How many fibres make a thread

A cover factor is a sett times a diameter. A jammed sett is a reciprocal of a diameter. A hole is a spacing less a diameter, a bending rigidity is a diameter to the fourth power, and a cloth’s thickness is two diameters and a crimp. There is hardly a number in this collection that did not pass through the diameter of a thread on its way to being computed.

That diameter has never been measured. It has been derived, every time, from a count and a packing factor — a mass per unit length divided by a density and an assumed fraction of solid, square-rooted into a width. The arithmetic is honest and the result is a good number. But it is worth asking what it is a number of, because the answer is not a width at all.

It is a count of fibres. A yarn is an assembly, and the assembly can be counted.

The claim #

A yarn’s cross-section holds a countable number of fibres, n = tex ÷ tex-per-fibre, and three quantities that appear in quite different parts of this collection are all functions of that one count.

• The diameter. A yarn is √(n/φ) fibre diameters across, exactly, where φ is the packing factor. For the yarn above that is fourteen fibres across; for a coarse 80 tex weft it is twenty-eight.
• The stiffness bracket. The ratio between a yarn whose fibres slide freely and one that bends as a solid rod is n/φ², which is a result this collection already had without noticing that it is the fibre count.
• The finest yarn that fibre can make. A yarn needs enough fibres in its section for twist to hold them; below that floor it breaks faster than it can be wound. The floor is a count, so the finest yarn is a count times the fibre’s own fineness — and belongs to the fibre rather than to the spinner.

The second and third run in opposite directions, which is the part worth carrying away and the subject of the last section.

The division, and why it is exact #

A count in the direct systems is a mass per unit length: one tex is a gram per kilometre. A fibre has a count too — the trade quotes it in decitex, and a cotton fibre is about 1.7, which is 0.17 tex.

Cut a yarn across. Every fibre in it runs along it, so a fibre crossing the plane contributes its own mass per unit length to the yarn’s. Add them up and the yarn’s tex is the sum of its fibres’ tex, which for fibres of equal fineness is

$$n = \frac{\text{tex}_{\text{yarn}}}{\text{tex}_{\text{fibre}}}.$$

There is no geometry in that and no assumption to argue about. It does not care how the fibres are arranged, whether they are twisted, or whether they lie straight — a helical fibre crosses the plane obliquely and presents more area, but it also runs further per unit of yarn, and the two cancel exactly in the mass. The count is a division and nothing else.

What the division does assume is that the fibres are alike. They are not: a cotton’s fineness varies from fibre to fibre by about forty per cent, so n is a mean and a section taken at random holds a number near it rather than the number itself. That is not a caveat to be waved away; it is the next rung of this ladder, and it turns out to set a floor on how even any yarn can be.

The count systems, read as fibre counts #

There are two families of yarn count and this collection has set them beside each other before: the direct ones, where a bigger number is a coarser yarn because it is a mass per length, and the indirect ones, where a bigger number is a finer yarn because it is a length per mass. The conversions between them are arithmetic and the argument for one over the other is historical.

Once the fibre is fixed, both are counts of fibres.

The reason this matters is not tidiness. A count is the quantity a weaver specifies, a merchant prices and a standard quotes, and it looks like an arbitrary unit with a historical accident in it. It is not arbitrary at all: it is the fibre count, scaled by whatever the fibre happens to be. Two yarns of the same count in two different fibres are two different assemblies, and the collection’s habit of computing from a count without naming the fibre has been getting away with something.

What the count does to the diameter #

The site’s own volume arithmetic says a yarn of tex, fibre density ρ and packing factor φ has

$$d_{\text{yarn}} = \sqrt{\frac{4\,\text{tex}}{\pi \rho \varphi} \times 10^{-3}}\ \text{mm},$$

and a single fibre obeys the same expression with its own tex and φ = 1, because a fibre is not a bundle. Divide one by the other and everything except the counts cancels:

$$\frac{d_{\text{yarn}}}{d_{\text{fibre}}} = \sqrt{\frac{n}{\varphi}}.$$

A relation with no material in it. A 20 tex cotton yarn is 14.0 fibre diameters across; a 6 tex hosiery yarn is 7.7; an 80 tex weft is 28.0. Each is the square root of its own fibre count over the packing, and the identity is asserted across counts, fibres and packing factors to twelve figures rather than remarked on, because two routes through the same volume arithmetic must not be allowed to drift apart.

