Textile Structure
how cloth is put together
What cloth is

Every fabric's thread lies in a plane

A woven thread's crimp wave lies in a plane at right angles to the cloth. A knitted loop lies in a plane twelve degrees off it. Both halves of this collection turn out to be one picture with one angle in it, and the angle decides how much of a fabric's contact force acts through its thickness.
15 min read 6 figures Structure before fibreA mechanism, not a materialMeasured, not claimed

Worth reading first: Every crossing is a force · A loop is a plane curve in another plane · A woven thread has no room to bend.

This collection has two halves and they have never had a common picture. The woven half draws threads in section: a warp end rising over a weft pick and dipping under the next, in a diagram whose vertical axis is the cloth’s thickness. The knitted half draws loops in plan: a wave running along the fabric, seen from the face.

Two conventions, two kinds of drawing, two vocabularies. It turns out they are the same picture at two values of one angle.

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. 1 The angle, doing its work. The contact force at an interlacing resolves into a part along the fabric and a part through it, and the split is fixed by the angle between the thread’s own plane and the fabric’s. A knitted loop is at a twelfth of a right angle; a woven thread is at a right angle.

Both threads are plane curves

Start with what the two models actually say, which is more alike than the drawings suggest.

A woven thread in Peirce’s geometry follows a path made of arcs and straights, and every point of that path lies in one plane: the plane containing the thread’s own direction and the cloth’s normal. That is what makes a thread diagram a section — the whole thread is in the plane of the paper, and nothing has been projected away.

A knitted loop in this collection’s account is also a plane curve. A half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and the solved curve is the flat solution rotated out of the fabric.

So both threads are flat, and the only question is which plane.

The angle

Measure the angle between the thread’s own plane and the fabric it is in.

A woven thread: ninety degrees. Its plane stands perpendicular to the cloth. The thread goes up and down through the cloth’s thickness while advancing along it, and it does not deviate sideways at all.

A knitted loop: about twelve. Its plane is the fabric’s, tipped by the arctangent of a yarn diameter over a course spacing and a diameter. For a 20 tex cotton at a 3.5 mm loop that is 11.75°.

Two fabrics, one parameter, and the values are at opposite ends of its range.

What the angle decides

The contact force at an interlacing is the derivative of the thread’s energy with respect to where the interlacing is, and the interlacing can move in three directions. So the force is a vector, and the vector lies in the thread’s own plane — because a rotation of the problem rotates its answer.

The plane’s angle therefore fixes how the force is shared between the fabric’s plane and its thickness.

At ninety degrees the whole force is through the cloth. A woven crossing’s normal force is entirely across the thickness; there is nothing left over to act along the cloth. That is why a woven cloth resists compression stoutly and why its thickness is the dimension its whole compression model is about.

At twelve degrees, four fifths of it is along the fabric and a fifth through it. A knitted loop’s contact force is mostly trying to make the fabric wider and longer, and only 7.8 millinewtons of its 38.3 is holding the two faces apart.

One angle, and it explains a difference in behaviour that has been described here twice from different directions without being connected.

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 What the angle costs. Increasing a thread’s climb — which is increasing the angle — lengthens the straight line it has to span and lowers its bending. A woven thread at ninety degrees is at the far right of a curve this only shows the first tenth of.

The range in between

The two extremes are not the only values, and the intermediate ones are all knitted.

angle
a flat fabric (impossible) 0°
a loose jersey, 5 mm loop 8.8°
an ordinary jersey 11.7°
a tight jersey, 2.6 mm loop 14.6°
a 1×1 rib, three-diameter gap, on average 16.2°
the steep half of that rib’s course 27.5°
a woven thread 90°

Reading down that column is reading the whole of this collection’s mechanics of fabrics. The fabrics near the top are extensible, soft through their thickness and held together by friction; the ones near the bottom are stiff, thick-resistant and held together by their own geometry.

Two things about the spacing of that column are worth noticing. The first is how narrow the knitted range is: every plain knit anybody makes, from the loosest to the tightest, lies between about nine degrees and about fifteen, which is a factor of well under two across the entire manufacturable range of loop lengths. The second is how far the gap to the last row is. Ninety degrees is not the next value after 27.5; it is six times the whole spread of everything above it, and there is nothing in between because there is no fabric in between. A thread either turns out of the plane of the cloth or it does not, and no construction turns a little.

