A knit is soft because it bends

Everybody knows a knitted fabric stretches and a woven one does not. The loop explains the geometry of it — a knit’s thread is bent into loops with slack in every direction, so the fabric can extend by changing the loops’ shape without any yarn changing length — and why a knit recovers explains what brings it back.

Neither can say what the extension costs, because until the yarn was given a bending rigidity the site had no stiffness anywhere. With one, the same energy question can be asked of both structures, and the answer is not a ratio.

The claim #

In the model that gives a woven cloth a definite crimp modulus, a plain knit has exactly none. Not a small one — none, to machine precision, over the whole range of extension the loop geometry admits.

The correct reading of that is not that a knit is infinitely soft. It is that a knit’s initial modulus is not a bending property, so whatever its first few per cent of extension costs is being paid to something else — and that something else is measured differently, varies more, and is why a knit’s load–extension curve is so much less repeatable than a woven’s.

One energy function, two structures #

The energy of a bent rod is ½·B·κ² integrated along it, and for a path made of circular arcs joined by straight runs that integral is one multiplication. If a path turns through a total angle Θ on arcs of radius r,

u = ½ · B · Θ / r

The straight runs contribute nothing at all.

For a woven thread, Peirce’s path turns through 2θ at each crossing on arcs of radius D/2, where D is the sum of the two diameters and θ is the weave angle. So u = 2Bθ/D. The radius is set by contact — it is decided by the two yarns’ diameters — and the angle is set by the spacing. Move the picks apart and θ falls immediately, so the energy changes and there is a force.

For a knitted loop, the head and the feet are arcs around the yarn beneath them, so their radius is one yarn diameter — again set by contact. And the total turning is a property of the loop’s topology: the yarn goes round and comes back, and how far round it goes does not depend on how far apart the courses are. So Θ is constant and r is constant, and the energy does not move.

That is the whole of it. The two structures differ in one respect: whether the turning angle is fixed by contact or by the fabric’s dimensions. A woven cloth’s is dimensional and a knit’s is topological.

What the loop does instead of bending #

The extension has to go somewhere, and in this model it goes into the legs.

A loop of thread of a given length is spent on two things: the arcs, whose length is fixed by the radius and the turning, and the straight legs, which get whatever is left. For a 20 tex cotton yarn at a loop length of 3.5 millimetres the arcs take 1.05 mm and the legs take 2.45.

Extend the fabric course-wise and the legs lengthen and the arcs do not change. Nothing bends further; the loops rotate and slide and the yarn redistributes. The model asks no force for it.

That redistribution is the same move the crimp-interchange argument makes for a woven cloth, and it is worth noticing that a woven cloth’s version does cost something while a knit’s does not. Both hold the thread length fixed. The difference is that a woven cloth cannot redistribute thread without changing an angle and a knit can.

And the available extension is enormous. The legs run out only when every millimetre of thread is in an arc, which for these numbers is at 233 per cent — more than three times the resting length. That is far more than a real knit gives, and the discrepancy is itself informative: a real plain knit reaches perhaps fifty to a hundred per cent before it stiffens sharply, so something other than running out of leg is what stops it, and that something is the loops meeting each other and jamming.

Where the nought stops being useful #

An exact zero over a whole range is a strong result and it has a boundary, and the boundary is worth locating because it is where a knit starts behaving like a woven cloth.

The arcs’ radius is fixed by contact only while the contact holds. A loop’s head wraps a yarn beneath it and cannot close tighter than that yarn allows — but it can open wider, and at a large enough extension it does: the head straightens, the wrap angle falls, and the turning stops being topological. At that point the energy starts to move and there is a force.

So the nought is a statement about the extension regime rather than about the structure. Below whatever extension keeps every arc in contact, a knit’s bending energy is constant. Above it, the loops have begun to open and the model’s two idealisations have both failed at once — the radius is no longer the yarn’s and the turning is no longer 2π.

Where that happens is not computed here and it is almost certainly before the 233 per cent the leg arithmetic gives, because a loop whose legs are nearly all straight has very little wrap left. So the regime the nought describes is the low-extension one, and the low-extension one is exactly where the conclusion matters — a fabric’s initial modulus is what a wearer feels and what a testing machine reads first.

That makes the result stronger rather than weaker. A zero that held everywhere would be a claim about a model with no room in it; a zero that holds over the useful range and fails where the idealisations fail is a claim with a stated domain, and the domain is the one anybody cares about. What it also says is that a knit’s load–extension curve should have a knee — a flat initial region governed by friction, then a rise as the loops open and bending begins to cost — and a knit’s measured curve has exactly that shape.

The knee’s position is therefore a measurement of where the wrap runs out, which is a geometric quantity this model has the pieces for and does not compute. That makes it the most tractable of the things left open here: everything needed is a loop’s three-dimensional path, and the payoff is a prediction of a feature every knitted load–extension curve already shows and nobody has a number for. A model that predicted where a jersey stiffens would be predicting the one thing about a knitted garment that every wearer notices and no specification records.

Why this makes a knit’s behaviour less predictable, not more #

The usual framing has a knit as the simple case — it stretches, a woven does not — and this rung inverts that.

A woven cloth’s first few per cent of extension is resisted by a bending stiffness, which is a material property of the yarn. It is known to a factor of three at best on this site, and that is bad, but it is a definite quantity that a specimen carries with it.

A knit’s first few per cent is resisted by friction between the loops and flattening at their contacts. Friction is not a material constant: it depends on the finish, the moisture, the pressure at the contact and how the fabric was handled last. Flattening depends on the yarn’s compressive behaviour, which this site does not model at all.

