Where a yarn is thinnest

Every section drawing in this collection shows a yarn with one shape. A circle, or a racetrack, or an ellipse — one outline, held constant along the thread and used everywhere the thread appears.

A yarn in a fabric is squashed by its neighbours, and its neighbours are only there in some places.

The measurement #

Take the same two adjacent courses whose closest approach gives the fabric’s demanded flattening, and instead of reporting the minimum, report the whole profile: for every point on one course, the distance to the nearest point of the other.

The result is not flat. It runs from 0.780 diameters at the tightest to 3.81 at the freest, and only twenty per cent of the course’s length is inside one diameter of its neighbour at all.

So four fifths of a yarn in a knitted fabric is not being pressed by anything.

What that means for the flattening #

The number this ladder has been quoting — a flattening of about four fifths — is the minimum of that profile. It is what the yarn has to be at its most squashed, in the fifth of its length where it is squashed at all.

It is not what the yarn is elsewhere, and elsewhere is most of it.

That is a real limitation on the previous rungs, and it is worth stating plainly rather than as a footnote: a single flattening ratio is an average over a profile that varies by a factor of five.

Which does not make the number wrong #

It makes it a different number from the one somebody might assume, and the difference matters for what it can be used for.

Where the flattening is used as a clearance — will this arrangement fit, does the yarn have to be squashed, how much room does the fabric leave — the minimum is exactly the right quantity, because a fit is decided at the tightest point. Every use on the previous rungs is of that kind.

Where it is used as a section — what is the yarn’s shape, what is its second moment of area, how wide does it look — the minimum is the wrong quantity, because the yarn only has that section in a fifth of its length.

The two uses have been conflated everywhere in this subject, including here, and the profile is what separates them.

Where the free length is #

The profile has a shape and it is worth reading rather than summarising.

The yarn is pressed at and around the interlacing — where a needle loop’s head passes the feet of the loop below — and that region is short. Either side of it the two courses diverge quickly, because both are curving away.

The freest point is at the crest of a loop, as far from the neighbouring course as the geometry allows, and there the clearance is nearly four diameters.

So the section a yarn takes is a flattened one at the interlacing and a round one at the crest, with a transition over a millimetre or so of arc.

What a varying section changes #

Four things depend on the yarn’s section, and the profile affects each differently.

Thickness depends on the section at the interlacing, because the fabric’s thickness is set by two yarns passing through one another’s neighbourhood. So the minimum is the right number and the thickness result stands.

Bending rigidity depends on the section everywhere, weighted by how much the yarn is bending at each place. The yarn bends most at the crest, which is where it is roundest, so a rigidity computed at the minimum flattening is too low.

Cover depends on the section’s width, everywhere, and a yarn that is flattened only at the interlacing covers less than a uniformly flattened one would.

And contact area depends on the section at the crossing alone, so the minimum is right again — which is what the relaxed cloth’s contact force is computed over.

Two of the four use the minimum and two do not, and no calculation in this collection has previously distinguished them.

The tight limit, where a single section is right #

The profile is not equally misleading at every construction, and the dependence runs in a useful direction.

In a tight fabric the courses are close everywhere, so the pressed fraction rises and the free maximum falls. The profile flattens towards a constant, and a single section becomes a good idealisation.

In a slack fabric the courses are far apart except at the interlacing, so the profile is extreme and a single section is badly wrong.

That means the assumption every geometry in this subject makes — a uniform section — is best exactly where fabrics are tightest, which is where jamming calculations are made and where the assumption matters most. That is a piece of luck rather than a design, and it is worth knowing.

What the trade measures #

The measurement everybody actually makes is a section: embed a piece of fabric in resin, cut it, and look at the threads.

A cut through a fabric crosses a yarn at whatever point of its profile the cut happens to fall on, so a set of sections samples the profile rather than measuring the minimum. And because the cut is usually taken through the interlacings — that is where the structure is legible — the samples are biased towards the pressed end.

