing simply be the one that has the least friction throughout, and so, requiring the least "loop-pulls" to both loosen and tighten the shoe?
You really the best man! 🙌 Stay cool! Your contents really hits me hard and inspired me to create my own channel and be an inspiration also to others. More power everyone
This is exactly the type of video I love! I didn’t even go through the step of wondering at myself for enjoying a shoelace video! I ponder this type of stuff too! Perhaps there a mathematician struggling to come out!😁 Thanks for the great video!
Fascinating example of modeling something from the real world into something formal :D. For me this is teaching a certain way of thinking. Thank you for your work! Cheers!
What about for lacings where the holes don’t have to be directly opposite each other? The zigzag lacing shown is very similar to spiral lacing, used for stays/corsets until some time in the 19th century, except for two things: the holes (except for the top and bottom pair) are offset so that the crossing angle stays constant, and the lace is simply tied off at the top and bottom, with no long lace running between them, but I suspect that’s also the case for the shoes. EDIT: after hearing about which lacing is strongest, I have to take a look at whether the typical distance between eyelet became denser over time, in which case it would make sense if the bones bodices of the renaissance were spiral laced, but the Victorian corset crisscross laced.
Was coming to mention spiral lacing myself, but the context I learnt of it was arming garments which bore the weight of armour (usually arm or leg) tied to them.
I've always laced my shoes the same way as your 'french shoe shop' example at 2:10 - it is not always about 'shortest' or 'strongest' but also 'neatest'. The top view of this lacing shows clean horizontal lacings - which are easy to grab and tighten. Unlike the zig-zag lacing, it gives equal tension to each half of the lace, while producing a similar overhead appearance. However, love the maths you demonstrate here and links to the TSP problem.
I love recreational maths problems they're wonderful, thanks for sharing your little passion! Consequently, I learned that I had actually given up tying the starting shoelace knot entirely! (Straight to the bow, no slipknot)
So essentially if we were to rank lacings on strength and length 1 being the best and infinite being the worst we're looking for the lowest sum of the two factors to have the optimal lacing. The natural choice here would be the crisscross lacing. The crisscross lacing at best has rank 1 in both situations for a sum score of 2 and at worst it falls behind in strength on the zigzag lacing for a sum score of 3. The zigzag lacing at best will be 1st on strength and 2nd in length for a sum score of 3 which is merely equal to the crisscross in the same case. The case where they do equal at a sum score of 3, it would be a competition of preference where some situations might demand short laces by lack of material but some may prefer a higher tensile strength because they want to climb a mountain today.
Never had the patience to do the math, but bushwhacking for over 50 years made me decide which worked best for me. I re-lace every pair of shoes I purchase.
One thing I noticed is that for example at 24:35 you need the fact that those two segments intersect each other. I don't know if that's something you can assume or you need a proof of that.
At this stage in the proof, the intersection animation is just a visual aid! Remember, we are no longer considering actual lacings but instead exploded lacings, which may or may not come from any actual lacing. What's actually happening is purely abstract: we argue by contradiction that the shortest exploded lacing (which is just a list of numbers!) cannot contain any number >1. To do this, we suppose to the contrary that the shortest exploded lacing does have a number >1. Remember that the shortest exploded lacing must contain a 0, and note that whenever we have 0 and n in an exploded lacing with n>1, we can replace these by 1 and an n-1 and still have a valid exploded lacing: the numbers are all still non-negative, they have the same sum, there are still 10 of them, there are still no more than 5 zeros, and none of them are larger than 4! But this new exploded lacing must be shorter than the one we started with - one way to prove this is to explicitly calculate the length of each edge (using the Pythagorean theorem) and compare length(0) + length(n) to length(1) + length(n-1). It turns out length(1) + length(n-1) is always smaller, so we have a contradiction! Thus, the shortest exploded lacing cannot contain any number >1. If it would help, I'm happy to work out the algebraic details to prove that length(0) + length(n) > length(1) + length(n-1) whenever n>1, but they are kind of tedious. Instead, Burkard used the animation to visually argue that length(0) + length(n) *must* be larger than length(1) + length(n-1)! But it's just a visual tool, and is not at all trying to say that such an intersection actually appeared in the lacing (again, these are exploded lacings, which don't need to come from *any* actual lacing at all, or if they do that lacing need not be unique). Hope this helps!
@@BenSpitz That makes sense. In the first version of my comment I wrote that it just could be a tedious part and it turns out it is. And I think I'm able to prove that fact using the Pythagorean theorem for myself. I'm going to try later.
I love this! I am curious about two parts of the dream proof -- How do we know that the TSP criss cross shortening trick can be applied to exploded lacings (maybe the TSP graph interpretation only applies to real lacings)? And, when carrying out the shortening trick, how do we know there will always be a "0" in the right position to un-cross with? 🤔❤️
because all lacings that provide the same exploded lacing are the same length, therefore any exploded lacing *that can be expressed* as a proper lacing in such a manner that there is a 0 crossed with a longer segment has a shorter version described by following the trick in the situation expressed by that proper lacing and re-exploding it into an exploded lacing
@@firebrain2991 Ah, thank you. The intuition I was missing is that we have a choice of how to express an exploded lacing (maybe even sub-lacing?) as a proper lacing, so we can set up the cross cross trick on purpose. Also, since it's length we care about and length is preserved by exploding/un-exploding, we are OK.
For the last couple of years I've relaced my new shoes like at 2:14. I do this for the esthetic reasons of symmetry and horizontal lines. I dont like horizontal lines with one long diagonal.. I dont see many others wear it so it was good to see it in the video
The animation at circa 18:15 is the best thing I've ever encountered on this channel. Please, give us more of your original research, beautiful dreamer.
Glad you like that animation so much :) Having said that I'd say dig around a bit more in other Mathologer videos and you'll find a lot fancier animations.
