Two thirds is where a fold weighs its singles

A cotton spinner who folds two singles of 40s together sells the result as 2/40s, and everybody reading the ticket knows what it weighs: a 20s, half the count of either single, twice the mass a metre. The yarn count systems lay out how many ways the trade has of saying how fine a yarn is, and in every one of them the folded count is the singles’ count with the number of singles folded in by plain arithmetic. That arithmetic is a claim about length. It says a metre of fold contains exactly two metres of single, and that a metre of single inside the fold contains exactly the fibre a metre of single held before it was folded.

Neither half is true, and the error in each is several per cent. What is surprising is what happens when the two are put together.

Two lengths change, and they change in opposite directions #

The fold’s helix lengthens every single. Two singles twisted round each other do not lie along the fold’s axis. Each centre line runs round it as a helix whose radius is half a single’s diameter, so there is more single than fold, by the secant of the angle that helix makes with the axis. For a 20-tex cotton at a twist factor of 3,800 — 850 turns a metre in the single, an ordinary weaving twist — folded at two thirds of that, the helix stands at 16.6 degrees, and every metre of fold holds 4.33 per cent more single than a metre.

The untwisting shortens what each single carries. Folding is untwisting: a single wound round its neighbour turns about its own axis once for every turn of the fold, so a Z single folded S at 566 turns a metre is left with 283 of its own. And a twisted yarn is shorter than the fibre in it, because every fibre away from the axis follows a helix and a helix is longer than its axis. At 850 turns the single’s surface angle is 24.0 degrees and its fibres are on average 4.8 per cent longer than the yarn they are in. At 283 turns the angle is 8.5 degrees and the excess is just over half a per cent. So a single that loses most of its twist gets longer for the fibre it holds, and its own count falls — by 4.07 per cent here.

The two land within a tenth of a per cent of each other. The fold is 0.08 per cent heavier than its notation says.

Why the two curves have to cross #

The shapes in that figure are not a coincidence of these constants, and it is worth seeing why before seeing where.

Both changes are second order in the twist. A helix of small angle is longer than its axis by half the square of its tangent; a twisted yarn is shorter than its fibre by a quarter of the square of its surface tangent, the quarter coming from averaging over a section in which the fibres at the axis are straight and only the outer ones are at the full angle. So each length is a square, and the two squares are of different things.

The ply’s helix grows as the square of the folding twist. Fold harder and its contribution rises without limit, as fast as the fold’s own angle rises.

The singles’ gain is the difference of two squares: the singles’ original twist squared, less what is left squared. Fold a little and the residual falls a little, and the gain is nearly linear in the folding twist. Fold hard and the residual approaches nothing, and the gain approaches all the retraction the single had in the first place — a ceiling it cannot pass, because the fibres can be no straighter than straight.

A curve that grows without limit and a curve that levels off at a ceiling, both starting from nothing, cross once. Below the crossing the untwisting wins and the fold is lighter than its notation; above it the helix wins and the fold is heavier.

The crossing is exactly two thirds #

The squares can be written out. With $\tan\alpha = \pi d T$ for a single of diameter $d$ at $T$ turns, and $\tan\beta = 2\pi o T_f$ for a ply whose singles’ centres sit a distance $o$ off its axis, the two lengths cancel where

$$\tfrac14\,\pi^2 d^2\left(T_s^2 - (T_s - T_f)^2\right) = \tfrac12\,(2\pi o T_f)^2,$$

and dividing through by $T_f$ leaves a straight line in the two twists,

$$\frac{T_f}{T_s} = \frac{2}{1 + 8\,(o/d)^2}.$$

For two singles lying side by side, each centre is half a diameter off the axis, $o/d = \tfrac12$, and the ratio is $2/(1+2)$: exactly two thirds.

That is the number a folder quotes. The folding rule is a surface angle found a different number in the same place — one over the root of two, 0.707, the ratio at which the fold’s surface runs at the angle its singles’ surfaces run at — and read the trade’s two thirds as that number rounded to a fraction. It was a fair reading. It is no longer the only one, because a second condition with no free constant in it gives two thirds without rounding anything.

For three singles in a triangle the centres sit on a circle of radius $d/\sqrt3$, and the ratio is $2/(1 + 8/3)$, six elevenths, 0.545. For four in a square they sit at $d/\sqrt2$, and the ratio is $2/(1+4)$, two fifths.

The exact calculation, without the small-angle expansion, puts the crossing slightly lower: 0.661 at a twist factor of 3,000, 0.658 at 3,800, 0.654 at 4,500. At those angles the retraction’s next term is no longer negligible, and it is the term that makes the singles’ ceiling arrive sooner. Even at a crêpe-like 6,000 the two-fold’s crossing is 0.645 — still inside the trade’s band, and still within a twentieth of the limit.

