Twist is one angle

Every geometric argument on this site has been allowed to write d for a yarn and get away with it. Peirce’s geometry bends a cylinder round another cylinder; the cover factor hides a surface with cylinders; the trellis lets cylinders rotate where they cross. Ninety-odd essays, one assumption, never examined.

A spun yarn is not a cylinder. It is a rope of fibres two or three centimetres long, held together by nothing but friction, and the only reason it has a diameter at all is that it is twisted. A helix under tension pulls a bundle towards a circle and holds it there.

So the assumption the whole site rests on is supplied by one operation, and that operation is a helix — which means the whole subject is trigonometry once the right angle is named.

The angle, in one line #

A fibre on the surface of a yarn of diameter d, twisted T turns per unit length, goes once round the circumference — πd — while advancing 1/T along the axis. So the tangent of its angle to the axis is the one over the other:

tan α = π d T

That is the whole of it. A 0.167 mm yarn at 800 turns per metre gives tan α = 0.42, so α is 22.8°.

The angle is not the same everywhere. A fibre nearer the axis goes round a smaller circle in the same distance, so its angle is smaller, and a fibre exactly on the axis is straight. The surface angle is the largest, and it is the one quoted, because it is the one that decides how the yarn meets the world.

Why the trade counts in twist factors and not in turns #

Substitute the site’s own diameter arithmetic and something falls out.

A yarn’s diameter comes from its count by conservation of volume: mass per length divided by density is an area, and an area gives a diameter. So d goes as the square root of the tex.

Then tan α goes as T√tex. That combination has a name in every spinning mill in the world — it is the twist factor — and it is quoted rather than the turns because a spinner setting a 40 tex yarn and a 10 tex yarn to the same factor gets the same yarn in every way that matters.

The claim can be checked exactly rather than believed. Five counts spanning a factor of sixteen, each twisted to the same factor of 4,000, give surface angles of 25.152°, 25.152°, 25.152°, 25.152° and 25.152° — equal to twelve decimal places, because the square root in the diameter and the square root in the factor are the same square root.

The twist factor is therefore not a convention that happened to stick. It is the only combination of count and turns that decides anything about the yarn’s geometry, and every practical rule in spinning — a hosiery yarn at 3,000, a warp yarn at 4,500, a crepe yarn at 7,000 in tex units — is a rule about an angle written in the units a spinner can set.

What the angle costs in length #

A fibre following a helix is longer than the yarn it is in. Its length per unit of yarn is the secant of its own helix angle, which is one on the axis and largest at the surface, so the yarn is shorter than the fibre in it by an amount that is an average over the cross-section rather than a single angle.

Taking the fibres as uniformly distributed over the section and each following a perfect helix of constant radius, that average has a closed form — 2(√(1 + tan²α)³ − 1)/(3 tan²α) — and it is worth having both because a quadrature nobody checked is a number nobody should quote and because the closed form is short enough to state.

Retraction is the reason a spinner’s yarn is shorter than the roving it came from, and it is why twist is added after drafting rather than before. At ordinary twist factors it is a few per cent; at the crepe factors in the last row it is a seventh of the length, which is a real cost in fibre and a real gain in liveliness.

What the angle costs in strength #

The obliquity column is the other half and it is exact within the helix model. A fibre lying at α to the axis contributes only the component of its tension along that axis, and fewer of them cross a section for the same reason, so the standard first-order account has the strength fall as cos²α.

That is the cost of twist. The benefit is not in this site’s reach at all.

Twist is what makes the fibres grip one another. A bundle with no twist has no strength whatever the fibre is — pull it and the fibres slide past each other and it comes apart with no fibre broken. How quickly the grip builds with twist depends on fibre length, fineness, crimp, surface friction and the migration of fibres between the core and the surface, none of which is geometry.

So the strength–twist curve has two halves and only one of them belongs here. The site’s rule for a case like this is to compute the exact half, model the other half explicitly, and say which is which in the caption — never to blend them and quote the maximum as a result.

At the stated cohesion scale the maximum is at 825 turns per metre and 23.4° — comfortably inside the range spinners actually use, which is reassuring and is not evidence. It would be evidence if the optimum were insensitive to the cohesion model, and it is not.

Where the angle shows up in the rest of the site #

The reason to compute this at all is that the angle is not confined to the yarn. It reaches into four arguments the site has already made and it was absent from every one of them.

The diameter. Peirce’s rule gives a cotton yarn’s diameter as one over twenty-eight times the root of the count, and the site inverted it to find the packing factor it implies. Packing depends on twist, so Peirce’s constant is a statement about a yarn at some particular twist factor — the ordinary one for cotton in 1937 — and using it on a hosiery yarn or a crepe yarn is an extrapolation nobody flags.

The section. Kemp’s racetrack flattens a yarn in the thickness of a cloth, and how far a yarn flattens depends on how hard it is twisted: a soft-twisted weft flattens readily and a hard-twisted warp does not. The flattening ratio the site has been passing as an argument is, physically, a twist consequence.

The spirality of a knit. The site already computes spirality from twist liveliness — a single yarn has residual torque, and knitted into a loop it rotates until the fabric resists — and the twist factor is the input there. That essay and this one are the same quantity used twice, and until now the quantity had no page of its own.

And the surface itself. A yarn’s fibres lie at α to its axis, so a cloth’s surface is covered in short parallel lines at that angle, and light reflected off parallel lines is a streak along them. That is the whole of the next rung.

What was counted, and how #

Three assertions carry the essay and each is written against its claim rather than against its numbers, which is a rule this site has had to learn three separate times.

The twist factor decides the angle. Asserted as an equality across five counts spanning a factor of sixteen, to twelve decimal places — and paired with the complementary check that the angle moves when the factor moves, because an assertion that nothing changes is satisfied trivially by a function that always returns the same thing.

