Textile Structure
how cloth is put together
Knits and other structures

The relaxed knit is not at a minimum

Differentiate a loop's bending energy along the fabric instead of across it and the answer should be zero, because a relaxed fabric is one nothing is pulling. It is not zero. It is tens of newtons a metre, downhill in both directions at once — and the three relaxation states everybody measures run the wrong way up the slope.
15 min read 7 figures Measured, not claimedA mechanism, not a materialExactly this many

Worth reading first: What a loop presses with · A knit's dimensions come from its loop · A dimension without a state.

A relaxed fabric is one that nothing is pulling. So if the model is any good, the derivative of its stored energy with respect to its own dimensions ought to come out at zero: nudge the wale spacing, nudge the course spacing, and the energy should not care.

It cares a great deal. For a 20 tex cotton jersey at an ordinary loop length the two derivatives are five millinewtons a stitch widening the fabric across its courses and thirty-nine lengthening it along its wales, and both point the same way. The fabric wants to be wider and it wants to be longer, and nothing in the loop’s own bending is stopping it.

Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left.
Fig. 1 Every state a plain knit can be in at one loop length: the two spacings on the two axes, the bending energy shaded, with the geometric edge — where the straight line between two interlacings reaches the yarn between them — drawn as the boundary. The three states Munden measured are marked. They are not in a hollow. They are strung along a slope, in order, with the most completely relaxed of them the furthest up it.

What is being differentiated

The same solve as before, read along the fabric instead of across it. A knitted stitch at a fixed loop length has exactly two macroscopic freedoms — how wide its wale is and how tall its course is — and the loop’s energy is a surface over those two.

The contact force was the slope across the fabric. The slope along the fabric is a different multiplier from the same small matrix, and it is the force per stitch tending to change the spacings. Neither derives from the other and both come free with the shape.

Both derivatives point downhill

Widening a wale at a fixed loop length straightens the yarn between the interlacings, so it costs less bending. Lengthening a course does the same. The energy falls in both directions and it does not turn round anywhere: the surface has no interior minimum at all.

What stops it is not elastic. It is geometric: the yarn between one interlacing and the next has a fixed length, so the two spacings are confined inside a quadrant of an ellipse, and the fall continues until the fabric reaches that edge and the yarn is straight.

The relaxed fabric is on a slope, not in a hollow. Bending energy per stitch of an unset 20 tex cotton yarn, against the wale spacing and against the course spacing, each varied through the relaxed fabric's own value at a constant 3.5 mm loop, and each divided by the energy the relaxed fabric holds. Both curves fall away from the relaxed state and neither turns round: by the right-hand edge the fabric holds 42 per cent of what it held, and the fall goes on until the yarn runs straight between its interlacings and the geometry stops. The slopes at the relaxed state are 4.98 mN and 39.0 mN per stitch, which is what something other than the yarn's own springing has to be supplying. A model whose energy minimum is nowhere near the fabric everybody measures is not nearly right; it is right about the yarn and wrong about the mechanism.
Fig. 2 The same surface as two sections through it, each passing through the fabric a knitter actually gets, and each divided by the energy that fabric holds. Both fall away and neither turns round: by twice the relaxed spacing the fabric holds seventy-seven per cent of its energy in one direction and forty-two in the other.

An identity worth noticing

The two forces are less independent than they look, and the relation is exact.

The loop’s height enters the geometry only as the course spacing plus one yarn diameter. So differentiating the energy with respect to the course spacing is the same operation as differentiating it with respect to the loop’s height, which is the transverse endpoint condition — and that derivative is the contact force.

The force along the wales and the contact force are the same number. Thirty-nine millinewtons appears twice above for that reason and not by coincidence, and the identity holds at every configuration rather than at the relaxed one. It is a small thing and it is a good sign: a model whose two answers were independently plausible and unrelated would be a model with two free ways to be wrong.

How big the slope is, in fabric units

Per stitch these are small forces. A fabric’s edge carries a great many stitches, so the conversion matters, and it is worth doing carefully because the two directions divide by different spacings.

The force lengthening the wales is carried by every wale across the fabric’s width, one per wale spacing, and comes to forty-eight newtons per metre. The force widening the fabric is carried by every course up its height, one per course spacing, and comes to eight.

A piece of jersey a metre tall, held at its relaxed width, would on this reckoning be pushing outwards with the weight of eight hundred grams hanging on it — and a metre-wide piece would be pushing its own length out with six times that. It manifestly does neither. Something is holding it, and the something is not in the model.

The three states, which decide it

The strongest evidence is not the size of the slope. It is its direction, set against a measurement made sixty years ago and repeated ever since.

