Textile Structure
how cloth is put together
Mechanics and drape

What leaving the plane costs

Every number the flat loop model produced, beside the same number with the climb in it. Four of the five fall, none moves by four per cent, and the estimate that priced the third dimension beforehand had the sign the wrong way round for a reason worth naming.
15 min read 6 figures Measured, not claimedA mechanism, not a materialExactly this many

Worth reading first: A loop is a plane curve in another plane · A loop is nine tenths free run · What a loop presses with.

A correction that turns out to be small is worth publishing, because the alternative is a collection whose readers do not know which of its numbers were provisional. This one is small, and it is small in a particular direction.

Letting a knitted loop out of the fabric’s plane lowers its bending energy by two and seven tenths per cent, lowers the force at its interlacings by one and eight tenths, and raises the tightest bend in it by three parts in a thousand. Every published number on this ladder survives. But the sign of the first of those was predicted the other way round, and the reason it was predicted wrongly is more useful than the size of any of them.

What leaving the plane costs, which is less than nothing. Bending energy a stitch against how far the yarn climbs between one interlacing and the next, for a 20 tex cotton jersey at a 3 mm loop. The curve falls. A climb is not an extra bend added to a curve that was otherwise unchanged; it is part of the straight line the thread has to span, so a longer climb leaves less slack to be spent on curvature. At a jersey's own climb of one diameter the energy is 27001 nJ against 27988 for the planar model — 3.5% lower, where an earlier estimate on this site put the ride at three per cent of the loop's bending and had the sign the other way round. At a rib's climb of three or four diameters the fall is 41%.
Fig. 1 The same cost on a tighter fabric. The energy falls as the climb grows here too, and it falls further — 3.5 per cent at a jersey’s own climb against 2.7 on the slacker construction. Leaving the plane is cheaper the tighter the fabric is knitted.

The estimate that was made in advance

Before the thread was solved in three dimensions, the excursion was priced. The argument ran like this: the interlacing needs two centre lines to pass a diameter apart, so the yarn rides out of the plane by about half a diameter over about a quarter of a loop length; a ride of that amplitude over that wavelength has a curvature of a certain size; compare it with the in-plane curvature and square the ratio.

That gave about three per cent, and it was reported as a cost — as the share of the loop’s bending that the third dimension adds.

Both halves of the answer are wrong. The share is not three per cent of anything, and it is not added.

Why the estimate had to fail

The estimate treats the excursion as a curvature laid on top of a curve that is otherwise unchanged. That is the natural thing to do and it is the wrong object.

A thread between two interlacings has a fixed length and a fixed pair of endpoints. Its bending is not something it chooses; it is what is left over after the straight line between its ends has been subtracted from its length. That leftover is the thread’s slack, and slack is the whole of what decides how hard a free thread bends.

Now move one endpoint out of the plane. The straight line between the ends gets longer, because it now has a third component. The length of thread has not changed. So the slack falls, and with it the curvature.

The excursion does not add a bend. It removes one.

The numbers, in one place

For a 20 tex cotton at a 3.5 mm loop, fully relaxed, with the loop’s own diameter as its climb:

quantity flat with the climb change
bending energy a stitch 25,074 nJ 24,395 nJ −2.71%
contact force 38.99 mN 38.30 mN −1.77%
its component along the wales 38.99 mN 37.50 mN −3.83%
tightest bend, in diameters 1.001 1.004 +0.30%
slack 48.5% 47.7% −1.81%

The slack row is the one that explains all the others. It falls by a little under two per cent because the chord grew, and everything else follows: less slack, less curvature, less stored energy, less force.

The peak curvature rising while the energy falls looks contradictory and is not. The energy is an integral of curvature squared over the whole thread; the peak is one point on it. A thread with slightly less room redistributes: the crowns bend a shade harder and the free run between them straightens more than enough to pay for it.

Where the two per cent goes

It is worth following the energy rather than only reporting it, because the accounting says something about how a loop stores what it stores.

A thread’s bending energy is an integral of its squared curvature, and in a relaxed loop that integral is dominated by two short stretches — the crowns, where the yarn wraps round the thread it interlaces with. Between them the thread is nearly straight and contributes almost nothing. So the energy of a loop is very nearly the energy of its two crowns, and the free run between them is a length of yarn doing no work at all.