I've always done a bow tie lacing with my oxfords, because its the only practical way you can guarantee that the crosses will happen on top of the shoe at regular spacing. The regular spacing also looks nice. Would recommend, not just because it's generally the shortest (but thanks for additional excuse Burkard)
It skips non-optimal lacing choices. Imagine a horizontal line. Now take a string and wrap it around that line once. Non-optimal lacings loop over or hook on existing laced strings. Kind of like Ian's "weave lacings". These non-optimal lacings give stretch flexations for mid-arch flexations (shoe top surface distorting while running). For a 12 hole system, string loops are best positioned in the center of the pattern and the very top of the pattern. These lacings are also much harder to represent by line segments in simple mathematical permutations, but obvious in topology systems.
Do the math considerations in question take into account the way the lace enters the eyelets: from up or from below (the 3-D model)? What if add some physics considerations of friction and so on? The real problem is an attrition of laces. What kind of lacing is the best to minimize the effect?
You’re adorable, Burkard. Each time I find you here, I see an excited little Boy, speaking of his passion, smiling whilst saying numbers; you’re absolutely beautiful, my dear Friend. Don’t you ever change. I so love your Soul’s most brilliant colours. I spend a lot of time here with you, thank you for the amazing and top calibre Company.
I would like to know which lacing has the least friction. I am generally using Ian's "over under lacing" and I am happy with that, but I wonder if it can be proven to have the least friction. I also wonder whether lacings can have asymmetrical friction, e.g. being easy to tighten and resistant to loosening, or vice versa.
Least friction is also least amount of cross over for the laces which build friction with every cross over, so the 'Bowtie' would be the best practical answer and the crisscross second (first if you want no vertical loops). As long as there aren't special materials or sliding knots (like a taut line hitch) placed mid lacing, the trip both ways is mirrored so the amount of friction is the exact same, making them directly proportional attributes to one another and unable to be asymmetrical in outcome. The one difference in mirroring (but not asymmetrically so) is that on odd-numbered eyelets, it would be slightly more secure starting with going under in the pattern because you will have more under loops by the end (it's even in even-looped shoes so it's moot) and they hold more friction on the contact points with the eyelets than over because the tongue of the shoe and pressure from inside the shoe create more friction. This also makes it harder to tighten though.
That's a really neat "dream proof" and great video as usual. I may go off to ponder what, if any, difference it makes when you have to miss one eyelet. Bunions, before you ask.
I love Ian's Shoelace Site. At one point I checked it whenever I got new shoes to find a new way to lace my shoes, nowadays I have my fave picked so I don't really need to anymore, but it's still nice
The proof diagrams (11:45) look like magic Runes. I think Bukard might be using some Norse magic here! the exploded lacing dream proof is fantastic. I think the criss-crossing of lacings is just inherently confusing to the human mind, and eliminating that factor by exploding them really does relieve the mind and make it less intimidating to tackle the problem. great stuff!
These days kids have velcro fasteners. When I was a kid, I laced all my running shoes. What I learned early on was that unusual lacings caused uneven wear. The worst was the zigzag form you showed at the beginning -- this always resulted in a worn out, then broken lace on the long diagonal lace. Today whenever I buy shoes they usually don't have criss cross lacings (no idea why) but I always re-lace them to criss-cross style so that I get the most even wear.
I'm a musician, and I sometimes dream of a new piece of music that sounds wonderful in my dream... and when I wake up, I can't remember it at all. I have a terrible suspicion that my brain is just *telling* me I'm hearing good music in the dream, rather than actually composing it...
I accidentally discovered shortness utility of the bow-tie lacing in my school days when I had a pair of artistic laces (I don't actually remember what the pattern was) which were too short for the shoes I wanted to put them on. A bow-tie pattern with the parallel pairs exposed (to show off the maximum amount of my cool laces) allowed me to tie the laces and keep my sneakers tight enough to wear.
"a closed path that visits each eyelet exactly once" can be modeled as a derangement of the set of eyelets, so the number of possible lacings is ((2n)-1)!/2, Where n is the number of eyelet pairs (so 2n is the number of eyelets), and the minus 1 accounts for the fact that {abc} is congruent to {bca} because it's a closed loop, divide by 2 because {abc} is congruent to {cba}. (This counts the non-lacing closed loops) The number of "tight" lacings where we alternate between the two sets of eyelets is (n!^2)/(2n) Because we create 2 separate lists {abc} {def} then weave them together {adbecf} then divide by 2n because of the congruence thing I said earlier.
I only searched for the number of tight lacings, and I found twice your result, i.e. n!×(n-1)! = (n!²)/n. Indeed, we have n choices for the first zig, (n-1) choices for the first zag, then (n-1) choices for the second zig, (n-2) for the second zag, and so on, until the second to last zag (one choice), the last zig (one choice) and the last zag (one choice). The result is n × (n-1) × (n-1) × (n-2) × ... × 2 × 1 × 1 × 1 = n!×(n-1)!.
I enoyed this video so much, thanks Burkard - the dream proof is beautiful! I might try picking up the book soon. Is there any obvious connection to other areas of maths you know, like juggling for example? Does adding an orientaion to each crossing affect things in any significant way?
Double helix lacing for the win! It's a variant of criss-cross, because In practical application you also have to decide for each island if you go from top to down or from down to top. Here the double helix makes less resistance than the criss cross. BTW: I love Ian's side, years ago after changing the laces of my dance shoes, I just pulled out the old ones, and then was stumped... how does one actually do this correctly? After some searching I found Ians site and it was a great resource to a) learn that there is no "one correct" way, b) helped me to decide for double helix.
I had a dream proof for an assignment once, never felt as productive as when working in my sleep, plus I had gone to bed the night before being stuck and within 10 minutes of waking up everything worked perfectly
9:45 the straight lines might not necessarily have a lower cost however. imagine that centre could have been a faster road, there might not be direct roads down the sides. etc.
Given a real shoe (with a given number of eyelets and fixed spacing) and given a particular lacing pattern (like for example the criss cross), is there a way to actually compute the overall lenght of the lacing? I always find myself struggling when I put laces on my shoes and the two loose ends of the string are either too short or too long to make a stable knot, or worse, of two different lenghts. It would be nice to know in advance how much string I'll need so that I can make sure that the one I have will fit.