A knitted fabric has no single set of dimensions; it has one per relaxation treatment. Left alone it is dry-relaxed. Wetted out and dried flat it is wet-relaxed. Wetted, tumbled and dried it is fully relaxed. Munden’s constants are three sets, one per state, and they are among the most reproduced numbers in knitting.

They go in a definite direction. From dry to wet to fully relaxed the fabric gets smaller in both directions at once — its wale spacing falls, its course spacing falls, its stitch density rises.

Which is uphill

Put those three states on the surface and they are in the order of increasing energy. Dry-relaxed holds the least, fully relaxed the most, and the rise from the first to the last is twelve per cent.

Relaxation moves a knit away from its own energy minimum. The bending energy an unset yarn would hold in each of Munden's three relaxation states, at a constant 3.5 mm loop. A fabric taken from dry relaxation to wet relaxation to full relaxation gets smaller in both directions, and the loop therefore holds more bending energy at every further stage — a rise of 12 per cent from the first to the last. The bars run from zero, so twelve per cent is a small difference on them and the rule marks the first state's value to make the ordering readable; the percentages beside each bar are the quantity the claim is about. If the yarn springing were what set a knit's dimensions the ordering would be the other way round, and it is strict in every published set.
Fig. 3 The bending energy an unset yarn would hold in each of the three states, at one loop length. Every further stage of relaxation leaves the loop holding more, not less. The ordering is strict, it is asserted rather than remarked on, and it is the opposite of what a fabric relaxing towards its own energy minimum would do.

If a knit’s relaxed dimensions were where its yarn’s springing put them, then the more completely a fabric is relaxed — the more thoroughly friction is overcome by wetting and tumbling — the closer it should get to the minimum. It goes the other way, in both dimensions, in every published set, by an amount far too large to be noise.

The edge, and what it is made of

The boundary on the field figure deserves a paragraph of its own, because it is the only hard limit in the whole picture and it has nothing elastic in it.

The yarn between two interlacings has a fixed length, half a loop length. The straight line between them is half a wale across and one loop height down. So the two spacings must satisfy a Pythagorean inequality, and the reachable states are the inside of an ellipse quadrant. Beyond it there is no configuration at any force whatever, because an inextensible thread cannot reach further than its own length.

Everything soft about a knit happens well inside that boundary and everything sudden happens as it is approached. The rung that plots the force shows both halves on one curve.

What that rules out

It rules out the story the model tells if the yarn is taken to be a straight rod bent into a loop. Not because the numbers are a bit off: because the sign is wrong, and a sign is not something a better rigidity or a finer basis will fix.

That is a stronger conclusion than “the model is approximate”, and it is the reason this rung is worth having. A model that agreed to within a factor of two would have been quietly believed and quietly used. One that predicts a fabric should expand where every measurement says it contracts has said something specific about which piece of physics is missing.

What is left

Two candidates, and they are not alternatives.

The first is friction. The loops touch, they touch under a load the model computes, and a fabric held by friction can sit anywhere it was last left. That would explain immediately why a knit has three sets of dimensions instead of one, and why the treatments that move it between them — wetting, tumbling, agitation — are exactly the treatments that reduce friction or supply the energy to overcome it. The rung that prices it finds that friction holds one direction and not the other, and that the direction it fails in has an exact arithmetic behind the failure.

The second is set. A yarn that has been wetted, heated and dried has taken the loop’s shape as its own natural shape, and a yarn whose natural shape is the loop presses with nothing. The rung that pursues that finds it explains the direction as well as the magnitude, because setting happens during the relaxation treatments and therefore increases with them.

Friction explains why the fabric can stay anywhere it is put across its courses. It cannot explain the wales at all, for a reason that turns out to be an identity. Set explains both, and explains why the fabric goes the way it does.

What the picture cannot show

The field figure shades an energy over two axes, and neither of the two things a reader most wants is in it.

It cannot show the third dimension of the state space, because there is not one: the loop length is fixed on that square, and a fabric that has been shrunk by felting or stretched on a stenter has changed its loop length and moved to a different square entirely.

And it cannot show which way a fabric would go. A shaded surface shows where the energy is; the path a real fabric takes across it is decided by friction and by the order in which things happen to it, and neither is drawn.

What a woven cloth does instead

The contrast makes the claim sharper, because a woven cloth really is close to an energy minimum and this collection has been treating it as one for several ladders.

The difference is structural rather than a matter of degree. A woven cloth’s two systems are tied together by a closure condition: the thickness one system takes is thickness the other cannot have, so pushing either way costs energy and the fabric sits in a hollow. A knit has no such equation — which is exactly what an earlier rung found, for a different reason and about a different question — and a structure with no closure condition has nothing to make a hollow out of.