So the prediction is that a knit’s load–extension curve should be less repeatable, more history-dependent and more sensitive to finishing than a woven’s — and that is exactly what knitted-fabric testing finds. A knit’s dimensions and its low-load behaviour are notoriously dependent on relaxation state, which is why the site’s own knit ladder insists that a knit’s dimensions have no meaning without a state attached.

The two facts have always been stated side by side. This rung says they are the same fact: the thing that would make a knit’s behaviour predictable is absent from it.

The comparison run honestly #

An assertion that comes out exactly zero deserves suspicion, so it is worth being explicit about what would have made it non-zero.

If the arcs’ radius depended on the fabric’s dimensions — if a stretched loop’s head wrapped more tightly — the energy would change and there would be a force. It does not, because the head is wrapped around a yarn of a definite diameter and cannot close further than that yarn allows.

If the total turning changed with extension, the same. It does not, because turning is a topological count.

If the legs were not straight, they would carry curvature and the energy would depend on their length. They are taken as straight, which is Peirce’s own idealisation of the knitted loop and is the same idealisation his woven geometry makes about the runs between crossings.

So the zero is a consequence of two idealisations, and both are the same ones the woven case uses. That is what makes the comparison worth anything: the same model applied to two structures, giving a definite answer for one and a nought for the other, rather than two models tuned separately.

The machinery asserts both halves at once — the knit’s spread must be exactly zero and the woven’s must be positive — because either one alone could be produced by a bug.

What was counted, and how #

The knit’s loop is Peirce’s idealisation: circular arcs of radius one yarn diameter at the head and the two feet, joined by straight legs, with the loop length taken as given. The diameter comes from the count by the site’s standing volume arithmetic at a stated packing factor.

The total turning is taken as 2π per loop, which is the count for a plain knitted loop: the head turns through π and each foot through π/2 into the loop below. That number is the one input a reader might want to argue with, and the argument would be about the loop’s three-dimensional path rather than about the arithmetic — every value of Θ gives a constant energy, so the conclusion survives any turning count at all.

The woven half is the site’s existing machinery: a Peirce state at the cloth’s quoted construction, its constant-thread-length locus, and the closed-form bending energy at each state.

The rigidity used is the free bound — the sum of the fibres’ — because it is the lower of the two and because where a real yarn sits between the bounds is known only to a factor of three. It scales both structures’ energies equally and cancels out of the comparison entirely, which is the point of comparing rather than quoting.

Two refusals are checked. A loop length that is not a length is refused, and a compression to nothing is refused, because both would return a plausible number from an impossible fabric.

Where the model stops #

Jamming is not modelled and it is what actually stops a knit. The model’s 233 per cent is where the legs run out of thread; a real knit stiffens at a third of that because the loops meet. Predicting where needs the loops’ three-dimensional path and their compressive behaviour, and the site has neither.

Friction between loops is absent, and it is the thing this rung concludes must be carrying the load. Saying that a force is not a bending force is a weaker statement than saying what it is, and this rung makes only the weaker one.

The loop is two-dimensional. A real loop’s legs cross behind the loop below, and the contact there is what makes a plain knit curl at its edges — which is an argument this site has already made and which cannot be made in this model, because a flat loop has no reason to curl.

And nothing here is about the wale direction. Extending a knit lengthwise and crosswise are different deformations with different available extensions, and the argument above is about the course direction. The conclusion — that the arcs’ radius is fixed by contact — holds in both, so the nought does too, but the available extension does not.

The generalisation #

The distinction that survives is about where a structure’s curvature comes from.

A structure whose curvature is set by contact deforms without bending energy; a structure whose curvature is set by its own dimensions cannot. That is a statement about mechanisms in general and it separates two families of soft structure that otherwise look alike: chain mail, a woven basket, a knitted stent, a coiled cable, a folded honeycomb. Some of them extend by rearranging contacts and some by flexing members, and the first kind is soft in a way the second cannot be however limp its material is.

The design consequence is worth stating in the direction a designer would use it. To make a structure whose stiffness does not depend on its material, make its curvature a contact condition. The knitted loop does this by accident of construction, and it is why a knit made of steel wire is still a stretchy fabric — which is a real object, is used for cut-resistant gloves, and behaves in a way that would be inexplicable if the extension had to bend the wire.

And there is a warning in the second half of the rung. A structure that is soft because of contact is soft in a way that is not a material property, so it inherits all the unpredictability of contact: friction, wear, moisture, history. Softness bought this way is bought at the price of repeatability, and no amount of care about the material buys it back.

Who found it, and when #

Peirce’s knitted-loop geometry is from 1947 and is the direct counterpart of his 1937 woven geometry: circular arcs and straight runs, an idealised loop, a loop length that everything else is measured against. It is a geometry and has no forces in it, which is exactly the position this site was in until the yarn acquired a stiffness.

The mechanics of knitted-fabric extension belongs to the work of Postle, Munden and others from the 1960s onwards, and that literature is very clear that a knit’s low-load behaviour is dominated by friction and by loop rearrangement rather than by yarn bending. So the conclusion here is not new.

What is this site’s is the route to it: running the same energy function over both structures and getting a nought, rather than arguing from what is known about knits. A nought obtained that way says something a statement about friction does not — that the bending contribution is not merely small but structurally absent, and that no refinement of the yarn’s stiffness will ever produce one.

Where the ladder goes next #

The obvious next rung is jamming: where a knitted loop actually runs out of extension, which needs the three-dimensional path and is where the site’s loop-length arithmetic would meet a force for the first time.

Sideways, the same argument applied to a two-bed fabric should give the same nought with a different available extension, and applied to a warp knit should not, because a warp knit’s laps are held by neighbouring wales rather than by a loop below.

Further out is the missing companion to this whole group: a model of what a yarn does when it is compressed rather than bent. It is what stops a knit extending, it is what this site’s racetrack section describes without a stiffness, and it is the largest single thing the mechanics field is short of.