The consequence for the rest of the collection is quiet and general. Wherever a diameter appears, a square root of a fibre count appears; wherever a diameter squared appears — an areal weight, a mass — the count appears bare, which is why a weight is the one quantity here that never had a diameter in it at all.

The floor: how fine a yarn can be #

A spun yarn is held together by nothing but friction between fibres, supplied by twist pressing them against one another. That mechanism needs fibres to press: a section containing five fibres has almost no fibre-to-fibre contact and nothing to hold, and a yarn made of such sections breaks at the thinnest of them before it can be wound onto a bobbin.

The floor is a count, not a mass. It is a property of the spinning machine and of how much irregularity the process can carry, and it is put at about thirty-five fibres for a ring frame and nearer a hundred for rotor spinning. Neither number is a physical constant, so the floor is carried here as a stated argument rather than as a table entry, and every result below quotes the floor it used.

What follows from it is not a machine property at all.

The finest yarn belongs to the fibre. This explains a fact about the trade that otherwise looks like tradition: the extra-fine cottons are the long, fine ones and the price of a fine yarn tracks the fineness of the cotton it is spun from rather than the skill applied to it. A spinner buying a finer cotton is buying access to a count, not an improvement in quality.

It also explains why the man-made staple fibres are sold in a range of finenesses at all. A polyester filament can be extruded at any thickness the spinneret allows, so the fineness is chosen — and it is chosen to reach a count. There is no analogous freedom in wool.

The same count, decided twice #

Here is the thing worth carrying out of the arithmetic, and it was not visible before the count was named.

The bending rigidity of a yarn cannot be computed. It can only be bracketed, between the fibres bending independently — the sum of n fibre rigidities — and the whole yarn bending as a solid rod of its own diameter. The ratio between the two bounds was shown there to be

$$\frac{\text{coherent}}{\text{free}} = \frac{n}{\varphi^2},$$

with no fibre, no count and no modulus in it. That is the fibre count. And the floor on a yarn’s evenness is 100/√n per cent, which is also the fibre count.

They pull opposite ways.

So: a coarse yarn is even and unknowably stiff; a fine yarn is uneven and nearly determined. That is a statement about what can be predicted rather than about what is good, and it has a practical edge. A collection like this one, which computes fabric properties from constructions, is on its firmest ground with fine yarns and its shakiest with coarse ones — precisely the opposite of where the measurements are easiest to make.

The two consequences are one relation #

The evenness floor and the stiffness bracket are drawn above on separate scales because they are in different units, and that hides how tightly they are tied. They are not merely opposed; they are the same quantity read at two exponents, and eliminating the count between them gives a relation with no count in it at all.

The floor goes as n to the minus a half and the bracket as n to the plus one, so

bracket = (100 ÷ floor per cent)² × (1 + CV_f²) ÷ φ².

The stiffness bracket is the inverse square of the evenness floor, times two constants that belong to the fibre and the spinning. No yarn count appears anywhere in it.

Check it against the ends of the spinnable range for cotton. At the spinning floor of thirty-five fibres the evenness floor is 18.1 per cent and the bracket is 97; at a coarse 80 tex with four hundred and seventy-one fibres the floor is 4.9 per cent and the bracket is 1,308. The floor has fallen by 3.7 and the bracket has risen by 13.5, and 3.7 squared is 13.7.

Three things follow that the two curves side by side do not say.

A yarn’s evenness certificate is a stiffness statement. The floor is not on a delivery note but the measured coefficient of variation and the count both are, so the floor follows, and the bracket follows from it — which means a spinner’s own quality figure carries, in a form nobody reads, how well the yarn’s bending can be known.

And the exponents are what make the trade unavoidable. Two quantities with the same sign of exponent could be improved together by moving the count; two with opposite signs cannot. That the ratio is exactly minus a half to plus one, rather than any other pair, is what makes the exchange rate a square rather than something a designer could argue with.

And the relation says which end of the range a model like this one should trust. A fine yarn is uneven and nearly determined in stiffness; a coarse one is smooth and unknowably stiff. So a collection that computes fabric properties from constructions is on firm ground exactly where the measurements are hardest to make — and the reason is one relation between two exponents, rather than a coincidence of two separate difficulties.

