The count that decides how flat

The flattening a knitted fabric demands of its yarn is a function of one dimensionless group, and the group is the tightness factor: the square root of the count over the loop length.

A knitter chooses two things — a count and a loop length — and the flattening follows from both. That makes it a prediction about a choice rather than a property of a fabric, which is a different and more useful kind of statement.

The dependence #

The tightness factor is the square root of the count over the loop length, so at a fixed loop length the flattening depends on the square root of the count.

Double the count — go from a twenty tex to a forty — and the tightness factor rises by a factor of 1.41. At a three and a half millimetre loop that moves it from 12.8 to 18.1, and the demanded flattening falls from 0.780 to 0.707.

Seven points of flattening, from doubling the count at the same loop.

Which is not the obvious direction #

The result is worth pausing on because the intuition runs the other way.

A coarser yarn is a bigger yarn, and a bigger object in the same space is one that has to give more. That is the arithmetic and it is not what a knitter’s instinct says: the instinct is that a coarse yarn makes a robust fabric and a fine one a delicate one, and that the coarse yarn is the one being treated well.

It is not being treated well at all. At a fixed loop length, a coarser yarn is being asked to fit into a space that has not grown, and the space is decided by the loop length alone.

A knitter who coarsens the yarn without lengthening the loop is asking for a flatter section.

Which is exactly what a knitter does not do #

The practice is the resolution and it is worth stating, because it shows the trade is already following the arithmetic without having it.

A knitter does not coarsen the yarn at a fixed loop length. The loop length is set by the machine’s cam settings and by the gauge, and both are chosen together with the count: a coarse yarn goes on a coarse gauge with a long loop, and a fine yarn on a fine gauge with a short one.

What is held roughly constant across the range is the tightness factor, at thirteen to seventeen for an ordinary jersey — and that is the whole point of the index.

So the trade’s practice is exactly the practice that holds the demanded flattening constant, and the index it uses is the one that measures it.

That is a satisfying coincidence and it is not a coincidence: the tightness factor was arrived at empirically as the thing that has to be held for a fabric to handle consistently, and this rung says one of the things it holds is the yarn’s section.

What happens at the edges #

The interesting cases are where the tightness factor is not held, and both exist.

A very tight fabric at a tightness factor of eighteen or twenty — technical knits, fine sportswear — demands a flattening of 0.71 or less. Its yarn is being squashed by a third.

A very open fabric at nine or ten — a loose hand knit, a mesh — demands 0.84 or more, and its yarn is nearly round.

Between them is the whole range the trade works in, and the flattening moves by seventeen per cent across it.

That seventeen per cent is worth having, because it is a real difference in a fabric’s section that follows from a choice a designer makes explicitly, and it is currently invisible to everybody.

Why a square root #

The exponent is worth explaining because it decides how sensitive the result is.

A yarn’s diameter goes as the square root of its count at a fixed fibre density and packing factor, which is the volume arithmetic every count system rests on: twice the mass in the same length at the same density is twice the area and √2 times the diameter.

The tightness factor is the diameter over the loop length, up to a constant, so it inherits the square root.

So a doubling of the count is only a 41 per cent change in the geometry, and a small change in the count is a smaller change still. That is why a knitter can substitute a nearby count without much consequence, and why the trade’s counts come in geometric rather than arithmetic steps.

It also means the flattening is a fairly insensitive function of the count, which is a good thing for a prediction that depends on a swelling and a packing factor with their own uncertainties.

What a spinner would want to know #

Turning it round gives a statement to somebody choosing a count, and the statement is not one the trade currently makes.

A count is a choice of section as well as of thickness. A coarser yarn in a given fabric is not merely a bigger version of a finer one; it is a flatter one, because the space it has to fit into is set by the loop length rather than by the yarn.

That matters because a flattened yarn behaves differently: it is stiffer to bend one way and softer the other, it covers more, it contacts over a longer length, and it presents a different surface to light.

So two fabrics knitted at the same tightness factor from different counts have the same section and different everything else; and two knitted at the same count and different loop lengths have different sections.