The retraction integral agrees with its closed form. Two thousand quadrature points against the analytic expression, to a part in a million. Either alone would be a number; the pair is a check.

The strength curve has an interior maximum. Asserted as a shape rather than a value: the curve must rise from zero, reach a maximum strictly inside the range, and fall — because a yarn with no twist has no strength and a yarn twisted to ninety degrees has no length along its own axis. A model whose optimum sat at an endpoint would be describing something else, and that is a failure the assertion catches while a tolerance on the optimum’s value would not.

Why the angle saturates, and what that does to a spinner’s choices #

The arctangent is the reason every consequence in this essay flattens, and it is worth reading as a design constraint rather than as a feature of a function.

The angle cannot pass ninety degrees, because a fibre at ninety degrees is going round the yarn and not along it, so it advances nothing. So however much twist is put in, the angle approaches a right angle and never reaches it — and the marginal angle per turn falls as the square of the cosine.

The table shows it plainly. The first four hundred turns a metre buy six degrees; the four hundred from twelve hundred to sixteen hundred buy eight, which looks like more until the retraction column is read beside it: those same four hundred turns cost five points of retraction where the first four hundred cost a quarter of one. The angle is bought at a steadily worse price in length and in strength.

That is what puts an upper bound on twist independently of anything about cohesion. A yarn twisted past about forty degrees is spending more than a seventh of its fibre on being twisted and keeping under three fifths of its fibres’ strength, and the angle it has bought is approaching a ceiling. Nothing in the trade goes much past that, and the reason is arithmetic rather than practice.

It also explains why the twist factor bands are so narrow. A hosiery yarn at 3,000 and a warp yarn at 4,500 differ by half, which is five degrees of angle — and five degrees near twenty is a substantial change in cohesion and a small one in retraction, because the two curves have different curvature at that point. The trade’s whole usable range sits on the stretch where the angle is still cheap and the grip is still climbing, which is a narrow window and is why the numbers are quoted so precisely for a quantity that is fitted to a fibre.

There is a symmetry in that worth naming. The window is narrow at the bottom because a yarn below it has no grip and falls apart, and narrow at the top because a yarn above it is spending fibre and strength on an angle it can barely increase — so the two bounds come from completely unrelated mechanisms and happen to leave a range of about one octave in the twist factor. Everything anybody spins lives in that octave, and the whole vocabulary of soft, medium and hard twist is a description of where in it a yarn sits.

The direction, which is a sign rather than a quantity #

Twist has a handedness and the trade names it after the letters it looks like. Hold a yarn vertically: if the surface fibres run like the middle stroke of a Z, up to the right, it is Z twist; if they run like an S, up to the left, it is S twist.

Everything in this essay is indifferent to the sign. The angle is the same, the retraction is the same, the obliquity is the same, and no property of a single yarn on its own distinguishes the two.

The sign matters the moment two things are put together, and it matters in three places the site can name. A folded yarn is normally plied opposite to its singles, so the folding twist takes some of the singles’ twist out and the yarn is balanced — a yarn folded the same way as its singles is “twist-lively” and kinks on itself. A knitted fabric leans, because an unbalanced yarn’s torque rotates every loop the same way and the wale line walks around the garment. And a woven cloth’s twill line either agrees with its yarn’s surface fibres or crosses them, which is the next rung and is the only one of the three where the answer is a subtraction of two angles rather than a sign.

Where the model stops #

Fibres do not stay at one radius. A real spun yarn has fibres migrating between the surface and the core along their length, and that migration is not a defect — it is what ties the layers of the yarn together and is a large part of why a spun yarn holds at all. The constant-radius helix has no migration in it, so it understates the cohesion and gets the retraction slightly wrong in a direction that depends on the migration’s amplitude.

The packing factor is an input. The diameter comes from the count and the packing, and packing depends on twist — a harder-twisted yarn is denser. So d and T are not independent, and treating them as independent, which everything above does, makes the angle at high twist an underestimate. The site has the packing arithmetic and does not couple it here.

A ply yarn is two helices. Everything here is a single. A folded yarn has the singles’ twist and the folding twist, usually in opposite directions, and its surface angle is a composition of the two that the trade balances deliberately. Nothing in this essay addresses it.

And the strength model is one sentence long. Cohesion as a saturating exponential with one constant is the crudest defensible shape. It is here to make the argument’s structure visible — two competing effects, one exact and one not — and not to predict a breaking load.

Who found it, and when #

The helix geometry is old enough to have no clear owner: rope-makers have known that twist shortens a rope and that too much of it weakens one for as long as there have been ropes, and the trigonometry is immediate once anybody writes it down.

The obliquity factor is Gégauff’s, from 1907, and it is one of the earliest quantitative results in textile mechanics. His argument is the one above — resolve the fibre’s tension along the yarn axis and count the fibres crossing an oblique section — and cos²α has survived a century of refinement as the leading term.

The twist factor is older than the explanation for it. Spinners were specifying twist per unit length divided by the square root of the count long before anybody connected it to a helix angle, because it was the number that made two different counts behave alike. That is the ordinary history of a trade constant, and this site’s job with one is always the same: ask what it assumes, and see whether the assumption is exact. Here it is. The twist factor is not an approximation or a rule of thumb; it is the exact invariant of the geometry, and the fact that it was found empirically first says something about how good spinners were rather than about how rough the number is.

Where the ladder goes next #

The angle is a direction on the surface of a yarn, and a woven cloth has another direction on its surface — the twill line, whose angle comes from the two setts. The next rung puts the two on the same square of cloth and subtracts them, which turns the oldest rule of thumb in weaving into a number.