The rib rows are the ones that show the discontinuity is a real boundary and not a gap in the table. A rib has two half periods with different angles — 16.2 degrees on average and 27.5 on the steep one — so a single fabric already spans a range wider than the whole of the jersey column, and it does it by putting some of its yarn between the beds. Push that further and the angle keeps climbing, and at the limit the yarn is going straight through the thickness and there is no course direction left. That limit is a woven cloth, reached from the knitted side, and the reason no fabric sits between 27.5 and 90 is that everything on that stretch has been built and is unstable.

Why zero is impossible

The top of the table is worth a sentence because it says what an interlacing is.

A fabric whose threads lay entirely in its own plane would have an angle of zero, and it cannot exist: two curves in one plane cannot pass through one another. Every fabric there has ever been has a nonzero angle for that reason alone, and the minimum angle is set by the interlacing — a yarn diameter of climb over whatever the thread has to travel between interlacings.

So the angle is bounded below by the thing that makes a fabric a fabric, and the loosest knittable jersey is near that bound.

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. 3 Why zero is impossible, at a second construction. The energy falls as the climb grows and it is still falling at the smallest climb the geometry admits — a thread that left the plane not at all would have to pass through the one below it. The angle has a floor set by the yarn’s own diameter, and the floor is where every real fabric sits.

Why the woven end is not on the same curve

There is a limit to how far this unification can be pushed and it is worth marking, because the table above invites a reader to interpolate.

A knitted loop and a woven thread differ in slack by two orders of magnitude — half a thread’s length against four parts in a thousand — and that difference is what decides whether the thread can be solved as a free elastica at all. A woven thread has no free run; its whole crimp is spent inside the wrap, and its shape is arcs and straights whether anybody likes it or not.

So the two ends of the table are computed by different methods and the middle is empty of anything but knits. The angle is a common coordinate and it is not a continuous family: nothing sits at forty-five degrees, and if something did, neither method would obviously apply to it.

Why nobody noticed

The two conventions hid it, and the hiding was reasonable each time.

A woven section diagram draws the thread’s plane as the page, so the angle is invisible: it is the whole picture. A knitted plan diagram draws the fabric’s plane as the page, so the angle is projected away: it is what has been left out.

Neither drawing is wrong. What was missing was a picture that had both planes in it, and that picture needed the knitted loop to have a third coordinate — which it did not, until it did.

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. 4 The angle as a setting. A rib’s average tilt is a bed gap over a drop, so a machine can be set to put a fabric anywhere between a jersey’s twelve degrees and something much steeper.

What the angle does not decide

Three things, and the list matters because a single unifying parameter invites over-reading.

Slack. A woven thread has four parts in a thousand of it and a knitted loop half its own length. That is why one can be solved as a free elastica and the other cannot, and it is independent of the angle: it is a length comparison, not a direction one.

Thickness. A woven cloth’s thickness comes from two thread diameters and a crimp height; a knit’s comes from two thread diameters and nothing else. Both are lengths in millimetres and neither is a function of the angle.

And how much the fabric holds together. That is connectivity for a knit and interlacing count for a weave, and it is combinatorics rather than geometry.

So the angle is one of several parameters and it is the one that decides where a force points. It is not a theory of fabric.

Which resolves an asymmetry that looked like a defect

There is an entry in this collection’s account of knitted compression that reads oddly on its own, and this rung is where it stops being odd.

A woven cloth’s crossings press with 185 to 851 millinewtons and a knitted loop’s with 38.3 — a factor of five to twenty-two. But the through-thickness comparison is worse than that, because the woven figure is entirely through the cloth and only a fifth of the knitted one is. Compare like with like and a woven cloth holds itself open with between 24 and 109 times the force a jersey does.

That is a much larger contrast than the raw force ratio suggests, and it is the honest one for anybody asking why a knit compresses so easily. Two thirds of the answer is that the forces are smaller; the remaining third is that they point the wrong way.