So published flattening figures are measurements of a biased sample of a varying quantity, and the scatter in them is partly real variation along the yarn rather than measurement error. That is worth knowing before comparing any prediction against them.

What a woven cloth’s profile would look like #

The same question can be asked of a woven cloth and the answer should be different in a way worth predicting before anybody computes it.

A woven thread crosses its neighbours regularly — every pick, in a plain weave — so the pressed regions are close together and the free lengths between them are short. A plain weave’s thread should therefore have a much flatter profile than a knitted loop’s, and a single section should be a much better idealisation.

A satin’s thread is the opposite: a four-end float passes four picks with nothing pressing it, so its profile has long free runs and short pressed ones, much like a knitted loop’s.

So the prediction is: the uniform-section assumption is best for a plain weave, worse for a twill, worst for a satin and for a knit. That ordering is the same as the float length, and the float length is the quantity this collection has found decides most things about a cloth.

Whether it holds is a computation on the site’s own Peirce geometry, and it is asked directly a rung along.

Why the free length matters more than it looks #

There is a reason to care about the free four fifths beyond bookkeeping, and it is about what a fabric does rather than what it is.

A yarn that is not pressed is a yarn that can move. It can slide along its own path, rotate about its own axis, and shift laterally, and all three are things a fabric does when it is deformed. The pressed fifth is where friction acts and the free four fifths is where the yarn is free to rearrange.

That is the geometry behind several things this collection has computed separately. A knitted fabric’s enormous extensibility is the yarn moving through the free lengths towards the pressed points. Its recovery is the same motion reversed. Its hysteresis is the friction at the pressed fifth resisting both, which is what a cloth does not give back.

So the profile is not only a caveat on a section. It is a map of where a fabric’s friction is, and that map has been assumed uniform everywhere in this collection’s knitted work.

What was counted, and how #

The profile is the distance from each sample of one course to the nearest point of the other, over the whole of both.

Nearest rather than same-index: two curves have their own parameterisations, and the distance between the points at the same parameter is a distance between two arbitrary bookkeeping choices. It means nothing, and it would have produced a profile with the right shape and the wrong values.

The sampling is a hundred and twenty points per half period over two wales, so about five hundred points a course and a quarter of a million pairs, which takes a moment.

The check is that the profile is not flat: the ratio of its maximum to its minimum must exceed two, and the pressed fraction must be a fraction rather than the whole length. Both would fail if the layout were wrong, and the second would fail if the courses were placed too close.

The number a specification should carry #

If a single flattening ratio is an average over a factor of five, a specification that quotes one is quoting a number whose meaning depends on where it was measured. It is worth saying what a better one would look like.

Two numbers rather than one. The flattening at the interlacing, which decides fit, thickness and contact; and the flattening between interlacings, which decides bending and cover. For a relaxed jersey those are about 0.78 and about 1.0.

And the pressed fraction, which is what decides how much of the yarn each number applies to. For a relaxed jersey it is a fifth.

Three numbers is not a burden and it removes the ambiguity entirely. It also makes the comparison between fabrics meaningful: a tight fabric and a slack one differ far more in their pressed fraction than in their minimum flattening, and a specification carrying only the minimum reports them as almost the same.

What this does to the previous rungs #

An honest accounting, because two rungs of this ladder quoted the minimum without qualification.

The demanded flattening stands. It is a clearance question and a clearance is decided at the tightest point.

Its dependence on the tightness factor stands, for the same reason, and the sweep is a sweep of minima throughout.

The cost of flattening does not stand as stated. It priced the deformation as though the whole yarn were squashed to the minimum, and only a fifth of it is. The true cost is about a fifth of the number quoted — which strengthens rather than weakens the conclusion, since the conclusion was that flattening is cheap enough for the fabric to do it.

And any use of the flattening as a section is wrong by whatever the profile’s spread implies, which is a factor of five at the extremes and much less in the pressed region.

That is one correction and two confirmations, and recording it here is cheaper than leaving a reader to find the inconsistency.