So the two fabrics differ in the one respect that decides whether “relaxed” names a state or names a history, and the same missing equation is behind both.

Why nobody noticed

Because the question was never asked in this form. Munden’s result is a dimensional one and was validated dimensionally: it predicts a fabric’s size from its loop length, it does so across fibres and counts and gauges, and it works. Nothing in that programme requires the state to be an energy minimum, and nothing in it would reveal that it is not.

The minimum-energy loop models of the same period asked a different question — what shape does a loop take — and answered it for a loop in isolation, with the dimensions imposed. Imposing the dimensions is exactly what hides the slope, because a slope is what appears only when they are not imposed.

So the two literatures each answered their own question correctly and the gap between them stayed empty. That is a common shape for a missing result and it is worth naming when it turns up.

What this does not overturn

Nothing dimensional. Munden’s constants are measurements and they are still measurements. Every dimensional prediction made from them is untouched, and so is everything this collection has computed from them.

What changes is their status. They are not the output of an equilibrium calculation and they never claimed to be. They are the record of where fabrics stop when they are treated in three stated ways, and the reason the treatment has to be stated is precisely that the fabric is not in equilibrium and therefore remembers.

That is why a dimension without a state is meaningless in a knit and merely careless in a woven cloth: a woven cloth’s dimensions really are close to an energy minimum, and a knit’s are a history — which is the same distinction a knit relaxing for as long as it is allowed arrived at from the other direction.

Two ways a fabric could be held

It is worth separating the two candidates properly, because they are usually conflated under the word “relaxation” and they behave differently.

Friction holds a fabric where it is. It is a threshold rather than a force: below it nothing moves, above it something does, and once it has moved it stays at the new place. A fabric held that way has no preferred state at all — it has a set of states it can be left in, bounded by how hard the restoring force pushes and how much friction is available to resist. Agitation helps a cloth relax for exactly this reason, and the argument transfers.

Set removes the restoring force. It is not a resistance; it is a change to what the yarn wants. A fully set yarn has no preferred state to be held away from, so nothing needs holding, and the fabric’s dimensions are whatever they were when the setting happened.

The two are told apart by an experiment. Release a fabric from friction — by wetting, by agitating, by adding a lubricant — and a frictionally-held fabric moves while a set one does not. That is the experiment the trade already runs every time it specifies a relaxation treatment, and the answer it gets is that both are happening.

A prediction this makes

If the fabric is held by friction against a real restoring force, then supplying energy without changing anything else should move it — and always in the same direction, outwards along the slope.

What a run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.
Fig. 4 What the holding force is made of, on the one quantity of it the site can compute: the friction holding one loop in the loop below, against how tightly the fabric is knitted. It runs from 14.7 to 35.1 millinewtons across the knittable range. The slope this essay measures is of the same order, which is why the fabric stays where it is — and why a fabric shaken while wet is the experiment, since anything that lowers that friction lets the slope act.

A knit agitated in water while under no load should get bigger, not smaller. It does not, and the reason is that wetting a cotton yarn also swells it, sets it, and changes both spacings for reasons that have nothing to do with this slope. Disentangling the two would take an experiment with a fibre that does not swell and does not set, run wet and dry, and it is the cleanest test of this rung available. Nothing here has run it.

The shape of the slope, along one axis

One more section of the surface, because the two directions behave differently and the difference is usable.

A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns.
Fig. 5 The loop’s tightest bend against the tightness factor, which is the same sweep read for a different purpose. Its relevance here is that the whole slope scales with it: a fabric knitted tighter sits on a steeper part of the surface, presses harder, and therefore needs more friction to stay put.
The relaxed fabric is on a slope, not in a hollow. Bending energy per stitch of an unset 20 tex cotton yarn, against the wale spacing and against the course spacing, each varied through the relaxed fabric's own value at a constant 3 mm loop, and each divided by the energy the relaxed fabric holds. Both curves fall away from the relaxed state and neither turns round: by the right-hand edge the fabric holds 43 per cent of what it held, and the fall goes on until the yarn runs straight between its interlacings and the geometry stops. The slopes at the relaxed state are 6.80 mN and 51.6 mN per stitch, which is what something other than the yarn's own springing has to be supplying. A model whose energy minimum is nowhere near the fabric everybody measures is not nearly right; it is right about the yarn and wrong about the mechanism.
Fig. 6 The same two sections through a tighter fabric — a three-millimetre loop against three and a half. Both still fall away from the relaxed state and neither turns round, and the fall is very slightly shallower: forty-three per cent of the relaxed energy left by the right-hand edge against forty-two. Tightening the fabric does not move it into a hollow, and it barely changes the shape of the slope it is on.