@@Mathologer Yes, I've found it! Thank you! It's going straight to my favourites and, if you don't mind, I'll share the link here too, in case anyone is interested: www.fieggen.com/shoelace/shoelacecalc.htm
Did you consider that the eyelets are separated into two different arrays who’s distance from each other changes as the lacing is tightened? This changes the spacing aspect dramatically.
Good idea. However in terms of shortest nothing changes since the best solutions apply to all spacings. In terms of strongest things may flip from crisscross to zigzag as the two sides of the shoe come together since as you pull you are creating a longer shoe spacingwise. Of course this will only happen if you hit the crossover point somewhere along the transformation.
I've got another way to evaluate the "best" lacing. How about most evenly distributed? Specifically, what arrangement would have the lowest standard deviation of each hole's horizontal factor in a vector that is the sum of the vectors, pointed the same way the laces are going. As for the magnitude, I can think of two options: A: Perfect maths land where friction is meaningless and thus all the "pulling vectors" have a magnitude of 1. B: We account for friction, where the more holes the string passes through from the knot the less any tightening affects it. And thus the magnitude of a string pulling is F^D1+F^D2, where F is our factor of friction and D1 is how many holes must be passed through to get to the knot in one direction and D2 in the other direction. I predict that for case A the common criss-cross will work, but I expect something different and interesting for case B. As with how it is the furtheraway you get from the knot the weaker it's effect it and so there's a gradient of strength down the lacing, but by using a longer and overall weaker lace we might discover a more evenly distributed lace. (And potentially a more convenient/comfortable lace ;D )
I have known Ian's website for ages, being myself a kind of knots geek. I have used Ian's knot for my shoelaces for almost 20 years (more than 15 for sure), and it really drives me crazy when I see someone who ties their laces with a granny knot. Lastly, my 4yo niece was proud that she could tie her shoes, alas, her mother learned her, and she always ties them the wrong way. My niece was so disappointed when I told her, as gently as I could, that she learned it incorrectly and that it would be better for her to learn it again... I think I will try to teach her Ian's knot. It's really easy and fast, once you have the hang of it. By the way, am I right to suppose that the boat illustration comes from the Ashley's book of knots?
I think the easiest way to change tying a granny knot into tying a reef knot is by simply changing the over and under of first half of the knot and leave the rest, the complicated bunny ears (or whatever) action unchanged. "By the way, am I right to suppose that the boat illustration comes from the Ashley's book of knots?" That's it, well spotted :)
@@Mathologer Of course, that's the easiest part to change, but when you're a little child, even learning such a simple knot is an achievement, and changing what you have learned is quite an achievement. That said, the "wrong" knot is not as solidly implanted in her little brain as for an adult, so I'm sure it will be OK. Ashley's book of knots will forever be a reference, and will have a prominent place in my library. As a teen, it was probably the most proud of owning. It is amusing to see that Ian's knot is kind of present in that book, but not as a shoelace, but as a party trick.
@@Mathologer I have the French translation, so I cannot tell you the page, but it's in chapter 33, n°2534. It is certainly not the same than Ian's knot, but there is definitely a kind of similarity.
Oh holy crap. Holy CRAP. Less than three hours ago I was poking around TH-cam thinking "why do i have these three 'magic trick' videos in my 'maffs' playlist, and why does the happy little kitty say 'QED' on him and oh wow REALLY now oh man HOW long has it been oh jeez a whole month i hope things are okay…" and now this video!
I may have missed something but does your strongest lacing take into account of the force vector of the lacing? Considering that all the crossings have the same amount of tension, the force vector horizontally would be reduced by a factor of the angle of the lace. Following.
Exciting proof.)) Especial longest. 😅 I even try it! 😄 In end of video I had remember the scene of the Beautiful Mind movie, when Nash say "There could a mathematical explanation for how bad your tie is". 😊
Here is something for you :). www.tcm.phy.cam.ac.uk/~ym101/tie4/nature_tiepaper.pdf en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie (a couple more links in the description)
Whats really funny about this is how we still subconsciously use the crisscross lacing across history like a tried and tested method even a shoe as old as 5000 yrs old has bigger eyeholes with a rather thick width and after tying firm the crisscross. This was highly education and just a fun exploration in general.
Have you considered which side the lace enters the eyelet (from top or bottom), esp under a constraint that both laces need to come from the bottom at the final eyelet?
My answer to the general form for the problem mentioned around 4:25 (apologies if it's not very well-written...) Let n be the number of pairs of eyelets. To construct a lacing, start at the top left vertex. Since our lacing must be tight and use all eyelets, the next vertex in our lacing must be an unvisited one on the right, for which there are n choices. Then, our next vertex must be an unvisited one on the left, for which we have (n-1) choices. Then our next vertex must be unvisited on the right, then an unvisited on the left, and so on, until our only possible next move is to go back to the vertex where we started. This gives the pattern (n) * (n - 1) * (n - 1) * (n - 2) * (n - 2) * ... * 2 * 2 * 1 * 1 = (n)!(n-1)!. However, consider the fact that this forms a closed loop that starts and ends on the top left corner. Consider the sequence of vertices we visited to arrive at this lacing. We could have arrived at the lacing in exactly 2 ways: either by following the sequence in order, or by following it in reverse order. Hence the form (n)!(n-1)! counts each possible tight lacing exactly twice, so we divide by 2 to get the answer (1/2)(n)!(n-1)!.
Some questions What about holes arranged at different distances? By couples or free. What about arrangements of dots free in the plane (or space)? Is it possible to relate travelling salesman and lacing problems?
If you have access to a library, have a look my book for various answers to these questions :) Short answer, it gets complicated. However, as I also already said in the video in terms of being the shortest tight lacing overall the crisscross can be shown to be remarkable "immune" to perturbing the position of the eyelets.
In terms of practicality, since the laces are tensioned from only one end, under extreme friction, why are we so married to the tradition of placing the eyelets at equidistant intervals? Maybe the distance between the lower holes could be increased, or maybe laces could be tapered to be thicker towards the ends, or maybe a lace of composite materials could be slicker in its mid-section but progressively coarser toward the tips for knot secureness? Personally, I've always tied the criss-cross on leisure footwear and the 'French' lacing, as you call it, on formal shoes. The problem with the french is that you'll soon find that one end lengthens disproportionately with every re-tie... unless you use decorative jam-knots or knotted beads across the lower 'bridge'.