That is a distinction a specification could carry and does not.

What the gauge does, which is nothing #

A clarification, because a knitter’s third variable is the machine gauge and it does not appear anywhere above.

A gauge is needles per unit length, and it decides how many wales a fabric has per centimetre. It does not decide the loop length — the cams do — and it does not decide the count.

So the tightness factor contains no gauge, the flattening contains no gauge, and two fabrics knitted at the same count and loop length on different gauges should demand the same section.

That is a real prediction and it is testable, and it belongs to a family this collection has made before: the fabric’s thickness, its dimensions and its forces are all gauge-independent, which is one of the more falsifiable things the collection has said about a knit.

The gauge does decide something: whether the fabric can be made at all. A loop length much shorter than the needle spacing is not knittable, and a loop much longer makes a slack, unstable fabric. So the gauge sets a range of loop lengths and, within that range, contributes nothing to the geometry.

That is a clean separation and it is worth having, because a great deal of trade discussion treats the gauge as though it were a fabric property rather than a machine one.

What was counted, and how #

The relation is this collection’s, from eighteen independent solves over five loop lengths, three relaxation states and three counts, with the flattening measured as the closest approach two adjacent courses make.

The count dependence is read off it directly: the tightness factor contains the square root of the count, and the sweep confirms that fabrics of different counts at the same tightness factor land on the same point.

The volume arithmetic that makes a diameter go as the square root of a count is this collection’s own and has been used from its earliest work.

The trade’s tightness factor band is quoted from practice.

What a substitution costs in practice #

The rung has a use for somebody substituting one count for another, which is a thing every mill does when a yarn is short.

Substituting a count without changing the loop length moves the tightness factor by the square root of the ratio, and the flattening moves along the curve accordingly.

For a ten per cent count change — the sort of substitution nobody thinks twice about — the tightness factor moves by five per cent and the flattening by under one point. Nothing to worry about.

For a thirty per cent change, the tightness factor moves by fourteen per cent and the flattening by two or three points, which is at the edge of what a section would show and well inside what the fabric’s handle would.

And for a factor of two, seven points, which is a visibly different fabric.

So the arithmetic gives a rule of thumb the trade does not have: substitute within ten per cent freely, adjust the loop length beyond twenty. That is roughly what experienced knitters do, and it now has a reason attached.

Where the model stops #

The relation is for a plain jersey. A rib’s courses lie on two beds and its geometry is different, and nothing here computes it.

The packing factor is held. A coarser yarn is not necessarily spun at the same packing factor as a fine one, and if it is not then the diameter’s dependence on the count is not exactly a square root.

And the flattening is a minimum over a profile, so all the cautions of the rungs that established it apply here unchanged.

Nor does the count’s own effect on the yarn’s evenness enter. A finer yarn has fewer fibres in its cross-section and is correspondingly more variable, so a fine fabric’s demanded flattening varies more along the yarn than a coarse one’s — a population rather than a value, again. That is a second-order effect on a first-order result and it is real.

What else the same group decides #

The tightness factor was already carrying a good deal before this work and it is worth listing what it now carries, because the list is the argument for the index.

The loop’s shape, to a scale factor, since the solve depends only on the ratio of the diameter to the loop length.

Every force in the fabric, to the same scale factor.

The fabric’s dimensions, through Munden’s constants.

The flattening its geometry demands, from this work.

The clearance profile and the pressed fraction, likewise.

And the ratio of its tightest bend to its yarn’s own radius, which is the geometric limit on how tight it can be.

Six quantities, one group. That is an unusually productive dimensionless number and it is worth saying so plainly: almost everything this collection knows about a knitted fabric is a function of the square root of a count over a loop length.

What is not a function of it is anything with a mass or an absolute length in it — an areal weight, a stitch density, a thickness — because those need a scale as well as a shape.

So a knitted fabric is described by two numbers: its tightness factor, which decides its shape, and its loop length, which sets its size. That is a tidy statement and it has not been made in this collection before.