The count does not appear, and neither does the fibre #

Nothing in the crossing depends on how fine the singles are. The angle a single’s surface makes is fixed by its twist factor — turns a metre times the root of the count — and not by the twist or the count separately, which is the whole reason the trade specifies twist that way. Both lengths are functions of angles alone, so a 10-tex fold and an 80-tex fold at one twist factor and one folding ratio differ from their notation by exactly the same fraction, to the twelfth figure.

Nor does the fibre appear, except through the angle. Neither length has a modulus or a friction in it. A fibre’s density does reach the angle, because a lighter fibre makes a fatter yarn of the same count, and a wool at a twist factor of 3,800 stands at 25.7 degrees where a cotton stands at 24.0 — but that moves the crossing from 0.658 to 0.656, the same small drift a harder twist gives. The small-angle limit has no angle in it at all. The only things it knows are how the singles are arranged round the axis, which is why the answer comes out as a pure number for each arrangement.

Every folding rule that lands in the trade’s bracket has now turned out to be geometry. The folding rule is not a torque balance found that a balance of moments, which does contain the fibre, lands at a fifth rather than two thirds. The surface rule and the weight rule both lack the fibre’s mechanics, and both land in the bracket. What a balanced yarn is balanced about separated a fold balanced for torque from one balanced for its surface; a fold balanced for its weight is a third meaning of the word, with a third number.

Two exact conditions, one bracket #

There are now two conditions in the folder’s bracket, and the three fold counts tell them apart.

For two singles, the surface condition is 0.707 and the weight condition 0.654 to 0.667, depending on the twist. Both are inside the trade’s 0.60 to 0.75, and the folder’s “two thirds” lies between them — at the second exactly.

For three singles, the surface condition is 0.577 and the weight condition 0.529 to 0.545. Both are inside 0.50 to 0.65.

For four singles, the surface condition is 0.500 and the weight condition 0.379 to 0.400. The trade’s bracket is 0.45 to 0.60. The surface condition is inside it and the weight condition is not.

So the four-fold decides it. A folder who folded for weight would fold a four-fold at two fifths; the trade folds it at a half or more. The surface rule remains the condition practice tracks. What this calculation adds is the reason a folder following that rule never had to correct the ticket: at the surface rule, the weight condition is close enough to be invisible.

That is a real finding rather than a consolation, and it is worth stating as one. The notation 2/40s is right because a two-fold folded for its surface happens to be folded within a few hundredths of the ratio at which it weighs its notation. Had the surface rule landed at a half for two singles, as it does for four, every two-fold in the trade would weigh about a per cent less than its ticket, and somebody would long ago have written a correction into the count tables.

What a ticket is out by, wherever the trade folds #

The useful number for a mill is not the crossing but the distance from it over the whole of practice.

Over the two-fold bracket and twist factors from 3,000 to 4,500, a fold weighs between 0.6 per cent less and 1.3 per cent more than its notation. At the surface rule itself, the range is 0.3 to 0.7 per cent heavy. A 2/40s cotton folded at one over root two at a twist factor of 3,800 weighs 29.66 tex against the 29.53 its notation implies: a 19.9s sold as a 20s.

Three singles behave almost as well: 0.3 per cent light to 1.7 per cent heavy over their bracket.

Four singles do not. Folded anywhere in their bracket, they are always heavier than their notation, by 0.4 to 3.9 per cent, because the bracket sits entirely above their crossing. At the surface rule a four-fold is 0.8 to 1.8 per cent over.

Against what a laboratory can see, these are small. A count is checked by reeling a lea and weighing it, and the spread between leas of one ordinary yarn is a couple of per cent. A systematic half per cent is invisible in any one test and would need tens of leas to resolve. A four-fold’s two or three per cent is detectable with care, and a cloth woven from four-fold warp and weft would weigh that much more than the weight arithmetic predicts from the ticket — an error a fabric designer would attribute to the crimp before suspecting the count.

Folded the same way, nothing cancels #

The cancellation depends on the fold going against the singles. Almost every fold does, and the reasons are the torque and the surface; the weight is a third reason nobody gives.

A fold twisted the same way as its singles — a crêpe fold, or a cord meant to be lively rather than stable — adds twist to each single instead of removing it. Now the singles’ fibres lie steeper, the singles retract further, and their count rises. The ply’s helix still lengthens them. Both lengths add.

At half the singles’ twist folded the same way, a two-fold weighs 8.0 per cent more than its notation; at seven tenths, 13.2 per cent. Those are not small, and they are the folds whose notation a designer can least afford to trust, because a crêpe’s liveliness is quoted through a twist factor, and a twist factor has the count in it. The tex system writes a folded yarn’s resultant count separately, prefixed R, precisely because it need not be the sum; for a same-way fold, it never is.

