The singles pay once and every level adds a helix

A cable weighs its singles at the twists the finished thread keeps ended on an observation that looked like the beginning of a law. Twisted where the cabling rule is usually quoted, a two-by-two cotton cable is three and a half per cent heavier than its four singles. Twisted so that every level’s surface sits at one angle in the finished thread — singles, folds and cable all presenting the same slope of fibre to the eye — it comes back to within half a per cent of its notation. Equal angles and an honest count arrived together, apparently for a second time, since a two-fold folded at its rule weighs its singles too.

If that were a law, it would reach as far as anyone twists. Rope is laid in three levels — yarns into strands, strands into a rope, ropes into a cable-laid rope — and a heavy sewing thread or cord can be built the same way. The question it left was whether a cable of cables repeats the agreement a third time.

It does not, and the reason shows that it was never a law. With every surface at one angle, the singles keep the same twist whatever sits above them, so the length they save by being untwisted is paid once. Every level then adds a helix, and in a thread of twos every helix is the same length. The count walks upward by one helix per level, and the cable is simply the level at which the walk passes zero.

What one more level does to the arithmetic #

A cord is built the way a cable is, one level higher. Two cables are twisted together the other way from the cables themselves, so the cord twist runs the same way as the folds and against the singles.

A cabled yarn is a fold of folds, and the cable essay found that every frame does to the level beneath it what folding is untwisting found the folding frame does to the singles: it winds each component round its neighbours without letting it turn freely, so each component loses one turn of its own for every turn of the winding. The arithmetic generalises without a new idea. The top level keeps what its frame put in; each level beneath it keeps its frame’s twist less whatever the level above kept; the singles keep their spun twist less whatever the folds kept. For a two-fold that is one subtraction, for a cable two, for a cord three.

The count is then the same product it has been since the two-fold. The singles’ factor is their saving: a single left with less twist than it was spun with has straighter fibres, and so holds more single for the fibre in it. On top of that, every level contributes the secant of its own helix — each component running round the axis of the level above it, at the touching-circles offset for that many components, so there is more component than yarn by the secant of the angle it winds at. One level is the two-fold exactly, two are the cable exactly, to the fourteenth figure; three are a cord.

Equal angles fix what the singles keep #

The cable essay’s rule in the finished thread is the condition that every level’s surface tangent — π times its diameter times the twist it keeps — equals the singles’. Write $R$ for the fraction of the spun twist the singles keep and $K_i$ for what level $i$ keeps, $N_i$ singles across. The condition is

$$R = \sqrt{N_1}\,K_1 = \sqrt{N_2}\,K_2 = \dots,$$

and the singles keep their spun twist less the folds’, $R = 1 - K_1$.

Those two lines contain the whole of the surprise. The first level and the singles settle $R$ between them, at $\sqrt n/(1+\sqrt n)$ of the spun twist, 0.586 for two singles a fold — and nothing above the first level enters it. A cable, a cord and a cord of cords all keep exactly the singles a plain fold keeps, and every level above keeps its share of $R$ divided by the root of its size.

The rule as quoted behaves quite differently. It sets each frame at one over the root of the singles in the yarn that frame makes — 0.707, 0.500 and 0.354 of the spun twist for a cord of twos — and what each level keeps then depends on everything above it. In a cable the folds keep 0.207 and the singles 0.793. In a cord the cord takes twist out of the cables, the cables keep only 0.146, and since the cables’ frame then takes less out of the folds, the folds get back to 0.561 and the singles drop to 0.439. Each level’s twist alternates with the number of levels stacked on it.

Read in the finished thread, the cable and the cord agree level for level as far as both go: singles 0.586, folds 0.414, cables 0.293, and the cord, 0.207, one root of two below the cables.

There is a neater way to say it. In a thread whose every level twists the same number together, the finished-thread reading sets every frame exactly where the rule as quoted sets it, except the top one. A frame puts in what its level keeps plus what the level above will take out, and with every level a root of two below the one beneath, that sum is $1/\sqrt{N_i}$ — the rule as quoted, exactly. The cable essay found that the two readings agreed about the folding frame and differed about the cabling frame; that was a particular case. A finished-thread cord is a quoted cable — folds at 0.707, cables at 0.500 — with a cord twist of 0.207 on top, against the 0.354 the rule as quoted would ask.

Each level costs the same length #

The singles’ saving is therefore fixed: the same 0.586 of their spun twist kept, the same straightening, at every number of levels. At a twist factor of 3,800 it takes 2.99 per cent off the count.