The only instances of the TSP where I can see something similar working are highly regular arrangements of points that translate into some easy usable restrictions like "at most 5 horizontals" or "once up and down" :)
I also had modelled with combinatory of the sum of numbers adding up to the spaces between holes (1,1,2;1,2,1;2,1,1,...) I got 15 for 5 and 25 for 6. Always odd because of the crisscross. Nice graphs puzzle and I suspect, lots of real world applications beyond shoes
So what about which lacing will most evenly distribute the force applied when pulling the loose ends? Criss-cross lacing tends to pull more effectively up top and very little at the bottom thanks to friction. Will this always be the same as the shortest lacing?
I used to use a z-type lacing. This was because shoes usually came with that one. I moved over to the criss-cross lacing as it was easier to ensure that both ends were of an equal length. I suppose the bowtie and devil's lacings would be useful if you had to replace the laces with ones that weren't of the ideal length for the criss-cross pattern. I've watched several videos about optimum lacing strategies. This one is by far the most rigorous. How about a video about the optimum knot? :) With particular emphasis on knots that won't spontaneously come undone!
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!
Hlo sir ,Is there any relationship between ln(m+n) to[ ln(m) or ln(n)].?
@@Manoj_b hay man what did mean could you elaborate?.I am also very interested in logarithimic
property
@@noone7692 well actually I found a way to expand ln(m+n) to [ln(m) or ln(n)] so, trying to know whether if any existed.?
Manoj Really? Which is it? I’m not aware of any such method, but I’m no expert.
@@Manoj_b whaddya mean by 'or' in the middle of ur expression
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.
Absolutely right, there are lots of other considerations that go into optimising something as seemingly trivial as lacing and tying your shoes. While I was obsessing about this material I did come across a lot of other interesting insights. Definitely also check out Ian's site :)
I just keep a mirror symmetry between my shoes XD
@@VaradMahashabde It's too much hassle to mirror the knots and bows, so I tie both shoes the same way. Thus, it makes more sense to lace them the same way too, rather than mirrored. If the lacing is mirrored but the knot is not, it's what we call "not elegant".
@@JohnDlugosz
You don't mirror the knot?
@@leif1075 The height matters cause it affects the center of mass differently and get affected by the topolgy of your foot differently.
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.
Am i really watching a 29 min video at 3am about the best way to tie your shoe laces ?
YES.
Y E S
It's a perfect time to watch such a video.
You can draw your conclusions over night and directly apply them the next morning ^^
No, and nor am I...
Mmm, 03:05. Nice way to spend the evening.
Currently, it's 3:14am here and I'm procrastinating - I should be doing maths homework but instead I'm watching Mathologer videos.
Arguably, the best way to procrastinate.
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.
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.
also speed of tightening, number of steps to execute?
So, velcro.
@@Fyreye that would just be the number of holes in yourshoe
>- Equally tight at the top in the middle and at the bottom.
I came here to say this. "Shortest" is not the "best" in my opinion. We have to consider the function of laces, to allow you to slip your foot in, and then tighten the shoe so it is evenly stable on your foot. None of that involves how short a lacing is.
Equal tightness is the most important to me. Second is that the force on the upper pair of holes is low. For shoes without eyelets, that is the weakest part. I use the "loop back" technique. (The lace on the right loops around the lace on the left and then returns to the right. The lace on the left returns to the left.) This forms a little pulley that halves the force on the top holes.
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.
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 😄👏🏼👌🏼
Definitely check out Ian's site :)
Mathologer thanks!
@@Mathologer Ian's site
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!
Yes - this is actually a practica onel lacing given the stupidly long laces shoes come with these days.
As a programmer, I sometimes wake up with "dream functions" lol
I sometimes dream of the halting function that would determine for any machine if it halts or not. Ahhh, such dreams, dreams that will never come true...
@@AgentM124 Sometimes I dream of finding the value of BB(5).
@@xXLanyuzAnlunXx ah yes good ol busy beavers
@@xXLanyuzAnlunXx it's at least 7
how is being a programmer? i have the intention of studying to becone a game designer/programmer and im truly interested into how the programming business and worklife is.
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.
I've been using it for about 5 years now and I haven't known it was called Ian's knot. I learned it from Matt Parker.
I've probably been using it for about 10 years and didn't know the name of it either.
@@stewartzayat7526 So did I.
0:06 Just started and I already pressed like for the t-shirt
Where can you get that shirt???:D
@@checkm8606 Try googling "only half evil tshirt" ya lazy ass.
Dark!!
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.
I have no idea why but this is one of the most interesting videos I've watched in a while.
Apart from all the extremely beautiful ideas, Your style always makes me happy. Wholesome content.
Wow, this is a great application using math. I also like math modeling myself, and this video is in the same context of my interest so much. Great job and nice explanation!
This is the best video about lacings ever. I love the production, the editing, and the segments. Thanks for the great content!
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!
I have shoes where the laces are about half a meter too long. So now I know the best way to keep the laces and be happier with the shoes.
It's mathematically pleasing when the answer is neat and easy
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.
I have always used that, the advantage is that it is easier to tighten more eyelets with one pull
When I saw Burkard with hair, my head turned into a black hole.
Hole black poogtfgu. Yes he at turtle how house?
Whay
@@madkirk7431 What kind of code is that?
@@ckv1985 Whay 💀
Whenever I feel smart, I watch Mathologer to knock my ego down a few pegs.
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.
Great stuff! I did find myself looking at the zig-zag, and thinking - ugh, that's the one that gradually de-centres the lace each time you tighten it until it's moved so far out of centre you have to take the time tease it back into position. It can also easily look wonky, as you can pull it out of shape pulling on the long diagonal.
Enter what you called the French lacing, which I use on boots, Doc Marten's, that kind of thing - horizontals are always on top, diagonals always underneath; you just see the horizontal pieces neatly perpendicular to the seam since the diagonals are hidden, but at the same time solving both the de-centring problem and the skew problem. 8-)
Yeah, it took me a while to see it from the diagram, but I started doing the French lacing since finding it in a store because it looks nice.