Lengthening the chord does not lengthen the crowns. It lengthens the free run. What falls, when the climb takes eight tenths of a per cent off the slack, is the amount of yarn that has to be got rid of somewhere — and the somewhere is the free run’s gentle curvature, which is cheap. That is why a two per cent change in slack buys a two and seven tenths per cent change in energy rather than something much larger: the expensive part of the loop was never negotiable.

The same accounting is why the peak curvature moves the other way. The crowns are set by the diameter of the thread being wrapped, not by the slack, and squeezing the free run slightly straighter transfers a fraction of a degree of turning back into them.

What that leaves standing

The one number on the ladder that had no business surviving a change of dimension is the tightest bend, and it is the number that moves least.

A relaxed loop bends hardest to a curvature of 1.004 over the yarn’s own diameter, against 1.001 for the flat model. The claim it supports — that a knitted loop is bent about as hard as its yarn’s thickness allows and no harder, at every tightness a knitter can reach — is untouched. Nothing put that coincidence there in the first place, and nothing in the third dimension takes it away.

What leaving the plane costs, which is less than nothing. Bending energy a stitch against how far the yarn climbs between one interlacing and the next, for a 20 tex cotton jersey at a 3.5 mm loop. The curve falls. A climb is not an extra bend added to a curve that was otherwise unchanged; it is part of the straight line the thread has to span, so a longer climb leaves less slack to be spent on curvature. At a jersey's own climb of one diameter the energy is 24395 nJ against 25060 for the planar model — 2.7% lower, where an earlier estimate on this site put the ride at three per cent of the loop's bending and had the sign the other way round. At a rib's climb of three or four diameters the fall is 33%.
Fig. 2 Bending energy against the climb, as a proportion of the flat model’s, swept from no climb at all to four yarn diameters. The curve is smooth and monotone downwards over the whole range, so there is no regime where the third dimension starts costing rather than paying. At a jersey’s own climb the fall is 2.7 per cent; at a rib’s it is nearer a fifth.

The whole sweep, not just the one point

A single correction at a single climb is a fact about one fabric. The sweep is a statement about the model, and it is worth having because it says there is no crossover.

From no climb to four diameters — which spans a jersey at one end and a rib on an open gap at the other — the energy falls monotonically and smoothly. It does not turn round, it does not have a stationary point, and it does not have a regime where the excursion starts to be paid for. The fall is 2.7 per cent at a jersey’s climb, 10 per cent at two diameters, and a third by four.

A model that has a sign error somewhere usually shows it as a curve that turns over where it should not. This one does not turn over anywhere, which is weak evidence and evidence all the same.

The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.
Fig. 3 The crossing the whole correction is about. Two centre lines pass at a diameter, so the two ends of a half period are a diameter apart through the fabric, and the straight line between them is longer than the flat model’s by the amount that line climbs. Everything in the table above is that one length, followed through.

What was being compared with what

There is a subtlety in reporting a correction of this kind and it is easy to get backwards, so it is worth being explicit.

The flat model and the solved one are not two models of the same fabric with one of them more careful. They are models of two different fabrics: one whose interlacings lie in a plane, which is impossible, and one whose interlacings pass at a diameter, which is what a knitted fabric is.

So the right reading of the table is not “the flat model was 2.7 per cent out”. It is “a fabric whose loops could lie flat would hold 2.7 per cent more bending energy than a real one does, and it cannot exist”. The correction is a comparison with a fiction, and the fiction is the useful thing: it isolates what the interlacing itself is worth.

Which numbers on the ladder change

Only two published figures move enough to restate, and both are forces rather than shapes.

The contact force at an interlacing was 38.99 millinewtons a stitch and is 38.30. That is the whole force, and it is the one to quote when the question is how hard the loops press on one another — the number that goes into the comparison against a woven cloth’s crossings, which press between 185 and 851.

The part of that force along the wales was the same 38.99 and is now 37.50, because the rest has gone through the fabric’s thickness. That is the one to quote when the question is what is trying to lengthen the fabric, and it is the one the friction balance uses.

Those two numbers were the same in the flat model and are not the same any more, and that is the substantive change. Everything else is a rounding.