The helices are fixed too. A level’s helix angle is set by how far its components sit from its axis times the twist it keeps, and in the finished thread both scale together: the components are a root of two fatter at every level and keep a root of two less twist, so the product does not change. In a thread of twos every level winds at 10.5 degrees, and every level adds 1.69 per cent.

The rest is subtraction. One helix against the singles’ saving leaves a two-fold 1.35 per cent light. Two helices leave a cable 0.32 per cent heavy. Three leave a cord 2.02 per cent heavy, and four a cord of cords 3.75. Each step is one helix, multiplied in rather than added, which is why the steps differ from 1.69 points in the second decimal.

So the cable’s agreement with its notation is a coincidence of the number two. Two helices of 1.69 nearly pay off one saving of 2.99, and that is all the cable essay’s result was. It happens at every weaving twist factor, because both lengths are squares of the same angle to first order and their ratio barely moves — at 3,000 the cable is 0.16 per cent heavy, at 4,500 0.54 — but a cord is out by 1.22 to 2.92 per cent across the same range, and a cord of cords by 2.29 to 5.35.

The two-fold was never part of the pattern either. Folded at its own rule as quoted, 0.707, a two-fold weighs its singles to within two thirds of a per cent, and the cable essay counted that as the first time equal angles met an honest count. But a two-fold folded at 0.707 matches the angle its singles were spun at, not the angle they keep inside it; with every surface at one angle in the finished fold it would be folded at 0.414 and come out 1.35 per cent light. The two coincidences happened under two different readings of the rule, which is not a law happening twice.

What the cord’s surfaces look like #

The finished-thread reading is the cable essay’s, applied once more, and it does what it did there: it puts every surface at one angle.

Twisted as quoted, a cord’s four surfaces sit at four different angles: singles 11.1 degrees, folds 19.5, cables 7.4 and the cord 24.0. Only the top matches the singles’ spun angle, as in the cable, because only the top level has nothing above it to untwist it. The cord has also moved its imbalance: in a cable as quoted it was the folds that were nearly flat, at 7.4 degrees; in a cord it is the cables, and the folds have been given back most of what they lost.

Read in the finished thread, all four sit at 14.6 degrees — the same angle as the cable’s three, and the same as a fold’s two would be, because $R$ does not depend on how many levels there are. That is the sense in which this reading is a single rule: it makes every thread, however built, look alike at every level. What it does not do is make every thread weigh alike.

The angle at which a cord would weigh its notation #

If equal angles do not give an honest count, what would? The cleanest question to ask keeps the reading’s shape — every level above the singles at one common angle — and lets that common angle be a multiple of the singles’ own, then asks where each thread crosses its sum.

Each thread has one crossing, and they fall steadily with the number of levels. A two-fold weighs its singles with its own surface at 2.72 times the singles’ angle. That is not as hard as it sounds: at that angle the fold keeps 0.658 of the spun twist and its singles 0.342, which is the two-thirds fold again, its surface steep only against singles it has nearly untwisted. A cable crosses at 0.87, just under one, which is why the equal-angle reading lands near its notation. A cord needs 0.52 — every level above the singles at about half their angle — and a cord of cords 0.37.

The cord’s number is the useful one. It says that an honest cord count needs its upper levels twisted noticeably softer than its singles, and it says by how much: about half, measured as the tangent of the surface angle. A cord twisted that way has fewer helix turns to pay for and lighter singles to pay with, and the two meet at the notation.

Whether such a cord is worth making is not a question about weight. A softly twisted upper structure is looser and more likely to open under abrasion, and a fold’s strength depends on the twist its singles keep, on an account this calculation does not touch.

Every cord construction is heavy #

The walk is not a property of threads of twos. The singles’ saving is set by the first level alone, and every level above it adds a helix whose length depends on how many components it twists; a level of three sits its components at a slightly different offset from its axis, relative to its own diameter, than a level of two, so its helix is a little different. The direction never changes.

Under the rule as quoted the four constructions are 2.6 to 6.6 per cent heavy across the weaving twist factors. Read in the finished thread they come down to 1.11 to 4.22. Reading the rule properly takes off between two fifths and three fifths of the error, and leaves every cord heavier than a cable read the same way, which sits at 0.16 to 0.54. There is no construction among them, and no twist factor, at which a cord with every surface at one angle weighs its notation.

For a 2×2×2 cord of 20 tex singles, notation 160 tex, that is 163.2 tex in the finished thread and 167.3 as quoted. A reel costed at the notation is two per cent light on its cotton at best and four and a half at the rule as written — larger, in both readings, than the cable’s error that a sewing thread costing has to absorb.