I like to imagine that it's less stressful for the lace to be getting pulled against the edges of each side, but when has a lace ever broken there?
Is it stronger than the regular cross pattern for boots?
You are the best math teacher I had in my 40 years of life :-)
Good to hear :)
In recent years, when I buy shoes they tend to be laced in strange ways so that I always have to relace them (to criss-cross lacing). I wonder why this is. Are there lacings that can be done FASTER in the factory? Have you ever considered speed of lacing as a criterion?
I don't think I have ever bought a pair of shoes that didn't come with some form of crisscross lacing.
Most shoes come unlaced, or partially laced. Any factory or store lacing other than criss-cross is usually done for aesthetic reasons.
I square lace my shoes; absolutely suboptimal for efficiency, but tops in comfort if you want control over where the shoe compresses. I could see a case for square lacing via machine as there's no need to weave the laces, they simply lay flat over each-other.
as an addendum; under your (Mathologer) function for calculating by rise/run, each vertical member of a square lace would be +-infinite?
This is something I do to every pair of shoes/boots I have bought in the 40 years. OCD for sure, but it pays off over time.
Finally a video I could actually follow the whole way through. The section on strength was particularly satisfying, because the first thing I thought was that not all tight lacings are created equal in real shoes.
Wouldn't the preferred lacing simply be the one that has the least friction throughout, and so, requiring the least "loop-pulls" to both loosen and tighten the shoe?
You really the best man! 🙌 Stay cool! Your contents really hits me hard and inspired me to create my own channel and be an inspiration also to others. More power everyone
So lit 💯
I agree
Wow i got a heart from @mathologer ! 😊
This is exactly the type of video I love! I didn’t even go through the step of wondering at myself for enjoying a shoelace video! I ponder this type of stuff too! Perhaps there a mathematician struggling to come out!😁 Thanks for the great video!
Fascinating example of modeling something from the real world into something formal :D. For me this is teaching a certain way of thinking.
Thank you for your work! Cheers!
This video skyrocketed to my top 10 favorite TH-cam videos!
What about for lacings where the holes don’t have to be directly opposite each other? The zigzag lacing shown is very similar to spiral lacing, used for stays/corsets until some time in the 19th century, except for two things: the holes (except for the top and bottom pair) are offset so that the crossing angle stays constant, and the lace is simply tied off at the top and bottom, with no long lace running between them, but I suspect that’s also the case for the shoes.
EDIT: after hearing about which lacing is strongest, I have to take a look at whether the typical distance between eyelet became denser over time, in which case it would make sense if the bones bodices of the renaissance were spiral laced, but the Victorian corset crisscross laced.
Was coming to mention spiral lacing myself, but the context I learnt of it was arming garments which bore the weight of armour (usually arm or leg) tied to them.
I've always laced my shoes the same way as your 'french shoe shop' example at 2:10 - it is not always about 'shortest' or 'strongest' but also 'neatest'. The top view of this lacing shows clean horizontal lacings - which are easy to grab and tighten. Unlike the zig-zag lacing, it gives equal tension to each half of the lace, while producing a similar overhead appearance.
However, love the maths you demonstrate here and links to the TSP problem.
That's a really beautiful and intuitive proof. Congratulations for that!
I love recreational maths problems they're wonderful, thanks for sharing your little passion!
Consequently, I learned that I had actually given up tying the starting shoelace knot entirely! (Straight to the bow, no slipknot)
I know many people dismiss lacing shoes, but anyone who has put many miles on Shank's Mare can appreciate how important this is.
So essentially if we were to rank lacings on strength and length 1 being the best and infinite being the worst we're looking for the lowest sum of the two factors to have the optimal lacing. The natural choice here would be the crisscross lacing.
The crisscross lacing at best has rank 1 in both situations for a sum score of 2 and at worst it falls behind in strength on the zigzag lacing for a sum score of 3.
The zigzag lacing at best will be 1st on strength and 2nd in length for a sum score of 3 which is merely equal to the crisscross in the same case.
The case where they do equal at a sum score of 3, it would be a competition of preference where some situations might demand short laces by lack of material but some may prefer a higher tensile strength because they want to climb a mountain today.
Never had the patience to do the math, but bushwhacking for over 50 years made me decide which worked best for me. I re-lace every pair of shoes I purchase.
One thing I noticed is that for example at 24:35 you need the fact that those two segments intersect each other. I don't know if that's something you can assume or you need a proof of that.
At this stage in the proof, the intersection animation is just a visual aid! Remember, we are no longer considering actual lacings but instead exploded lacings, which may or may not come from any actual lacing. What's actually happening is purely abstract: we argue by contradiction that the shortest exploded lacing (which is just a list of numbers!) cannot contain any number >1.
To do this, we suppose to the contrary that the shortest exploded lacing does have a number >1. Remember that the shortest exploded lacing must contain a 0, and note that whenever we have 0 and n in an exploded lacing with n>1, we can replace these by 1 and an n-1 and still have a valid exploded lacing: the numbers are all still non-negative, they have the same sum, there are still 10 of them, there are still no more than 5 zeros, and none of them are larger than 4! But this new exploded lacing must be shorter than the one we started with - one way to prove this is to explicitly calculate the length of each edge (using the Pythagorean theorem) and compare length(0) + length(n) to length(1) + length(n-1). It turns out length(1) + length(n-1) is always smaller, so we have a contradiction! Thus, the shortest exploded lacing cannot contain any number >1.
If it would help, I'm happy to work out the algebraic details to prove that length(0) + length(n) > length(1) + length(n-1) whenever n>1, but they are kind of tedious. Instead, Burkard used the animation to visually argue that length(0) + length(n) *must* be larger than length(1) + length(n-1)! But it's just a visual tool, and is not at all trying to say that such an intersection actually appeared in the lacing (again, these are exploded lacings, which don't need to come from *any* actual lacing at all, or if they do that lacing need not be unique).
Hope this helps!
@@BenSpitz That makes sense. In the first version of my comment I wrote that it just could be a tedious part and it turns out it is. And I think I'm able to prove that fact using the Pythagorean theorem for myself. I'm going to try later.