How the two lengths were computed #

Each single is the ideal helical yarn twist is one angle uses: fibres spread evenly over a round section, each following a helix of constant radius at the yarn’s twist, so that the fibre’s length per unit of yarn is the secant of its own angle and the yarn’s retraction is that secant averaged over the section. The average has a closed form, $2\big((1+\tan^2\alpha)^{3/2}-1\big)/(3\tan^2\alpha)$, and the calculation checks a quadrature against it.

The single’s diameter comes from its count by the volume arithmetic, at a packing of 0.6. The twist left in each single after folding is the spun twist less the folding twist, or plus it for a same-way fold. The ply’s centre lines sit as touching circles round the fold’s axis — half a diameter off for two, $d/\sqrt3$ for three, $d/\sqrt2$ for four — the same arrangement the ply’s own geometry uses for its occupied diameter.

The resultant is the number of singles times the single’s count, times the ratio of the two mean secants, times the secant of the ply’s helix. The calculation is required to agree with the ply arithmetic’s own residual angle and helix angle to the ninth figure; to give the same ratio at four counts to the twelfth; to reach its small-angle limit, within a part in a thousand, as the twist factor falls to 200; and to refuse a fold of five singles, a folding ratio at or past the singles’ own twist, and a single with no count or no twist.

What the ideal helix leaves out #

Migration. A real spun fibre does not hold its radius; it wanders between the core and the surface of the yarn, which is what holds a staple yarn together at all. A migrating fibre spends part of its length at every radius, so its average excess is close to the section’s average that the ideal helix already uses, but it also travels radially, which adds length the ideal helix lacks. Both effects are small at weaving angles, and their net sign on the singles’ gain is not computed here.

The compound helix. A fibre in a folded yarn follows a helix round its single’s axis, which itself follows a helix round the fold’s. The calculation multiplies the two as though they were in series. The true path differs from that product where a fibre is on the outside of both helices at once or the inside of both; the correction is fourth order in the angles, and neither its size nor its sign is computed.

Flattening. Two singles pressed together by the fold’s tension are not round, and a flattened pair sits closer to the axis than two touching circles. That shrinks the ply’s helix and moves the crossing up. The four-fold, whose square arrangement is the loosest, is the case most moved by it.

Folding tension. A single is folded under tension and is slightly stretched while the twist goes in; the count is measured on the relaxed fold. The two lengths here are both relaxed lengths, which is the right comparison, but a fold that keeps some of its folding stretch is lighter than the calculation says.

So two of the four omissions raise the crossing, and the other two are of unknown sign and higher order. The two-fold’s crossing at a weaving twist factor is 0.658 on the ideal helix; on a real yarn it is likely a little higher, towards the surface rule’s 0.707 — and still inside the bracket either way.

Who wrote the arithmetic down #

The folded notation is as old as cotton counts, and in the indirect systems it is written as division without comment. The retraction of a twisted yarn, and its closed form for an ideal helix, belong to the structural mechanics of yarns worked out in the middle of the last century and collected by Hearle, Grosberg and Backer in 1969. Treloar worked out the geometry of multi-ply yarns in the 1950s, including the compound helix this calculation simplifies. The tex system’s habit of quoting a resultant count separately, rather than computing it, is the trade’s acknowledgement that something happens to the length.

What is added here is the observation that the two length changes oppose for any fold made against its singles; the crossing ratio, $2/(1+8(o/d)^2)$, with no count, twist or fibre in it; its identification with the folder’s two thirds; the four-fold’s crossing at two fifths, outside its bracket, which settles that weight is not what practice folds for; and the size of the error over the trade’s whole range, which is under the count’s own testing spread for two and three singles and reaches it for four.

Still open: what a fold of folds weighs #

A cabled yarn is a fold of folds: two-fold or three-fold yarns twisted together again, usually in the singles’ own direction, so that the cabling untwists the folds and adds twist back to the singles. Each level now has a helix that lengthens what it carries and a twist change that alters its retraction, and the singles’ change runs the opposite way to the folds’.

Whether a cabled yarn’s three levels cancel as the two-fold’s two do, or whether the cable is heavy the way a same-way fold is, is the same arithmetic with one more factor, and the cabling rule that sets its twists already fixes every input. It would say whether a sewing thread’s ticket, where the resultant count matters for every metre a reel claims to hold, is right for the same reason a 2/40s is.

And the two-fold’s crossing has a measurement that isolates it. Fold one pair of singles at five ratios from a half to four fifths, at one twist factor, and weigh a long length of each. On this account the counts cross the singles’ sum near 0.66 — not at the surface rule’s 0.71 — and the size of the departure at four fifths, 1.5 per cent heavy at a weaving twist, says how much of the ideal helix a real fold keeps.