The folding rule is a surface angle

The trade folds a two-fold yarn at about two thirds of its singles twist, a three-fold at rather under six tenths, and a four-fold at a half. Those three numbers have been in this collection’s tables for a long time as brackets copied out of practice, and the mechanical explanation everybody gives for them is impossible.

So the numbers need an explanation and there is one. It contains no mechanics at all, it reproduces all three exactly rather than approximately, and every ingredient of it has been on this site since its second phase.

The condition #

A yarn’s twist is one angle: the helix angle its surface fibres make with its axis. Everything else about twist follows from that, and this collection established it several ladders ago.

For a yarn of diameter d at T turns per unit length, the surface helix angle satisfies

tan α = π · d · T

Now fold n singles into one yarn. The fold’s own surface angle is the same expression with the fold’s diameter and the fold’s twist.

Ask for those two angles to be equal — for the fold’s surface to run at the same angle as the singles’ — and the condition is immediate:

T_fold / T_singles = d_single / d_fold

Why that is one over the root #

A fold of n singles carries n times the mass per unit length of one. If it is packed as densely as its components, its cross-sectional area is n times theirs, so its diameter is √n times theirs.

This collection’s volume arithmetic says exactly that, and it says it as an identity rather than an approximation: a diameter goes as the square root of a count at fixed density and packing. It has been asserted for a two-fold, to twelve figures, since this work that built the ply machinery.

So the ratio is

T_fold / T_singles = 1/√n

which is 0.7071 for two folds, 0.5774 for three, and exactly 0.5000 for four.

Against the trade #

The trade’s brackets are 0.60 to 0.75 for two folds, 0.50 to 0.65 for three, and 0.45 to 0.60 for four.

0.7071 is in the first. 0.5774 is in the second. 0.5000 is in the third.

Three for three, and not narrowly: the two-fold value sits seven tenths of the way up its bracket, the three-fold value halfway, the four-fold value a third of the way. They are inside without being at an edge, which is what a rule that practice is scattered around looks like.

Why a maker would want it #

A rule that fits three numbers is a coincidence until somebody says why anybody would follow it, and the reason here is not obscure.

Appearance. A yarn’s surface angle is what the eye reads as twist. A fold whose surface runs at its singles’ angle looks like a bigger version of the same yarn rather than like something else, and the cloth woven from it looks consistent with the cloth woven from the singles.

Lustre. This collection has a whole ladder about how a yarn’s helix angle decides where light goes. A fold that changes the surface angle changes the highlight, and matching the angle keeps it.

Hand. The surface angle decides how much of each fibre points along the yarn and how much across it, which is most of what a yarn feels like and is what the other half of the twist curve is about. Matching it keeps the hand.

And economy of specification. A folder who folds every yarn at one over the root of the fold count has one rule for every fibre, every count and every twist. A folder who balanced torque would need the fibre’s stiffness ratio, which nobody has and which varies by a factor of eighteen across the collection’s table.

That last is not a small consideration. A rule that requires a constant nobody measures is a rule nobody can follow.

The three tests it passes that the balance fails #

The previous rung established three structural mismatches between the trade’s rule and the torque balance. The surface rule matches on all three.

Magnitude. 0.707 against a bracket of 0.60 to 0.75, rather than 0.20.

Fold-count dependence. The trade’s ratios fall with the fold count and the surface rule falls with the fold count, as one over its square root. The torque balance does not depend on the fold count at all.

Fibre-independence. The trade quotes one set of ratios for every fibre; the surface rule contains no fibre property; the torque balance varies by a factor of three across the collection’s table.

Three for three again, and the second and third are the ones that matter. A number can be matched by luck. A dependence cannot.

What the rule assumes, and where it breaks #

The derivation has one assumption in it and it is worth being explicit, because it is the place the rule will fail.

The fold packs as densely as its singles. That is what makes the diameter go as the square root, and it is not exactly true. A fold’s singles are round objects packed into a round envelope, and round objects do not tile, which is the same packing arithmetic the collection uses for a yarn’s own fibres: two circles in a circle occupy about eighty-three per cent of it, three about seventy-seven, and the packing gets worse before it gets better.

So a real fold is somewhat larger than √n times its single, and the ratio d_single/d_fold is somewhat smaller than 1/√n.