The course-wise slope is eight times the wale-wise one, so a fabric released from friction would spread along its courses far faster than it lengthened along its wales. That is testable and it matches a familiar observation from the wrong direction: a jersey that has been washed and dried without restraint comes back narrower and shorter, not wider — which is the contraction the setting supplies, running against a spreading tendency the model says is there.

Two effects of opposite sign, one of which is computed here and one of which is not. That is the honest position, and it is why the next rung is about the one that is missing.

The size of the thing being claimed

It is worth being careful about how much is being asserted, because the headline is dramatic and the content is narrower.

What is computed is that the bending energy of an unset yarn has no interior minimum over a knit’s two spacings. What is measured is that the relaxation states run against that gradient. What is inferred is that the loop’s dimensions are not set by the yarn’s bending, and that friction and set are what stand in the gap.

What is not shown is that friction and set are the whole of the explanation, or in what proportion. Torsion is not in the model; the out-of-plane ride is priced and dropped; the loops of adjacent courses are not stopped from crowding each other in the fabric plane. Any of those could carry part of the load, and the honest position is that the gradient is real, large, and not accounted for.

How the slope was measured

The arithmetic is worth stating so that anybody can disagree with a specific step.

The energy is differenced across a four-tenths-of-a-per-cent change in each spacing, at a fixed loop length, with the shape re-solved at every point. Differencing is used here rather than the multiplier because the two quantities are not the same thing: a multiplier is a force at one thread’s end, and what is wanted is the force on the fabric, which is a different object with a different sign convention. Conflating them would be the easiest mistake in the whole ladder.

The two are then checked against each other through the identity above, which they satisfy, and the direction of both is asserted every time the site is built: a run in which the energy failed to fall in either direction would stop the build rather than print a smaller number.

The residual has also been checked for basis dependence, which is the obvious way for a numerical artefact to masquerade as physics. It moves by under a per cent between eight terms and sixteen, where the effect being claimed is a slope of the same size as the contact force itself and a twelve per cent rise across the three states.

How much of the answer the basis is. The solved energy against the number of terms in the tangent-angle expansion, as a percentage above the value at sixteen terms. Enlarging a Ritz basis can only lower the minimum, so this curve has to fall, and it is asserted to. Eight terms are within 0.21 per cent on the energy and 0.3 on the transverse force. The independent check is elsewhere and is stronger: the force fitted from the solved curve's own equilibrium agrees with the multiplier the solve returned to 0.10 per cent, by a route with nothing in common with it.
Fig. 7 That check, drawn. The solved energy against the number of terms in the tangent-angle expansion, as a percentage above the value at sixteen terms: enlarging a Ritz basis can only lower the minimum, so the curve has to fall and is asserted to. Eight terms are within 0.21 per cent on the energy. The slope this essay is about is two orders of magnitude larger than that, which is the only reason it can be claimed at all.

Where the ladder goes next

Straight into the gap. The next rung asks what happens if the yarn’s natural shape is taken to be the loop rather than a straight rod, and the answer is that everything downstream becomes computable — a load–extension curve, a modulus, a transverse response — with the relaxed fabric as the origin rather than as an embarrassment.

Then the friction, which turns out to have enough margin to hold the fabric wherever it was left, which is what a fabric with three sets of dimensions requires. After that the ladder can spend the forces: what a knit gives when it is pulled, what holds a nap in one, and why a run travels.

Where this goes next

A knit's dimensions come from its loop length. Courses and wales per centimetre against loop length, for a plain weft-knitted fabric in one relaxation state. Neither axis carries a yarn count, a fibre or a machine gauge, and that is the finding: every plain knit measured sits on these two curves whatever it is made of. A knit's dimensions come from its loop A relaxed plain knit's courses, wales and stitch density depend on the loop length and on nothing else — not the yarn count, not the fibre, not the machine gauge. The constants are measured rather than derived, and the interesting thing about the published set is that it does not quite satisfy its own arithmetic. How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it. A dimension without a state Every essay in this field turned on the same omission, and collecting them produces a result none of them had on its own — the quantities a cloth's geometry determines divide cleanly into those a finishing works can move and those it cannot, and the division is not the one the vocabulary suggests. 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.

Reads more easily once this is understood

Essays that name this one as worth reading first.

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 energyContact forceDimensional stabilityElasticaEnergy minimumFrictionMunden constantsRelaxationStitch density