A cord hangs still below both readings of its rule

The singles pay once and every level adds a helix followed a surface rule up to a cord and left one number undecided. The rule’s two readings agree on every frame but the last. For a cord of two cables, each two two-folds, the top frame is 0.207 of the singles’ spun twist if every level’s surface is to sit at one angle in the finished thread, and 0.354 if each frame is set to one over the root of its singles, as the rule is usually quoted. Neither number came from what a cord is for. A sewing thread or a cord that leaves a package has to run without twisting itself up, and whether it does is a moment, not an angle.

A folding rule was already shown not to be a torque balance. A two-fold cotton has no net moment at a fifth of its singles’ twist, and the trade folds it at seven tenths. The question here is what a stack of levels does to that disagreement, and it turns out to do something a single level gave no hint of. It does not accumulate. It alternates.

Every level pushes the other way #

A single with some twist left in it carries a torsional moment, of the hand of its twist. A folding frame lays that single on a helix of the opposite hand, and bending the single onto that helix is a moment of the fold’s hand. That was the whole of the fold’s balance: one moment of each sign, set equal. It gives the fraction $C/(B+C)$, with $C$ and $B$ the torsional and bending stiffness of the single.

A cabling frame then does to a fold what the folding frame did to the singles. It lays the fold on a helix of the opposite hand again. So bending the fold onto that helix is a moment of the same sign as the single’s own, against the fold’s. And each component carries its net moment upward with it, tipped by the cosine of the helix it now sits on. A thread is a chain of one recursion:

$$m_i = n_i\left(m_{i-1}\cos\alpha_i \pm B_{i-1}\frac{\cos^2\alpha_i\,\sin\alpha_i}{R_i}\right),$$

with the sign changing at each level, $B_{i-1}$ the bending stiffness of a component and $R_i$ the radius it is laid on. A fold’s own helix is the first bend, and a cord’s is the third.

Folding is untwisting still applies at each step, so a level above takes twist out of the one below, and what enters here is the twist each level keeps. Write $R$ for the share of the spun twist the singles keep, $K_i$ for the share level $i$ keeps. In the small-angle limit the radii cancel, as they did for a fold, and with every component’s stiffness being its singles’ stiffness added up, the moment per single in units of one single’s spun moment is

$$\frac{m}{N} = R - \frac{B}{C}\bigl(K_1 - K_2 + K_3 - \dots\bigr).$$

The whole of a thread’s torque is one fibre number times an alternating sum of what its levels keep. It has no count in it and no twist factor; a cotton cord’s top frame comes out at 0.150, 0.149 and 0.148 at twist factors of 3,000, 3,800 and 4,500. The rest of this account is reading that one line.

What the surface rule does to the sum #

Put every level’s surface at one angle and the kept shares are fixed: $K_i = R/\sqrt{N_i}$, with $N_i$ the singles in the yarn that level makes. So for a thread of twos the moment factorises:

$$\frac{m}{N} = R\left[1 - \frac{B}{C}\left(\frac{1}{\sqrt2} - \frac{1}{2} + \frac{1}{2\sqrt2} - \frac{1}{4} + \dots\right)\right].$$

The bracket is a walk of 0.707, 0.207, 0.561, 0.311 and on, closing in on a limit. A thread hangs still at the level where that walk equals $C/B$, the fibre’s stiffness ratio, and on the level’s other side it has the opposite sign or a different size. The ratio is 0.17 for a polyester, 0.25 for a cotton, 0.30 for a viscose, 0.40 for a nylon and 0.80 for a wool, in the fibre table this rests on.

So cotton’s 0.25 and polyester’s 0.17 fall nearest to the walk’s 0.207, which is the second level. A cotton cable, with every surface at one angle, very nearly hangs still: it carries +0.13 of a single’s moment per single, where the fold under it carries −1.00 and the cord above it −0.66. A polyester does the same at −0.08. Wool’s 0.80 lies nearest the first entry, 0.707, so a wool fold carries only +0.09, and the wool’s cord at +0.18 is closest after it.

This may be part of why the cabled sewing thread has its reputation. A cable of a stiff synthetic fibre sits nearly on the fibre’s own balance, under a rule nobody derived for that purpose, and a three-level construction in the same fibre falls well away from it. That was not something the rule could have known. It follows from where 0.207 happens to fall.

