A cabled yarn is a fold of folds

A cabled yarn is a fold of folds. Take singles, fold them; take the folds, fold those. Sewing thread is often built that way, so is much cord and twine, and so is the yarn in a good many carpets.

The trade quotes folding ratios for two, three and four folds and quotes nothing at all for cables. A rule that has been shown to be geometric rather than mechanical should extend upwards without any new ideas, and it does — but it also runs into the thing that limits it, which is worth reaching.

The rule, restated in the units that make it obvious #

A fold’s twist ratio to its singles is one over the square root of the number of singles, and the reason is that a fold of n components is √n times the diameter, so matching the surface helix angle divides the twist by √n.

In the trade’s own units it is shorter than that. A twist factor is turns per unit length times the square root of the count, and it is proportional to the tangent of the surface helix angle. So the rule is:

A fold has the same twist factor as its components.

Written that way there is no division to do and nothing to remember, and the rule extends to any number of levels without a moment’s thought.

Two levels #

A cable of m folds, each of n singles, has three twist levels. Write the singles twist as T and take the rule at each step.

The fold carries n singles, so its twist is T/√n and its count is n times the singles’ count, by the same arithmetic that makes it √n across.

The cable carries m folds, so its twist is the fold’s over √m, which is T/√(nm), and its count is nm times the singles’.

So a cable’s twist ratio to its own singles is one over the square root of the total number of singles in it, whatever way they are grouped.

That is the first prediction and it is slightly surprising: a six-fold cable made as three folds of two and one made as two folds of three should have the same overall twist ratio. Their intermediate twists differ — T/√3 against T/√2 — and their final twists do not.

The numbers #

For the common constructions:

A two-by-two cable — two folds of two singles — has a fold twist of 0.707 of the singles and a cable twist of 0.500. A three-by-two — three folds of two — has a fold twist of 0.707 and a cable twist of 0.408. A two-by-three has a fold twist of 0.577 and a cable twist of the same 0.408. A three-by-three has 0.577 and 0.333.

None of those is in any table this collection has found. All of them follow from one rule with no free constants and two integers.

The direction of each level #

The rule fixes magnitudes and says nothing about hands, and hands are where cabled yarns are actually distinguished.

The convention is that each level reverses: singles Z, folds S, cable Z. That is the arrangement that keeps every level’s twist working against the level below, and it is why a cabled thread lies flat and does not kink.

Reversing at every level also has a consequence for the surface rule that is worth noticing. Matching the magnitude of the surface angle is what the rule requires; the sense alternates, so a cable’s surface runs the opposite way from its folds’ surfaces at the same angle. That is visible in a good sewing thread and is one of the ways to tell a cable from a fold by eye.

Where the arithmetic starts to fail #

The rule rests on one assumption, and at two levels it is under more strain than at one.

A fold of n components is √n times the diameter only if it packs as densely as its components do. It does not, because round objects do not tile: two circles in a circle occupy about eighty-three per cent of it, three about seventy-seven, and the fraction gets worse before it improves.

At one level that costs perhaps ten per cent on the diameter, which moves the predicted ratio from 0.707 to about 0.64 — comfortably inside the trade’s bracket and, as it happens, closer to its middle.

At two levels the losses compound. A cable of three folds of two is packing imperfect circles into imperfect circles, and its diameter is meaningfully more than √6 times a single’s. So the cable ratio predicted at 0.408 should in practice be lower, perhaps nearer 0.35.

That is a prediction rather than an excuse, and it is testable: the departure from the ideal rule should be larger for cables than for folds, and larger for cables of many components than of few.

What a cable is for #

The reason to build a yarn this way is worth stating, because it explains which of the rule’s predictions matter.

Roundness. A two-fold yarn is not round; it has a visible groove between its components and a section that is closer to a figure of eight than to a circle. Three or more components are rounder, and a cable of folds is rounder still, because each level averages the last. A sewing thread has to pass through a needle eye and round a bobbin case thousands of times a minute and roundness is what makes that survivable.