Where the model stops #

The profile is a clearance, not a section. It says how much room the fabric leaves; it does not say what shape the yarn takes in that room, and a yarn that is free to be round may not be, because it has been squashed elsewhere and its fibres do not spring back.

That last point is not a quibble. A yarn whose fibres slide has no shape memory in cross-section, so a section that was squashed at the interlacing may stay squashed for some distance along the yarn rather than recovering immediately. How far is a question about friction between fibres and is not computed anywhere.

The neighbours are only the adjacent courses. A yarn is also near the yarn on either side within its own course, and that clearance is not in the profile.

And the loop is the free one. A yarn that is flattened at the interlacing follows a slightly different path from a round one, and the profile is computed on the round path.

A fifth is a suspiciously familiar number #

The pressed fraction comes out at about a fifth, and it is worth noticing that this collection has met that number before in a different guise.

A relaxed knitted loop is nine tenths free run: only a small part of the yarn between two interlacings is doing the interlacing, and most of it is running freely between them. That was computed from the loop’s own curvature distribution and has nothing obvious to do with clearance.

The two numbers are not the same quantity and they are measuring the same structural fact from two directions. A knitted loop is mostly free length with short regions of engagement, whether “engagement” is measured by curvature or by proximity to a neighbour.

That is worth flagging rather than claiming. Two independent measurements of a fabric agreeing on a rough proportion is either a real structural property or a coincidence, and the way to tell is to compute both across the tightness range and see whether they move together. Nobody has, and it would take a few minutes.

The generalisation #

This collection has a habit and this rung is another instance of it: a number is a summary of a distribution, and the summary is only right for the questions the distribution’s shape does not matter to.

The site’s own founding example is a yarn’s diameter. Every calculation here uses one diameter for a yarn whose thickness is a population with a coefficient of variation of fifteen per cent, and a cloth is a population rather than a thread works out where the difference bites: a mean is right for a mass, an order statistic is right for a jam, and using the mean for the jam is wrong by a great deal.

The flattening is the same structure one level along, and the same resolution applies. Ask which end of the distribution the question is about. A fit is decided by the tightest point; a stiffness is decided by an average weighted by curvature; a cover is decided by a plain average.

Three different numbers from one profile, and this collection has been quoting one.

Why the profile is asymmetric #

A detail visible in the plot and worth explaining, because it is not obvious and it says something about the fabric.

The profile is not symmetric about its minimum. The clearance rises more steeply on one side of the interlacing than on the other, and it does so consistently at every construction.

The reason is that the two courses are not symmetric partners at the crossing. One is arriving at a crest and the other is arriving at a trough, and a crest and a trough of the solved loop are not mirror images in the fabric’s plane — the loop’s own asymmetry is a property this collection has measured rather than assumed.

So the yarn is squeezed harder on the side its neighbour is climbing towards than on the side it is falling away from, and the pressed region is offset from the geometric crossing rather than centred on it.

That is a small effect and it has one practical consequence: the section a cut through the interlacing measures depends on which side of the interlacing the cut falls on, by rather more than the measurement’s own precision. Anybody comparing published sections should expect that to be part of the scatter.

Who found it, and when #

That yarn in cloth is flattened at its crossings and rounder between them is visible in any longitudinal section and is not news to anybody who has made one.

What is not usually done is to treat the flattening as a function of position in a calculation. Every cloth geometry from Peirce onwards uses a uniform section, including the ones that use a racetrack rather than a circle, and the reason is arithmetical convenience rather than belief.

What is this collection’s own is computing the profile from its own solved fabric, so that the size of the approximation is a number rather than an acknowledged worry.

Where the ladder goes next #

The profile says the yarn’s section varies along its length. The obvious next question is what a varying section does to the two stiffnesses, because a flattened section has two different flexural rigidities and the collection’s own headline ratio was derived for a circle.

That is the section that changes both stiffnesses, and it puts a correction on the cleanest result of the other half of this work.