The walk’s limit is also exact. It is a geometric series with ratio $-1/\sqrt2$, so its sum is $\tfrac{1}{\sqrt2}/(1 + \tfrac{1}{\sqrt2}) = \sqrt2 - 1 = 0.414$. The surface rule balances a thread of very many levels only in a fibre with a stiffness ratio of 0.414, and a nylon’s 0.40 is the nearest in this table. No real thread has so many levels, but it does say which way a fibre goes: a ratio under that comes closest at an even number of levels, and one over it at an odd number.

As quoted, nothing comes close #

The usual form of the rule sets frames and not finished twists, and that is a different thread.

The moments now run −2.17, +1.71, −2.21, +1.37 for cotton, and no fibre has a level near nought. Set to the quoted rule, the kept shares swing from level to level rather than falling steadily, 0.561, 0.146 and 0.354 at a cord. A cotton cord built to the quoted rule carries more than three times the moment of a cord built to the finished-thread reading, and in the same direction. Whatever the rule as quoted is good for, a thread that runs without twisting is not it, and the earlier essay’s “liveliness” is the part of that reading this arithmetic does not support.

Where the top frame should go #

That is the question as it was left. Keep a cord’s lower frames where a folder and a cabler would set them, 0.707 and 0.5 of the singles’ twist, and move the top one.

A cotton cord hangs still with its top frame at 0.149 of the singles’ spun twist. That is below both readings, and it is much nearer the finished-thread reading than the quoted one, by 0.06 against 0.20. So of the two numbers the earlier essay could not separate, one is near the answer and the other is not in the same place at all.

Top-frame twist is steep work because of what it moves. Raising the top frame by a tenth raises what that level keeps by a tenth, lowers what the cable keeps by a tenth, and raises what the fold keeps by a tenth, because folding is untwisting all the way down. All three of those push the alternating sum the same way, and the singles’ share falls too, so the moment changes by three times the bending stiffness plus the torsional one for every unit of frame. A frame twist off by 0.06 is therefore about two-thirds of a single’s whole moment.

It should be said what the moment is in. Those units are a single’s own moment at its spun twist, and a single at twist factor 3,800 will snarl readily, in one characteristic size. A moment of two-thirds of that, per single, is not a small residue. Friction between components can hold some of it with nothing turning, and this account has no term for how much.

Fibre moves it in opposite directions #

$C/B$ is the only fibre number present, so the fibre table runs straight through. It also splits by level, and that is the part a rule of thumb could not carry.

A stiffer-twisting fibre wants more top twist at an odd level and less at an even one. A fold goes from 0.143 in polyester to 0.456 in wool. A cord goes from 0.130 to 0.250. A cable goes the other way, 0.300 to 0.155, and a cord of cords 0.166 to 0.079. It follows from the sign in the sum: more torsional moment in the singles needs more of the helix that opposes it, and that is the top level’s helix only when the top level has the fold’s hand.

So no single fraction belongs to a level, and any table of frame twists that does not name a fibre is wrong for most of them at one level or another.

What a cord that hangs still weighs #

The earlier essay’s concern was the count, and the two questions turn out not to fight.

With the lower frames held, the torque-free top frame is very nearly the lightest cord they can make: 1.84 per cent over its notation, at the curve’s bottom, with its minimum near 0.16. The finished-thread reading weighs 2.02 and the quoted one 4.54. There is no shared cause in this, since one is a moment and one a sum of secants. What they share is that setting the top frame low leaves the cable and the fold with moderate twist, and moderate twist is cheap in both.

A cord at those lower frames is still heavy. That comes from the lower frames, and the count’s account of the levels said so: everything above the singles has to sit at about half their surface angle to bring a cord back to its notation. So free the lower frames too. There are three frames and only two conditions, a zero moment and a zero excess, so a cord can do both, along a whole line of solutions.

Three points on that line, for cotton: frames of 0.344, 0.290 and 0.146; of 0.442, 0.323 and 0.131; and of 0.528, 0.333 and 0.105. Each hangs still and weighs exactly its singles. All of them are far under the rule’s 0.707 and 0.5 for the first two frames. A cord built to hang still and to weigh what it says is a noticeably softer-twisted thread than a rule-book cord, and what it gives up is whatever those lower frames were buying, which this arithmetic does not price. Two thirds is where a fold weighs its singles and a torque-balanced fold is a fifth; the cord’s solutions sit between those for its first frame.