Evenness. Folding averages the thin places of its components, and this collection has computed how much: the improvement goes as the square root of the number of components, so a cable of six averages more than a fold of six because it does it in two stages with different groupings.

Strength. More components means more fibres sharing the load, but the gain is not linear because of the obliquity: fibres at a helix angle contribute the cosine of that angle. A cable’s fibres are at two helix angles compounded, which costs more obliquity than a single fold of the same total count.

And torque. A cable with alternating hands has less residual torque than a fold, because each level partly cancels the last — which is why cabled sewing threads are the ones used at speed, and why what a yarn is balanced about matters more here than anywhere.

The obliquity, which is the real cost #

The strength point deserves its own paragraph because it is the one that limits how far cabling is worth taking.

A fibre lying at an angle to the yarn’s axis contributes to the yarn’s strength by the cosine of that angle, and this collection has computed the cost. In a cable, a fibre is at its own helix angle within its single, that single is at a helix angle within the fold, and the fold is at a helix angle within the cable.

Those angles compound. A fibre near the surface of a single near the surface of a fold near the surface of a cable is at a substantial angle to the cable’s axis, and it contributes correspondingly less.

So each level of cabling costs strength, and the surface rule makes that cost predictable: since each level runs at the same surface angle, each level costs about the same fractional loss, and the losses multiply.

Two levels is common. Three is rare, and the reason is arithmetic rather than manufacturing: a third level costs another factor and buys very little roundness that the second did not already buy.

What the grouping does decide #

The overall twist ratio does not depend on how the components are grouped, and almost everything else does. It is worth listing, because a maker choosing between three-of-two and two-of-three is choosing among these.

Roundness. Three components are rounder than two at every level, so a two-by-three — two folds of three — is rounder in its intermediate stage and less round at its final one, and a three-by-two is the reverse. Which matters depends on which surface the yarn presents.

Evenness. Averaging in two stages of three and two is not the same as two and three, because the improvement at each stage goes as the square root of the number averaged and the stages are not commutative in what they average over. This collection has the arithmetic for one stage and has not composed it.

Obliquity. The compounded helix angles differ between the two groupings, because the intermediate twist differs — T/√2 against T/√3 — and the angles add up differently.

And the residual torque. Each level cancels part of the level below, and how much depends on how much twist that level carries. The groupings therefore leave different residuals even though they end at the same overall ratio.

So the rule fixes one number and leaves four, which is a fair description of what a geometric rule can do.

Why sewing thread is the worked example #

Sewing thread is where all of this is actually decided, and it is worth saying why, because it explains the constructions that exist.

A sewing thread passes through a needle eye at a few thousand stitches a minute, is bent round a hook, is abraded against the fabric at every stitch, and must not snarl, spin, kink or fluff. Almost every property this ladder has discussed is being pushed at once.

The constructions that survive are cables: three-by-two for a fine cotton, two-by-three or three-by-three for heavier work, and always with alternating hands. A single, however strong, is unusable — it is lively, it is not round, and it abrades.

So the class of yarn where the rule matters most is the class where the trade quotes no rule, and that is not a coincidence. Sewing thread is made by a small number of specialists to constructions arrived at over a century, and the numbers live in their process sheets rather than in textbooks.

That makes the prediction here worth something: it says what those process sheets should contain, and it says it from two integers.

What was counted, and how #

The surface rule is an identity and is checked as one: matching a fold’s surface angle to its components’ divides the twist by exactly the square root of the component count, to twelve figures, across the counts the trade quotes.

Applying it twice is composition and needs no separate check, because the second application takes the fold as its component and the fold’s count and diameter are the ones the first application produced.

The diameters come from counts, densities and packing factors by this collection’s own volume arithmetic, which has asserted the √2 relation for a two-fold since the ply ladder.

The packing losses are quoted from the standard circle-packing fractions and are not computed here: they are named to say which direction the correction runs and how large it might be, not to produce a corrected prediction.