I love this! I am curious about two parts of the dream proof -- How do we know that the TSP criss cross shortening trick can be applied to exploded lacings (maybe the TSP graph interpretation only applies to real lacings)? And, when carrying out the shortening trick, how do we know there will always be a "0" in the right position to un-cross with? 🤔❤️
because all lacings that provide the same exploded lacing are the same length, therefore any exploded lacing *that can be expressed* as a proper lacing in such a manner that there is a 0 crossed with a longer segment has a shorter version described by following the trick in the situation expressed by that proper lacing and re-exploding it into an exploded lacing
@@firebrain2991 Ah, thank you. The intuition I was missing is that we have a choice of how to express an exploded lacing (maybe even sub-lacing?) as a proper lacing, so we can set up the cross cross trick on purpose. Also, since it's length we care about and length is preserved by exploding/un-exploding, we are OK.
For the last couple of years I've relaced my new shoes like at 2:14. I do this for the esthetic reasons of symmetry and horizontal lines. I dont like horizontal lines with one long diagonal.. I dont see many others wear it so it was good to see it in the video
Thanks for the link to the Ian's site. I really like the quick release ladder.
0:35: "We're always on the lookout for the mathematical sole of things"
Nice cringeworthy one, right ?
Ha-ha.
Love your videos!
- A mathematically inclined electrical engineering student
"Well I've shown you the shortest lacings, now how do we prove such thing?"
Engineers:"Do what now?"
Only the very best nerds can pull it off to write a book about shoe lacing! Congratulations for that.
The animation at circa 18:15 is the best thing I've ever encountered on this channel.
Please, give us more of your original research, beautiful dreamer.
Glad you like that animation so much :) Having said that I'd say dig around a bit more in other Mathologer videos and you'll find a lot fancier animations.
I've always done a bow tie lacing with my oxfords, because its the only practical way you can guarantee that the crosses will happen on top of the shoe at regular spacing. The regular spacing also looks nice. Would recommend, not just because it's generally the shortest (but thanks for additional excuse Burkard)
That's great, I've finally heard from someone who actually uses bowtie lacings :)
This channel is an absolute treasure. Even for a smooth brain lump like me.
Must agree on the fact that there's math in everything. But an half an hour explanation for just tying shoe laces and a boon as well😳😳
It skips non-optimal lacing choices. Imagine a horizontal line. Now take a string and wrap it around that line once.
Non-optimal lacings loop over or hook on existing laced strings. Kind of like Ian's "weave lacings". These non-optimal lacings give stretch flexations for mid-arch flexations (shoe top surface distorting while running). For a 12 hole system, string loops are best positioned in the center of the pattern and the very top of the pattern. These lacings are also much harder to represent by line segments in simple mathematical permutations, but obvious in topology systems.
This video is NOT about tying laces :)
Do the math considerations in question take into account the way the lace enters the eyelets: from up or from below (the 3-D model)? What if add some physics considerations of friction and so on? The real problem is an attrition of laces. What kind of lacing is the best to minimize the effect?
It makes me happy that there are at least two people on the planet who have thought deeply about laces
Honestly, your videos are F**KING AWESOME!
You’re adorable, Burkard. Each time I find you here, I see an excited little Boy, speaking of his passion, smiling whilst saying numbers; you’re absolutely beautiful, my dear Friend. Don’t you ever change. I so love your Soul’s most brilliant colours. I spend a lot of time here with you, thank you for the amazing and top calibre Company.
I would like to know which lacing has the least friction. I am generally using Ian's "over under lacing" and I am happy with that, but I wonder if it can be proven to have the least friction. I also wonder whether lacings can have asymmetrical friction, e.g. being easy to tighten and resistant to loosening, or vice versa.
Least friction is also least amount of cross over for the laces which build friction with every cross over, so the 'Bowtie' would be the best practical answer and the crisscross second (first if you want no vertical loops).
As long as there aren't special materials or sliding knots (like a taut line hitch) placed mid lacing, the trip both ways is mirrored so the amount of friction is the exact same, making them directly proportional attributes to one another and unable to be asymmetrical in outcome. The one difference in mirroring (but not asymmetrically so) is that on odd-numbered eyelets, it would be slightly more secure starting with going under in the pattern because you will have more under loops by the end (it's even in even-looped shoes so it's moot) and they hold more friction on the contact points with the eyelets than over because the tongue of the shoe and pressure from inside the shoe create more friction. This also makes it harder to tighten though.
As a sneakerhead and mathematics student I greatly appreciate this.
That's a really neat "dream proof" and great video as usual. I may go off to ponder what, if any, difference it makes when you have to miss one eyelet. Bunions, before you ask.
I love Ian's Shoelace Site. At one point I checked it whenever I got new shoes to find a new way to lace my shoes, nowadays I have my fave picked so I don't really need to anymore, but it's still nice
The proof diagrams (11:45) look like magic Runes. I think Bukard might be using some Norse magic here!
the exploded lacing dream proof is fantastic. I think the criss-crossing of lacings is just inherently confusing to the human mind, and eliminating that factor by exploding them really does relieve the mind and make it less intimidating to tackle the problem. great stuff!
It's been months since I have watched your work. Great video as always.
Not much new to watch sadly. Have been super busy surviving the COVID-related mess at the university where I teach :(
The one I always use is the lacing at 2:09. I just like the way it looks. I first learned it at a Summer camp decades ago.
This got an instant subscribe, awesome video, super fun to watch, perfect pacing and i love the tshirt
These days kids have velcro fasteners. When I was a kid, I laced all my running shoes. What I learned early on was that unusual lacings caused uneven wear. The worst was the zigzag form you showed at the beginning -- this always resulted in a worn out, then broken lace on the long diagonal lace. Today whenever I buy shoes they usually don't have criss cross lacings (no idea why) but I always re-lace them to criss-cross style so that I get the most even wear.
I'm a musician, and I sometimes dream of a new piece of music that sounds wonderful in my dream... and when I wake up, I can't remember it at all. I have a terrible suspicion that my brain is just *telling* me I'm hearing good music in the dream, rather than actually composing it...