The contact force turns as the climb grows. The two components of the contact force against the climb, for a 20 tex cotton jersey at a 3.5 mm loop. The force along the wales is what friction has to hold and the force through the thickness is what holds the fabric open, and the second is bought at the expense of the first. At a jersey's own climb of one diameter they are 37.50 mN and 7.81 mN; at four diameters, which is a rib on an open gap, they are 22.39 mN and 18.61 mN. The friction balance is the ratio: friction has the whole force to work with and only the along-the-wales part to hold, so the coefficient a relaxed knit would need falls from a half to 0.490.
Fig. 4 The two components of the contact force, swept over the same range of climbs. They are one force resolved two ways, and the split moves from four-fifths-to-a-fifth at a jersey’s climb to nearly half and half at four diameters. The total falls slowly; what changes fast is where it points.

The bracket that swallows all of it

Every force here is a bending stiffness over a length squared, and a spun yarn’s bending stiffness is not a number. It is a band a hundred and thirty wide, running from the case where the fibres slide freely past one another to the case where they cannot.

Against a band of that width, a correction of two per cent is not visible. It does not follow that the correction is pointless — a ratio of two forces carries no bracket at all, and this collection’s strongest knitted claims are all ratios — but it does say what the correction is for. It is for the ratios, and for the direction the force points, neither of which the bracket touches.

Anybody quoting an absolute force in newtons from this ladder is quoting the free end of a wide bracket, and the essays that do it say so where they do it.

What the correction does not reach

Three things on this ladder are untouched by the third dimension, and it is worth listing them so that nobody looks for a revision that is not coming.

The geometric ceiling. How far a knit can be pulled before its yarn runs straight between interlacings is a statement about lengths, and the chord that reaches the limit is the same chord whether it lies in the fabric or across it. The ceiling moves by the same fraction as the slack and nothing about the argument changes.

Munden’s constants. They are measurements of a relaxed fabric’s spacings and were never predictions. The model takes them as input at both dimensions, and a fabric measured flat is measured in the two directions they are about.

The energy surface’s slope. The finding that a relaxed knit sits nowhere near a minimum of its own bending energy — that the energy falls away from the relaxed state in both directions at once — is about the shape of the surface, and moving every point on it down by three per cent moves no slope’s sign.

The one place where a small number is not small

There is an exception, and it is the reason the next rung exists.

The through-thickness component of the contact force is 7.81 millinewtons a stitch. Beside the 38.30 it was resolved out of, that is a fifth — a modest fraction of a modest force, and easy to read as another rounding.

It is not a rounding, because there is nothing for it to be a correction to. The flat model does not have a smaller version of this quantity; it has no version of it at all. A model with no thickness has no direction for a force to act through, so the comparison is not 7.81 against something, it is 7.81 against nothing, and a quantity that goes from absent to present is not a two per cent change.

What a reader should now quote

Three sentences, for anybody carrying numbers away.

For the shape of a relaxed loop — its slack, its peak curvature, its occupancy, its extension ceiling — the flat figures and the solved ones agree to within a couple of per cent and either will do.

For the force at an interlacing, quote 38.3 millinewtons a stitch for a 20 tex cotton at a 3.5 mm loop, at the free end of the stiffness bracket, and say which end.

And for anything that depends on where that force points — the friction balance, the fabric’s thickness, its resistance to being pressed — the flat figures are not wrong by a percentage. They are silent.

The comparison with the woven side

This collection’s woven threads have almost no slack at all — four parts in a thousand between one crossing and the next, against a knitted loop’s half — and that is why they cannot be solved as free elasticas and are drawn as Peirce’s arcs and straights instead.

It follows that a woven thread cannot have the correction this rung is about. There is no free run for a longer chord to take slack away from: the whole of a woven thread’s crimp is spent inside its wrap, which is set by the diameter of the thread it crosses and by nothing else.

That is a real asymmetry between the two halves of the site and it points the same way as everything else on this ladder. A knitted fabric’s mechanics is about a thread with room; a woven fabric’s is about a thread with none. The third dimension is one more thing that only the fabric with room can afford, and the price it pays for it is negative.

What it would take to move these numbers

The corrections above are small because the climb is small: one yarn diameter against a drop of nearly five. Two things would make them large.