Where the model stops #

The packing correction is not made. Everything above uses the ideal √n, and the honest prediction for a real cable is that the ratio should be somewhat lower. Making the correction properly needs a packing model for folds inside a cable, which this collection does not have and which is not straightforward, because the components deform against one another rather than staying round.

The components are treated as identical. A great many cables are not: a fine binder cabled round a coarse core is a different object and the rule has no obvious form for it.

And nothing here is a measurement. The trade quotes no cable ratios, so there is nothing to check against, which is why the rung produces predictions rather than confirmations.

Nor does the rule say anything about how many levels are worth having. That is a question about roundness, evenness and obliquity together, and only the third of those has an arithmetic here.

The generalisation #

The point of the rung is not cables. It is what a geometric rule buys over a mechanical one, and the difference is composability.

A mechanical rule has constants in it. Applying it at two levels means asking whether the constants are the same at both levels, which for a torque balance they are not — the fold’s torsional rigidity is not simply its singles’ summed, because the components are at an angle.

A geometric rule has integers in it, and integers compose. One over the square root of n, applied twice, is one over the square root of nm, and there is nothing to check.

That is a general reason to prefer a geometric account when one is available, and it is stronger than the usual reason. The usual reason is that geometry is more certain. The better reason is that geometry extends to cases nobody has measured, and a mechanical rule with fitted constants extends only as far as the fitting went.

This collection has one other rule of that shape and has not exploited it. The satin count — which satins exist at a given number of ends, and why six has none — is a purely arithmetic condition, and it composes upwards to compound and multi-layer structures in a way the site’s own derivation ladder has not followed.

The measurement that is available and has not been made #

There is a cheap measurement here and it is worth describing, because unlike most of this collection’s proposals it needs no equipment at all beyond what a mill already has.

Take a commercial cabled sewing thread of known construction — a three-by-two cotton, say. Measure its twist at the cable level with an ordinary twist tester. Untwist it to its folds and measure one of those. Untwist that to its singles and measure one of those.

The rule predicts two ratios: the fold’s twist to the single’s should be 0.707, and the cable’s to the fold’s 0.577 — or, in the units the tester’s own arithmetic produces, all three twist factors should be equal.

If they are equal, the rule holds at two levels and the packing losses are small. If the cable’s twist factor is systematically below the fold’s, which is what the packing argument predicts, the departure measures the packing loss directly.

Either result is worth having, and the second is worth more: it would give this collection its first measurement of how much a fold departs from ideal packing, which is a number it has needed in three other places and has always assumed.

What it would mean if the twist factors were not equal in the other direction #

A result in the opposite direction — the cable’s twist factor above its fold’s — would be more interesting still, and it is worth saying what it would imply.

The surface rule’s whole content is that the maker is matching an angle. If a cable is twisted harder than the rule says, the maker is deliberately putting more angle on the outside of the cable than on its components, and there is an obvious reason to want that: a steeper surface angle grips better, and a cable’s job is partly to keep its folds from separating.

That would mean the rule is a floor rather than an identity: match the angle at least, and go beyond it where the structure needs holding.

Which of the three outcomes obtains is exactly the kind of thing three measurements on one spool would settle, and it is the reason to prefer a prediction with a direction attached over one that merely quotes a number.

Who found it, and when #

Cabled yarn construction is old and the conventions for it are trade practice rather than published rules. Gégauff’s surface-angle relation is from 1907 and the square-root diameter relation is elementary.

What is this collection’s own is applying the rule at more than one level, observing that it composes into one over the square root of the total component count, and noticing that the grouping cancels — so that two ways of making a six-component yarn should be twisted the same overall and differently in the middle.

Where the ladder goes next #

The torsion ladder ends here and the reckoning is owed: what a second stiffness bought, what it cost, and which of this collection’s standing questions it left exactly where they were. Where a torsion model stops is that accounting, and it is shorter than the list of things the ladder found.