The bracket, and why a cord is less exposed to it #

Everything above uses the free end of the bending bracket, every single bending alone. A real fold has its singles pressed together, and a stuck-together component is stiffer, up to $N^2$ singles’ worth. A thread’s second stiffness carried a bracket like this at one level. It is wider here, because the upper levels’ bending counts for more.

A cable’s answer moves by a third across that bracket and a cord’s by about a ninth. A cord’s top helix is a larger stiffness at the stuck end, but the cable’s helix in the same thread is also stiffer and of the opposite sign, so most of the change cancels. That also says where the conclusion above can be trusted. The cotton cord’s frame stays below both readings at every point of the bracket, so that ordering holds wherever a real cord sits. A cable’s closeness to balance at the finished-thread reading, by contrast, is a closeness at the free end only.

How this was computed #

A thread is a stack of levels of twos, from 20 tex cotton at a twist factor of 3,800, using the kept twists the count’s account already uses, in which each frame puts in what its level keeps plus what the level above takes out. Components sit on touching-circle radii and their helices are exact at every level, cosines and sines both, so the small-angle sum above is the reading and the recursion is the number.

A top frame that hangs still is found by bisection on the recursion, inside the range where every level keeps some twist. The five fibres take their stiffness ratios from the same table as the fold. The solutions that both hang still and weigh the notation come from a scan of the first kept share in steps of a twentieth, with the second solved on a fine grid and the third by bisection.

Required of it, not shown: that one level return the fold’s own balance for cotton, wool and polyester; that the alternating closed form match the recursion at low twist; that more twist in a cable’s own level push back against the fold below it; and that a cord at 0.528, 0.333 and 0.105 have both a moment and an excess near nought.

Where the helices stop being the thread #

Friction is absent, and it is the largest missing term. A moment in a thread is held by contact between components, and the thread only turns when the moment beats the friction it has to overcome to rearrange. So “hangs still” here means zero moment, and a real cord may hang still across a band around it. That band’s width is not computed, and it may make the 0.06 between the answer and the finished-thread reading unobservable at a low enough tension. The prediction for the quoted rule’s 0.20 is a larger thing to hide.

Moments are taken at a free thread with no tension. Bending and twisting a component under load changes its radius and its helix, and a package unwinding at speed is not at rest.

A real level is not a tidy two. Three- and four-component levels have different offsets, cords of mixed levels have different sums, and a sewing thread’s own bonding or lubricant changes the bracket’s position. The arithmetic takes any level sizes, but these have been run only for threads of twos.

And residual moment relaxes. A thread steamed or set has given much of it up, and a yarn that has been set has no torque. So all of this describes a thread before setting. It is exactly a statement about how much a setting operation has to do, and nothing about how much it can.

Who worked out which part #

Cabling and cording rules, S-on-Z alternation and the term “balanced” are all long-established practice; a thread’s frame twists are trade tables and sewing-thread makers keep their own. The torque of a two-fold yarn in terms of its bending and torsional stiffness is standard yarn mechanics.

What is added here is the alternating sum: a thread’s net moment as one fibre ratio times what its levels keep, with the signs set by the hands; the walk under the surface rule, 0.707, 0.207, 0.561, 0.311 and on towards 0.414, against which a fibre’s ratio says which level comes closest; the cotton cord’s top frame at 0.149, below both readings; the fibre’s opposite effect at odd and even levels; and the line of cords that do both jobs.

Still open: how wide a band a thread will hold #

Everything here says where the moment is nought. What a reader of a package wants to know is how far from nought a thread can be before it does anything visible, and that is a friction result with the parts already to hand: a component’s contact force on its neighbours, and a coefficient.

The prediction is a band in frame twist, proportional to the friction and to the contact force, inside which a thread hangs still without being balanced. If it is wide, the finished-thread reading sits inside it for a cotton cord and the distinction between 0.149 and 0.207 is one no user can feel. If it is narrow, the trade’s reasons for setting a thread after cabling become an exact measure of how far the rule missed. A cord hung in a loop at a series of top-frame twists from 0.1 to 0.35, and its loop’s rotation counted, would read the width directly.