I accidentally discovered shortness utility of the bow-tie lacing in my school days when I had a pair of artistic laces (I don't actually remember what the pattern was) which were too short for the shoes I wanted to put them on. A bow-tie pattern with the parallel pairs exposed (to show off the maximum amount of my cool laces) allowed me to tie the laces and keep my sneakers tight enough to wear.
13:43 "Fun, no? ... Well I don't care. I think it's fun."
Holy shit. How long have you been back? I thought you were gone. Thanks for another awesome video.
"a closed path that visits each eyelet exactly once" can be modeled as a derangement of the set of eyelets, so the number of possible lacings is ((2n)-1)!/2, Where n is the number of eyelet pairs (so 2n is the number of eyelets), and the minus 1 accounts for the fact that {abc} is congruent to {bca} because it's a closed loop, divide by 2 because {abc} is congruent to {cba}. (This counts the non-lacing closed loops) The number of "tight" lacings where we alternate between the two sets of eyelets is (n!^2)/(2n) Because we create 2 separate lists {abc} {def} then weave them together {adbecf} then divide by 2n because of the congruence thing I said earlier.
I only searched for the number of tight lacings, and I found twice your result, i.e. n!×(n-1)! = (n!²)/n.
Indeed, we have n choices for the first zig, (n-1) choices for the first zag, then (n-1) choices for the second zig, (n-2) for the second zag, and so on, until the second to last zag (one choice), the last zig (one choice) and the last zag (one choice). The result is n × (n-1) × (n-1) × (n-2) × ... × 2 × 1 × 1 × 1 = n!×(n-1)!.
Oh, yes I understand: if we count as the same lacing when we begin above or below, my laces are counted twice. So I think you're right.
Finally, a mathematical proof that devils and angels are equally evil
Idk why I’m so drawn to this comment
I enoyed this video so much, thanks Burkard - the dream proof is beautiful! I might try picking up the book soon. Is there any obvious connection to other areas of maths you know, like juggling for example? Does adding an orientaion to each crossing affect things in any significant way?
Double helix lacing for the win! It's a variant of criss-cross, because In practical application you also have to decide for each island if you go from top to down or from down to top. Here the double helix makes less resistance than the criss cross.
BTW: I love Ian's side, years ago after changing the laces of my dance shoes, I just pulled out the old ones, and then was stumped... how does one actually do this correctly? After some searching I found Ians site and it was a great resource to a) learn that there is no "one correct" way, b) helped me to decide for double helix.
I had a dream proof for an assignment once, never felt as productive as when working in my sleep, plus I had gone to bed the night before being stuck and within 10 minutes of waking up everything worked perfectly
Simple.
Get velcro instead.
Witchcraft!
Can we trust that a Mathologer analysis of Velcro / hook-&-loop would be short?
The bunny comes out of the hole ....
Personally I recommend (Birkenstock) sandals. It's the footwear of choice for many mathematicians :)
yes, using binary is easier for such problems.
9:45 the straight lines might not necessarily have a lower cost however. imagine that centre could have been a faster road, there might not be direct roads down the sides. etc.
Best video I could have watched before going to bed :)
Given a real shoe (with a given number of eyelets and fixed spacing) and given a particular lacing pattern (like for example the criss cross), is there a way to actually compute the overall lenght of the lacing? I always find myself struggling when I put laces on my shoes and the two loose ends of the string are either too short or too long to make a stable knot, or worse, of two different lenghts. It would be nice to know in advance how much string I'll need so that I can make sure that the one I have will fit.
I seem to remember that Ian has a lacing calculator somewhere on his website. Have a look. You may even be able to google this.
@@Mathologer Yes, I've found it! Thank you! It's going straight to my favourites and, if you don't mind, I'll share the link here too, in case anyone is interested:
www.fieggen.com/shoelace/shoelacecalc.htm
Did you consider that the eyelets are separated into two different arrays who’s distance from each other changes as the lacing is tightened? This changes the spacing aspect dramatically.
Good idea. However in terms of shortest nothing changes since the best solutions apply to all spacings. In terms of strongest things may flip from crisscross to zigzag as the two sides of the shoe come together since as you pull you are creating a longer shoe spacingwise. Of course this will only happen if you hit the crossover point somewhere along the transformation.
What about the Champernowne constant? The continued fraction is interesting.
I've got another way to evaluate the "best" lacing. How about most evenly distributed? Specifically, what arrangement would have the lowest standard deviation of each hole's horizontal factor in a vector that is the sum of the vectors, pointed the same way the laces are going.
As for the magnitude, I can think of two options:
A: Perfect maths land where friction is meaningless and thus all the "pulling vectors" have a magnitude of 1.
B: We account for friction, where the more holes the string passes through from the knot the less any tightening affects it. And thus the magnitude of a string pulling is F^D1+F^D2, where F is our factor of friction and D1 is how many holes must be passed through to get to the knot in one direction and D2 in the other direction.
I predict that for case A the common criss-cross will work, but I expect something different and interesting for case B. As with how it is the furtheraway you get from the knot the weaker it's effect it and so there's a gradient of strength down the lacing, but by using a longer and overall weaker lace we might discover a more evenly distributed lace.
(And potentially a more convenient/comfortable lace ;D )
1st video I see with named segment on the video time line. Very usefull !
Yes, it's a great feature but the way TH-cam has implemented it is a bit buggy. I've tried this on four videos so far. Two worked and two didn't :(
I have known Ian's website for ages, being myself a kind of knots geek. I have used Ian's knot for my shoelaces for almost 20 years (more than 15 for sure), and it really drives me crazy when I see someone who ties their laces with a granny knot. Lastly, my 4yo niece was proud that she could tie her shoes, alas, her mother learned her, and she always ties them the wrong way. My niece was so disappointed when I told her, as gently as I could, that she learned it incorrectly and that it would be better for her to learn it again... I think I will try to teach her Ian's knot. It's really easy and fast, once you have the hang of it.
By the way, am I right to suppose that the boat illustration comes from the Ashley's book of knots?