What a twist tester should see #

The cable essay proposed its own test: untwist the cable and record the lengthening, then untwist a fold and record it again, and the ratio of the two tells the readings apart. A cord gives three readings.

Twisted as quoted, the cord should lengthen 4.86 per cent as the cord twist comes out, 0.43 as each cable is untwisted to its folds, and 3.08 as each fold is untwisted to its singles — a large step, a small one and a large one, the alternation of the kept twists showing up directly as lengths. Read in the finished thread, the three steps are 1.69 per cent each. This is a stronger test than the cable’s, because the quoted reading’s middle step is seven to eleven times smaller than its neighbours, and nothing in the finished reading is smaller than its neighbours at all.

The equal steps are also the clearest statement of this essay’s argument. Every level of a finished-thread cord gives back the same length when it is taken out, which is the same as saying every level cost the same length going in. The singles’ saving appears in none of the three steps, because it was paid when the first level went in and never changed after.

How the levels were computed #

The singles are the ideal helical yarn of twist is one angle: fibres spread evenly over a round section at the volume arithmetic’s diameter for a 0.6 packing, the retraction the closed-form mean secant over the section. Each level is a set of touching round components about its axis — half a component’s diameter off for two, the circumscribed offset for three — each component taken as $\sqrt N$ singles across, the diameter the folding rule’s surface angle uses. Each level keeps its frame’s twist less the level above’s, the singles their spun twist less the folds’.

The calculation is required to reproduce the fold at three folding ratios and three twist factors and the cable in all four constructions under both readings, to fourteen figures; to give the same thread whether it is read by its frame twists or its kept twists; to drop the count to the twelfth figure; to put every level’s surface at the singles’ angle in the finished reading, to nine figures; to give every level of a thread of twos one helix to twelve; to find each added level exactly one helix heavier on the same singles; to find the crossing angle falling with every level; to set every frame but the top one where the rule as quoted does, in every square thread; and to refuse a thread of no levels or five, a level of one component or five, a thread given both ways or neither, a level left with no twist of its own, and singles left with none.

What three or four ideal helices leave out #

The components are taken round. A fold inside a cable is pressed out of round, and a cable inside a cord more so. The cable essay found that taking a fold as the smallest circle round its singles, rather than at its count’s diameter, roughly doubles the cable’s helix; a cord has two such levels, and flattened components would move both its helices the other way. Neither is placed here, and the cord’s 2.02 per cent is a statement about round components.

The helices are taken in series. A fibre in a cord follows four nested helices, and the product of four secants is the first approximation to its path. At ten degrees a level the correction is of higher order, and its sign is not computed.

And the levels are assumed to settle as the arithmetic says. A real cord is twisted under tension, relaxes on the package, and may migrate twist between levels as it does. The kept twists here are what the frames impose, not what a cord holds after a month on a cone.

None of the three changes the shape of the result. The singles’ saving is set at the first level whatever happens above it, and every level above adds a positive length. A cord can be brought to its notation only by taking twist out of its upper levels, never by reading the same twists a different way.

Who worked out which part #

Multi-level threads with alternating hands are older than any arithmetic for them, in rope as much as in thread, and the trade’s reason for alternating is torque — what a balanced yarn is balanced about separates the things that word means. The untwisting identity is classical; Treloar gave the geometry of multi-ply and cabled structures in the 1950s; the retraction of an ideal helical yarn is in Hearle, Grosberg and Backer’s collected structural mechanics of 1969.

What is added here is the level arithmetic for any number of levels; the observation that with every surface at one angle the singles keep the same twist however many levels sit above them, so their saving is paid once while every level adds one helix of the same length; the walk that follows, light fold, honest cable, heavy cord; the crossing angle for each, 0.52 of the singles’ for a cord of twos; the finding that the two readings of the rule set every frame alike but the top one in any square thread; and the three equal untwisting steps that tell a finished-thread cord from a quoted one.

Still open: what a cord’s top frame is for #

The finished-thread reading and the rule as quoted agree on every frame but the last, and the last is exactly where the weight is decided: 0.207 or 0.354 of the singles’ twist for a cord of twos. Neither number came from weight. Both came from a surface rule, and a surface rule is a statement about appearance and, in the trade’s own account, about liveliness.

The question this leaves is what the top frame should be set to if the aim is a cord that does not twist itself up when it is let go. That is a torque calculation, level by level, with each level’s torque set by the twist it keeps and its own bending and twisting stiffness — the same nested arithmetic as this one, with moments where this has lengths. It would say whether a torque-free cord sits near the finished-thread reading, near the rule as quoted, or somewhere else, and so whether a cord built to hang still is built to weigh what it says.