That pushes the predicted ratio down, which is the direction the trade’s brackets extend: the two-fold bracket runs from 0.60 up to 0.75, and 0.7071 sits in its upper half. A packing correction of ten per cent on the fold diameter would move the prediction to 0.64, which is the middle of the bracket.

So the imperfection of the derivation lands the prediction more comfortably inside practice rather than less, which is a good sign and not a proof.

What this collection already had #

Nothing in the derivation is new machinery, and it is worth listing what was already here, because the rule could have been derived at any point at any time in the last few years.

The surface-angle relation has been here since twist is one angle, which is one of the site’s early rungs and is the essay the whole twist ladder rests on.

The √n diameter relation has been here since the ply ladder, where it is asserted as an identity rather than assumed: a two-fold yarn is not twice a single is largely about that arithmetic and its consequences.

The brackets have been in the tables since the same phase, copied from practice with a note saying so.

All three ingredients sat in the same repository for a long time and nobody put them together, because nobody had a reason to doubt the mechanical explanation. The doubt came from acquiring a torsional rigidity and finding that it gave the wrong answer — so the useful thing the previous rung did was not to compute a number but to make a question worth asking.

What happens if the singles are not alike #

The rule as stated folds n identical singles, and a great many real folds are not.

A grandrelle folds two singles of different colours; a corkscrew or spiral folds two of markedly different counts; a slub or fancy yarn folds a fine binder round a thick effect thread. In each case there is no single “singles twist” to take a fraction of.

The condition generalises without difficulty, because it was never about the singles being alike: it asks the fold’s surface angle to match, and what it should match is whatever the maker wants the surface to look like. For a fold of two unequal singles the natural target is the coarser one’s angle, because the coarse component dominates the surface.

That predicts something checkable and slightly unobvious: an unequal fold should be twisted at a ratio set by the coarse component alone, and the fine component’s twist should be irrelevant to the fold ratio. Whether folders do that is not something this collection knows, and it is the kind of question a fancy-yarn spinner would answer in a sentence.

Why the number is not exactly what a folder would say #

Anybody who folds yarn would say two thirds rather than 0.7071, and that difference is worth a paragraph rather than an apology.

A rule of thumb is quoted in the coarsest fraction that is close enough. Two thirds is 0.667, one over root two is 0.707, and the difference is six per cent — well inside the spread of what different folders do and well inside the bracket the trade itself quotes.

What makes the identification worth making is not that 0.7071 is more accurate than 0.667. It is that 0.7071 comes with the other two numbers attached: 0.577 and 0.500, which are also what the trade uses, and which two thirds does not predict at all.

A single number can be matched by any rule with a free constant. Three numbers from one expression with no free constant is a different kind of claim.

The prediction that would settle it #

The two accounts differ most where the trade quotes nothing, and that is where a measurement is worth making.

Fold a wool at the ratio a folder would use and measure the residual torque. The surface rule says the ratio should be 0.707 regardless of fibre; the balance account says a wool balances at 0.444. If the folder’s wool at 0.707 carries a substantial residual torque in the same direction as a cotton’s does, the surface rule is right and the balance account is not merely wrong about the number but wrong about the variable.

Or fold a five-fold and a six-fold cord. The surface rule says 0.447 and 0.408; the balance account says 0.20 for both. The two differ by more than a factor of two and nobody has published a bracket for either.

Or fold two yarns of the same fibre at very different counts. Both accounts say the ratio does not depend on the count, so this one is a control rather than a discriminator — which makes it the measurement to make first, because a result that moved with the count would mean something was wrong with both.

What was counted, and how #

The surface-angle relation and the √n diameter relation are both this collection’s own and both are asserted rather than assumed. The check on the rule is that matching the two surface angles divides the twist by exactly the square root of the fold count, to twelve figures, across the fold counts the trade quotes — which is an identity checking itself and would catch an algebraic slip rather than a physical error.

The comparison against the brackets is a containment test on three intervals, and it is stated as such: it is not a fit, there are no free parameters, and the rule was not adjusted to land inside anything.

The torque balance beside it is recomputed from the previous rung rather than quoted, so the two rules are drawn from the same arithmetic on the same figure.