I think the easiest way to change tying a granny knot into tying a reef knot is by simply changing the over and under of first half of the knot and leave the rest, the complicated bunny ears (or whatever) action unchanged. "By the way, am I right to suppose that the boat illustration comes from the Ashley's book of knots?" That's it, well spotted :)
@@Mathologer Of course, that's the easiest part to change, but when you're a little child, even learning such a simple knot is an achievement, and changing what you have learned is quite an achievement. That said, the "wrong" knot is not as solidly implanted in her little brain as for an adult, so I'm sure it will be OK.
Ashley's book of knots will forever be a reference, and will have a prominent place in my library. As a teen, it was probably the most proud of owning. It is amusing to see that Ian's knot is kind of present in that book, but not as a shoelace, but as a party trick.
@@etienneschramm83 "Ian's knot is kind of present in that book, but not as a shoelace, but as a party trick" Interesting, what page is that on?
@@Mathologer I have the French translation, so I cannot tell you the page, but it's in chapter 33, n°2534. It is certainly not the same than Ian's knot, but there is definitely a kind of similarity.
Fascinating, I would have expected the devil and angel lacings to be the strongest.
Oh holy crap. Holy CRAP. Less than three hours ago I was poking around TH-cam thinking "why do i have these three 'magic trick' videos in my 'maffs' playlist, and why does the happy little kitty say 'QED' on him and oh wow REALLY now oh man HOW long has it been oh jeez a whole month i hope things are okay…" and now this video!
Sir, put some videos of IMO questions because your explanation are unique and deep
hello from greece what a nice and honest mathematical experience love
I may have missed something but does your strongest lacing take into account of the force vector of the lacing? Considering that all the crossings have the same amount of tension, the force vector horizontally would be reduced by a factor of the angle of the lace. Following.
Exciting proof.)) Especial longest. 😅 I even try it! 😄
In end of video I had remember the scene of the Beautiful Mind movie, when Nash say "There could a mathematical explanation for how bad your tie is". 😊
Here is something for you :). www.tcm.phy.cam.ac.uk/~ym101/tie4/nature_tiepaper.pdf
en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie
(a couple more links in the description)
Whats really funny about this is how we still subconsciously use the crisscross lacing across history like a tried and tested method even a shoe as old as 5000 yrs old has bigger eyeholes with a rather thick width and after tying firm the crisscross.
This was highly education and just a fun exploration in general.
Have you considered which side the lace enters the eyelet (from top or bottom), esp under a constraint that both laces need to come from the bottom at the final eyelet?
Have a look at Ian's site :)
I love your work ❤️. Keep igniting minds.
Burkhardt, what's the font of the chapter headings?? It's extraordinary
My answer to the general form for the problem mentioned around 4:25 (apologies if it's not very well-written...)
Let n be the number of pairs of eyelets. To construct a lacing, start at the top left vertex. Since our lacing must be tight and use all eyelets, the next vertex in our lacing must be an unvisited one on the right, for which there are n choices. Then, our next vertex must be an unvisited one on the left, for which we have (n-1) choices. Then our next vertex must be unvisited on the right, then an unvisited on the left, and so on, until our only possible next move is to go back to the vertex where we started. This gives the pattern (n) * (n - 1) * (n - 1) * (n - 2) * (n - 2) * ... * 2 * 2 * 1 * 1 = (n)!(n-1)!.
However, consider the fact that this forms a closed loop that starts and ends on the top left corner. Consider the sequence of vertices we visited to arrive at this lacing. We could have arrived at the lacing in exactly 2 ways: either by following the sequence in order, or by following it in reverse order. Hence the form (n)!(n-1)! counts each possible tight lacing exactly twice, so we divide by 2 to get the answer (1/2)(n)!(n-1)!.
Full marks :)
what is the best most fashionable lacing? possibly with most uniform parallel patterns
Some questions
What about holes arranged at different distances? By couples or free.
What about arrangements of dots free in the plane (or space)?
Is it possible to relate travelling salesman and lacing problems?
If you have access to a library, have a look my book for various answers to these questions :) Short answer, it gets complicated. However, as I also already said in the video in terms of being the shortest tight lacing overall the crisscross can be shown to be remarkable "immune" to perturbing the position of the eyelets.
In terms of practicality, since the laces are tensioned from only one end, under extreme friction, why are we so married to the tradition of placing the eyelets at equidistant intervals? Maybe the distance between the lower holes could be increased, or maybe laces could be tapered to be thicker towards the ends, or maybe a lace of composite materials could be slicker in its mid-section but progressively coarser toward the tips for knot secureness? Personally, I've always tied the criss-cross on leisure footwear and the 'French' lacing, as you call it, on formal shoes. The problem with the french is that you'll soon find that one end lengthens disproportionately with every re-tie... unless you use decorative jam-knots or knotted beads across the lower 'bridge'.
Love your videos man. Wish I had mathematician and programmer friends 😪
Great vid as always. Have you tried to expand the idea of exploded lacings to the tsp?
The only instances of the TSP where I can see something similar working are highly regular arrangements of points that translate into some easy usable restrictions like "at most 5 horizontals" or "once up and down" :)
I also had modelled with combinatory of the sum of numbers adding up to the spaces between holes (1,1,2;1,2,1;2,1,1,...) I got 15 for 5 and 25 for 6. Always odd because of the crisscross. Nice graphs puzzle and I suspect, lots of real world applications beyond shoes
So what about which lacing will most evenly distribute the force applied when pulling the loose ends? Criss-cross lacing tends to pull more effectively up top and very little at the bottom thanks to friction. Will this always be the same as the shortest lacing?
Ian's site has a lot more about these sorts of very practical considerations. Check it out :)
I used to use a z-type lacing. This was because shoes usually came with that one. I moved over to the criss-cross lacing as it was easier to ensure that both ends were of an equal length. I suppose the bowtie and devil's lacings would be useful if you had to replace the laces with ones that weren't of the ideal length for the criss-cross pattern. I've watched several videos about optimum lacing strategies. This one is by far the most rigorous. How about a video about the optimum knot? :) With particular emphasis on knots that won't spontaneously come undone!
2 dream proofs that actually worked? What, you're op!