A second bed. A rib’s yarn crosses the whole gap between the beds rather than a single diameter, so its climb is three to five diameters and its energy falls by a fifth rather than by a fortieth. That is a different fabric rather than a better model of the same one, and it is where the machinery earns its keep.

A much tighter fabric. The climb is a diameter whatever the gauge, so tightening the loop shortens the drop and raises the ratio. At the tight end of what a knitter can reach the tilt is fifteen degrees rather than twelve, and the through-thickness share of the contact force rises with it. The forces themselves rise far faster than the angle does — the through-thickness component runs from 3.0 millinewtons a stitch at a 5 mm loop to 16.3 at a 2.6 mm one, a factor of five across a range of tightness a single machine can knit.

Neither is a correction to a jersey. Both are the same arithmetic asked about a fabric further from the flat case.

The general shape of the error

It is worth stating the trap in a form that travels, because it is not about knitting.

An estimate that adds a term to a functional is not the same as a solve that re-minimises it. Adding the excursion’s own curvature to the flat solution’s answers the question what would this curve cost if it also did that. The thread does not also do it; the thread does it instead, and re-minimising is what finds out what it stops doing.

The same shape has appeared on this site before in different clothes. A cloth whose crimp ratio was predicted from an energy minimum turned out to have had the minimum taken over the wrong set of states — the energy was re-minimised, but under a constraint the fabric does not obey. And a thickness estimated by treating threads as rectangular blocks, rather than letting a coating find its own way down between round ones, came out sixty-eight per cent low.

All of them are the same failure: reasoning about a modified object without letting the object re-settle. The remedy is always the same too, and it is always more work — re-solve, do not adjust.

A rib is quietest at a gap of two diameters. The through-thickness force of a two-by-two rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 11.3 mN at 5 diameters against 7.4 mN at two.
Fig. 5 And on a structure that leaves the plane at half its sinker loops. The through-thickness force behaves the same way and is smaller throughout, which is what makes the cost a property of how far the yarn climbs rather than of the fabric’s name.
A rib is quietest at a gap of two diameters. The through-thickness force of a one-by-one rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 14.8 mN at 5 diameters against 7.0 mN at two.
Fig. 6 Where the correction stops being a correction. Once the climb is a bed gap rather than a diameter, the same arithmetic moves the through-thickness force by factors rather than by percentages — and it does not move it monotonically, because a crossing’s climb and an interlacing’s diameter can cancel. That is a fabric with two beds and the whole of the next ladder.

What is genuinely new here

Two things.

The third dimension is a relief, of measured size. Not an adjective and not an estimate: 2.71 per cent, downwards, for the fabric this collection has been quoting throughout, with the whole sweep behind it.

And the contact force splits. One number became two, and the two do different work: one is held by friction and one holds the fabric open. Everything in the two rungs after this follows from that split rather than from any of the percentages above.

What the pictures cannot show

The bar chart shows five changes and reads as though the model has been checked in five places. It has been checked in one: every row is a consequence of the chord being longer, and knowing any one of them determines the rest.

A figure with five bars in it looks like five pieces of evidence. It is one piece of evidence, drawn five ways, and a reader who takes the agreement of the rows as corroboration is counting the same fact several times.

Where the ladder goes next

The force that turned out of the fabric is the whole subject of the force that holds a knit open — what it is worth as a pressure, what it says about a knit’s thickness, and the small correction it makes to a friction condition that had come out at exactly one half.

After that, a fabric with a thickness can be asked whether it curls. It cannot be asked that question at all without one, and how little asymmetry a curl needs is what happens when it finally is.

Where this goes next

A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted. A loop is a plane curve in another plane A knitted loop was solved as a flat curve because two curves in one plane cannot pass through one another and a loop must. Letting it out of the plane turns out to change nothing about its shape: the loop is still planar, and its plane is the fabric's turned through twelve degrees. A knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings. A loop is nine tenths free run Half the yarn in a knitted stitch is slack — the straight line between two interlacings is a little over half the thread available to span it. That is two orders of magnitude more room than a woven thread has, and it is why a knitted loop is a shape that can be solved rather than a shape that has to be constructed. How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip. What a loop presses with A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Bending bracketBending energyContact forceCurvatureElasticaJammingLoopLoop lengthTightness factor