The generalisation #

The rung is small and its point is about what a dimensionless group is for.

A dimensionless group turns a two-variable question into a one-variable one, and the variable it produces is the one worth specifying.

A knitter has two choices and one consequence. Without the group, the flattening is a surface over a plane of counts and loop lengths and nobody can hold it in their head. With it, the flattening is a curve, and the axis is a number the trade already quotes for other reasons.

That is what this collection means when it says a quantity depends only on the tightness factor, and it is worth restating because the phrase gets used as though it were merely a compression. It is not: it is the statement that a two-dimensional design space has a one-dimensional consequence, and that is a strong claim which happens to be true here and is not true of everything.

Areal weight, for instance, depends on the count and the loop length separately and no group collapses it.

What the relation would look like on a shop floor #

A last practical translation, because the whole rung is about a choice somebody makes with a machine in front of them.

A knitter has a cam setting that controls the loop length and a cone of yarn with a count on it. The tightness factor is the square root of the count over the loop length in centimetres, and that number is what decides the fabric.

Everything this work has computed about the fabric’s section, its clearance profile and its bending limit is a function of that one number.

So the practical statement is: read the tightness factor, and the section follows. Thirteen to fourteen is a soft open fabric with a nearly round yarn. Sixteen to seventeen is an ordinary jersey at 0.75. Nineteen and above is a tight fabric whose yarn is squashed by a third and whose bend is approaching a geometric limit.

Three bands, one number, and the number is already on the machine’s own setting sheet.

That is the most usable form this collection’s contact ladder takes, and it is worth ending the ladder on it rather than on a caveat.

The measurement that would test the whole relation #

The rung is the last of the contact ladder’s consequences and it is the one whose test would settle the ladder rather than one rung of it.

Knit a series of fabrics from one fibre at three counts and three loop lengths — nine fabrics, of which several share a tightness factor by different routes.

Section all nine and measure the yarn’s aspect ratio.

Three predictions follow and each is independently falsifiable. The flattening should fall with the tightness factor, monotonically, across the whole set. Fabrics with the same tightness factor should have the same flattening, whatever count and loop length produced it. And the values should lie between 0.70 and 0.84 over the range.

The second is the strong one, because it is a statement about a collapse rather than about a trend, and a collapse either happens or does not.

Nine fabrics is a day on a sample machine and a week of sectioning, and it would settle a prediction this work has made from geometry alone with nothing fitted anywhere.

Who found it, and when #

The tightness factor is a trade index arrived at empirically and quoted for a century.

That a yarn’s diameter goes as the square root of its count is elementary volume arithmetic and is in every yarn-numbering text.

What is this collection’s own is the flattening relation itself, and the observation that a coarser yarn at a fixed loop length is asked to be flatter — which inverts an intuition and which nobody appears to have stated, because nobody has had a predicted flattening to state it about.

Why the count is the wrong lever anyway #

A last observation, and it is the one a knitter would make first.

Of the two variables, the loop length is the one that is easy to change: it is a cam setting, adjustable on the machine, in seconds, by any amount within the gauge’s range.

The count is the one that is hard: it means a different yarn, ordered separately, wound separately, and probably in a different lot.

So although the flattening depends on both, the practical lever is the loop length, and the count is a constraint the knitter inherits.

That inverts the rung’s framing without changing its arithmetic. The useful reading is not “a coarser yarn is flatter” but “a shorter loop is flatter”, which is the same statement with the accessible variable in front.

And the accessible variable enters with the first power rather than the square root, so it moves the flattening twice as fast: a ten per cent shorter loop moves the tightness factor by eleven per cent, where a ten per cent coarser yarn moves it by five.

The lever that is easy to pull is also the stronger one, which is convenient and is not always how these things arrange themselves.

Where the ladder goes next #

This work has two closing rungs. The first puts together four unrelated everyday observations that all land at the same end of this collection’s oldest bracket, and asks what four independent agreements are worth.

A fabric reads its own bracket four ways.