Where the model stops #

It is a rule about the yarn’s surface and says nothing about its inside. A fold’s fibres are at every angle from nothing at the axis to the surface angle at the outside, and matching the surface angle does not match the distribution. Two yarns with the same surface angle and different internal structures behave differently under load, and this collection has a rung on exactly that.

It assumes the fold’s twist and its singles’ twist are measured in the same sense, which is a bookkeeping point that matters: the fold twist is turns per unit length of fold, and the singles twist per unit length of single, and the single inside the fold is longer than the fold by one over the cosine of its helix angle. At the angles involved that is under one per cent and it is ignored.

And it does not say the trade chose this rule for these reasons. Practice arrived at three numbers over a long time by trial, and the claim here is that the numbers are 1/√n rather than that anybody computed them. The reasons given above are why the condition is a sensible one to have converged on, not a history.

The generalisation #

Two things generalise and the second is the more useful.

The first is the specific habit: when a rule of thumb depends on an integer, look for the integer in a geometry rather than in a mechanics. A mechanical balance rarely depends on a count; a packing arithmetic almost always does.

The second is about what a refuted explanation is worth. The previous rung produced no number that anybody can use: it computed a balance point and showed that the balance point is not what folders use. That looks like a wasted rung and it was the necessary one, because the question “what else has this number in it” is only worth asking after the obvious answer has been ruled out.

This collection has a refutation index for that reason. What it has not had until now is a case where the refutation immediately produced the replacement — and the replacement was sitting in the same tables, unassembled, the whole time.

What it says about the word “twist factor” #

There is a tidier way to state the rule and it makes the point sharper.

A yarn’s twist factor is its turns per unit length times the square root of its count, and it is the quantity the trade specifies rather than the twist itself — precisely because it is the combination that keeps the surface angle fixed as the count changes.

Write the folding rule in those terms. The fold’s count is n times the singles’, so its square root is √n times theirs; and the rule says the fold’s twist is 1/√n of the singles’. Multiply:

The fold’s twist factor equals its singles’ twist factor.

That is the whole rule in five words, and it explains why nobody in the trade thinks of it as a rule at all. A folder working in twist factors does not divide by anything: they use the same twist factor for the fold that they used for the single, and the division by the square root happens automatically in the units.

So the rule is invisible in the units practice actually uses, which is the best possible explanation for why it has never been written down as a geometric condition. It only looks like a rule when somebody converts it into turns per metre — which is what a textbook does when it explains folding to a beginner, and is where the two-thirds figure comes from.

And what that says about the mechanical account #

The twist-factor form makes the mechanical account look worse rather than better, and it is worth saying why.

A twist factor is a geometric quantity: it is proportional to the tangent of the surface helix angle and to nothing else. It contains no modulus, no rigidity and no fibre property.

So the rule the trade follows, expressed in the trade’s own units, is manifestly a statement about shape. Reading it as a torque balance requires converting it out of the units that make it obvious and into units that make it look like a coincidence, and then explaining the coincidence with a mechanism that turns out to be impossible.

That is a general hazard worth carrying: a rule stated in the wrong units invites the wrong explanation. The same thing happened to this collection with a fabric dimension quoted without its state, and with a critical tension quoted in millinewtons rather than in metres of hanging yarn.

Who found it, and when #

The folding ratios are old trade practice. The surface-angle relation is Gégauff’s, from 1907, and is the foundation of every twist calculation in the subject. The square-root diameter relation is elementary and is in every yarn-numbering text.

Putting the three together to derive the folding ratios is not something this collection has found in the literature, which is surprising given how elementary each ingredient is — and is probably explained by the mechanical account being satisfying enough that nobody looked further.

Where the ladder goes next #

Two meanings of “balanced” have now been separated: a yarn balanced for torque, which is what the textbooks say folding achieves, and a yarn balanced for surface, which is what folding actually achieves. They are different conditions with different numbers and different dependences.

Keeping them apart is worth a rung of its own, because the word is used for both and a specification that says “balanced” says nothing about which: what a balanced yarn is balanced about.

And the arithmetic applies again one level up. A cabled yarn is a fold of folds, with three twist levels and two conditions, and 1/√n applies at each level — which is a cabled yarn is a fold of folds.