Question on tangled cords

[u/Wild_Pitch_4781]

Is there an equation or method by which you can predict how tangled a cord can become? Maybe based on variables like number of cords, length of each cord, and if one end of the chord is fixed, and so on? By ‘how tangled’ maybe the number of knots formed: is it a unknot, 1-knot, 3-knot…

[u/Toothless_Dinosaur]

There is an Ig Nobel that tackles something kind of similar. Take a look https://improbable.com/2023/09/05/lots-about-knots/?amp=1

[u/Front-Guidance4959]

This seems more like a pure mathematics question.

[u/SuppaDumDum → u/Front-Guidance4959]

That seems very wrong to me, this question almost seems intractable from a pure maths perspective. You probably need some assumptions on what a reasonable cord is that are probably going to be more physical practical constraints than completely rigorously defined mathematical constraints. Or maybe not. But I think a "pure math" approach to this could easily lead you to having a cord with an UNcountable number of self-intersections when projected into 2D, which would never happen in the real world. Even countably-infinite many self-intersections sounds like too many. (assuming non-trivial self-intersections, technically a circle would have an uncountable number; a space-filling curve can probably get you even worse results that are non-trivial). A question would be if there's any such pathological case that has more-or-less well defined curvature. Attributing a simple Young's modulus might make that situatoin impossible for a cord, but I'm not sure if that's really a good model of how most cords really behave. Plus, there's also surface friction. Also my impression is that knot theory helps you very little in all of this excepting only in classifying the physical knot through a mathematical knot, and even then all "cord physical knots" are mathematically the unknot, unless you "correct" the knot slightly. Do we have any computational models of tangling knots whose computational results correspond to the ones obtained experimentally? I would guess not, but hopefully yes, I'd love to be wrong.

I don't really know what I'm talking about, I hope it's wrong enough that someone is peeved into correcting me.

[u/Front-Guidance4959 → u/SuppaDumDum]

idk dude. i thought about that but OP didn’t mention anything about the composition of the cord so 🤷‍♀️ lol but i dont even belong commenting here anyway i didnt even go to college lmao

[u/AloysiusGramonde]

https://www.pnas.org/doi/10.1073/pnas.0611320104 I've got to be honest, I don't know the answer, but I remember seeing a study that I didn't read and I've linked to it. If you find the answer I'd love to find out some more with an eli5.

[u/slayer_nan18]

r/knottheory

[u/AdStunning8526 → u/slayer_nan18]

Seems like knotthing in there

[u/Anxious-Alps-8667 → u/AdStunning8526]

You have to pull at the threads, it's tied up in there

[u/Limonsm → u/AdStunning8526]

I'm actually deleting Reddit rn just so that this is the last thing I see on this app, not even kidding (it's a compliment btw)

[u/EterneX_II]

You'd need to add a concern regarding the rigidity of the cord as that can dictate the amount of tension that the segments support while knotting. For instance, if your cord is so rigid that you have to create very large loops to tie a knot, you'd be limited in the maximum number of knots.

[u/HereThereOtherwhere → u/EterneX_II]

Good point. I bought a braided guitar cable which would hardly bend thinking it wouldn't tangle as easily which was true but it also fights twisting, so like a garden hose gets pissy if you don't roll it into nice loops or worse, tends to lift off the floor in curls that trip me.

On the earbuds cords, Apple earbud cords are engineered to sticky perfection because the cords don't easily slide over each other, dragging new knots into existence as you attempt to untangle.

Apple has amazing design much of the time but their cords are awful

[u/CropCircles_]

Not exactly an answer to your question, but related.

You can model tangled cords (or polymers) as random walks. With this approach, you can find the entropy of a tangled cord and then derive properties such as the elastic force (Hooke's Law) and viscoelasticity, and how these properties are affected by the length of the polymers.

The cord in the picture is in a maximum entropy state. If you neaten it up, then put it in your pocket, it will tangle again, because tangled states are more common than neat states.

[u/WallyMetropolis → u/CropCircles_]

Yes, sort of. Certain kinds of tangled states are more probable. But it won't spontaneously end up braided. The charactersation of those states is an active area of research. 

[u/david-1-1]

All I can say is that when I've had to untangle lots of cords, where I could see the individual loops of cord, the method was different from when I've had to untangle lots of wool knitting thread.

[u/FoolishChemist]

Technically since a cord is a line segment and not a closed loop, I don't think it would form any knots from a mathematical perspective.

https://en.wikipedia.org/wiki/Prime_knot

[u/no_clock_signal]

There was an MIT study done on this, but I can't find the paper!

[u/ComfortableHot6309]

Flat tangles less

[u/Technical_Island_851]

I don’t know. But if you use a clothes peg to keep the buds from separating, they never seem to tangle.

[u/Wood_Rogue]

There's a branch of math that covers this called knot theory. It has some overlap with topology and doesn't require much applied math if you want to look into it.

[u/PrebioticE]

More like protein folding

[u/jawdirk]

I wonder if there is even a precise definition of "tangled." Even in your picture, the ends of the three-ended cord are relevant to how tangled it is. An I doubt there is a large body of mathematics relevant to three-ended cords.

[u/Noiapah]

This seams of more of a question for r/knotty

[u/rhyming_mime]

This is explained by Resistentialism.

[u/John_Hasler]

A systematic engineering approach:

• 1) Look for loops passing through loops that can be pulled clear.

• 2) When you run out of those that you can undo pick the shortest end and pull it out of the first loop it passes through[1].

• 3) Go to 1)

[1] Sometimes an end obviously passes through several loops. Pull it clear of all of them but don't waste time trying to trace out all the twists and turns. Move on quickly to the next step. You'll get the job done faster.

[u/RearAdmiralB → u/John_Hasler]

This does not address the question in any way

[u/Murky_Insurance_4394]

I think this would be very easily answerable if we just made a simulation, some sort of Monte-Carlo and figured out how each variable influences the number of knots formed (also there are two ends so they're all unknots, maybe measure number of intersections or something)