The caution is that both halves rest on the packing factor, which enters the floor not at all and the bracket as a square. So a yarn whose packing is not the assumed 0.6 has a bracket wrong by the square of the error while its floor is untouched, and the relation above is the place that error would show up.

What a spinner is choosing #

Put the three consequences together and the shape of the spinner’s problem appears, and it is not the shape the trade’s language suggests.

A spinner does not choose a diameter. A diameter is a consequence — of the count, which the customer specifies, and of the packing, which is mostly decided by the twist. A spinner does not choose an evenness either: the floor is set by the count and what is left is the index, the factor by which the process falls short of it, and that is the only quantity in the whole assembly that measures how well the job was done.

What a spinner chooses is the fibre and the twist. The fibre sets the floor on the count, and through the count it sets the floor on the evenness and the width of the stiffness bracket. The twist sets the packing, and through the packing it moves the diameter — and with the diameter every cover factor, every jam and every hole the cloth will ever have.

So the two levers are one level below everything this collection has been computing, and one of them — the fibre — is bought rather than made. That is why the finest and most even yarns come from the longest and finest cottons and why no amount of care substitutes: the arithmetic in this essay does not have a term for care in it.

What was counted, and how #

The division is asserted against the geometry, not alongside it. The identity d_yarn/d_fibre = √(n/φ) is checked over thirty-six yarns — four counts, three fibres, three packing factors — with the worst relative departure at three parts in 10¹⁶. That is not a check that the formula is right; it is a check that the two routes through the same volume arithmetic have not drifted, which is the failure a site with one diameter function used in twenty places is actually exposed to.

The staple table is checked for completeness against the density table, and specifically for the property that a fibre has a staple length exactly when it is not a filament. A filament has no staple length and the entry says so rather than carrying a plausible number, because the whole of the grip argument further up this ladder is about fibre ends — and a filament yarn has two, at the ends of the package.

The opposed pair is asserted as an ordering at every step, not at the extremes: the bracket must rise and the floor must fall between each adjacent pair of counts. An assertion about the endpoints of a monotone claim is satisfied by a curve that wanders in between.

Where the model stops #

The count is a mean. Fibres are not all alike, they do not all run parallel to the axis, and a real cross-section holds a number that varies from place to place. Everything above uses n as though it were the number, which is right for a diameter — a mass per unit length really is an average — and wrong for anything that depends on the thinnest section.

The arrangement is not modelled at all. The figures draw a hexagonal lattice because something has to be drawn, and real fibres are not on a lattice. Nothing here says how the fibres are distributed across the section, whether the packing is uniform from core to surface, or which fibres are where. A yarn’s packing factor is treated as a single number for the whole section, and it is not: the core of a ring-spun yarn is denser than its surface, which is one of the reasons the packing factor this collection uses is a weaker constant than it looks.

The spinning floor is a machine property and is stated rather than derived. Thirty-five is the number usually quoted for the ring frame and it is not a constant of nature; a different process has a different floor, and rotor spinning’s is roughly three times higher. What is derived is only that the floor is a count and therefore that the finest achievable yarn scales with the fibre’s fineness.

Nothing here has any twist in it. The count, the diameter and the floor are all properties of an assembly at rest. What holds the assembly together — and what it costs to hold it — is an angle, and everything about it is further up this ladder.

Where the ladder goes next #

Two ways, and they are the two halves of what a count implies.

Upward into the spread: n is a mean, a real section holds a Poisson count near it, and that single fact puts a floor under a yarn’s irregularity that no spinner can beat and that gets worse as the yarn gets finer. That floor is what makes fineness a cost rather than a virtue, and it settles what the coefficient of variation on a delivery note actually means.

Downward into the fibre in the yarn: n fibres of finite length have 2n ends per unit length to be held, and holding them is what twist is for. That is where the strength comes from, and where the load a straight fibre cannot share and the length of fibre a twist can grip both live.

And sideways, back into the cloth. The diameter this essay decomposed is the same diameter that decides how closely threads may be set, what a cover factor means and what a cloth weighs. None of those arguments changes. What changes is that the number they all share has stopped being a measurement and become a count, and a count can be reasoned about.