Where the angle comes from in each case

The two mechanisms are worth putting side by side because they are not the same mechanism.

In a woven cloth the angle is ninety degrees by construction: the two thread systems are perpendicular in the plane, so a warp end’s excursion to clear a weft pick is purely through the thickness. There is nowhere else for it to go.

In a knit the interlacing is between two parts of the same thread system — a loop’s feet and the head of the loop below, both courses — running roughly parallel. So the excursion is a small climb superimposed on a long run along the fabric, and the angle is the ratio of the two.

Perpendicular systems give a right angle; parallel ones give a shallow one. That is the structural statement behind the whole table, and it says the angle is a property of the topology rather than of the geometry: it would take the same two values for any yarn and any gauge.

What a third fabric would look like

The obvious question is whether anything sits in the middle of the range for a reason other than being a knit, and there are candidates.

A braid interlaces two thread systems that cross at an angle less than ninety degrees in the plane, so its threads’ planes should stand at less than a right angle to the fabric — by an amount set by the braid angle rather than by any yarn. This collection draws braids and has never asked.

A warp knit’s underlap runs between wales at a shallow angle in the plane while the overlap wraps a needle, which suggests two very different plane angles within one thread. That is a genuinely different arrangement from either fabric here.

Neither is computed. They are named because a unifying parameter is only worth having if it can be asked about things it was not built from.

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. 5 The knitted crossing, in section, with its contact force drawn at the true angle. A woven crossing’s force would be drawn straight up the page — which is the same picture at the other end of the range.

What the angle says about the two thicknesses

Both fabrics have a thickness and both get it from the same place, which is a further piece of the unification and one that runs the other way from the forces.

A woven cloth’s thickness is two thread diameters plus a crimp height — how far a thread’s wave rises above the cloth’s mid-plane. A knitted fabric’s is two thread diameters and no crimp height at all, because a knitted loop’s excursion through the fabric is a monotone climb from one face to the other rather than a wave with an amplitude.

So the woven fabric has a term the knitted one lacks, and it is the term that makes a woven cloth’s thickness depend on its sett while a knitted fabric’s does not depend on its gauge. How thick a knit is is that independence, and this is where it comes from: a thread at ninety degrees oscillates through the fabric and one at twelve degrees crosses it once.

Whether the parameter earns its place

A unifying parameter is worth having only if it does work, so it is fair to ask what this one has done that was not already known.

It has connected two facts. The first is that a woven cloth’s crossings press between five and twenty-two times harder than a knitted fabric’s. The second is that a woven cloth is far stiffer through its thickness than a knitted fabric is — which everybody knows from handling both and which nobody had a number for.

Those look like the same fact and they are not, and the angle is the difference. Two thirds of the contrast is in the size of the forces and the remaining third is in their direction, and separating the two is the only way to say which fabric property follows from which structural feature.

That is a modest amount of work for a parameter to do. It is more than nothing, and it is the kind of connection a collection with two halves ought to be making.

What would falsify the picture

One measurement, and it is not on either fabric.

The angle predicts the split of a fabric’s contact force, so it predicts the ratio between a fabric’s resistance to being stretched and its resistance to being squashed. For a woven cloth that ratio should be very large — everything through the thickness, nothing along. For a jersey it should be about four to one the other way.

Measuring both on one apparatus is standard fabric testing. A jersey whose two resistances came out in a ratio near ninety, or a woven cloth whose in-plane resistance was comparable with its through-thickness one, would say the angle is not doing what this rung says.

Neither has been measured here, and the collection’s woven and knitted compression accounts are not directly comparable for reasons of their own.

What is genuinely new here

Two things, and the first is a picture rather than a number.

Both halves of this collection draw plane curves, and the difference between them is the angle between the thread’s plane and the fabric’s. That was not sayable while one of the two threads was drawn without a third coordinate.

And the angle decides how a contact force is shared. Ninety degrees puts all of it through the fabric; twelve puts a fifth. Every difference between a woven cloth’s and a knitted fabric’s behaviour under a press follows from that, on top of the difference in the forces themselves.

What the pictures cannot show

Neither figure here draws a woven thread beside a knitted one, and the reason is a scale problem rather than a drawing one: at the same magnification a woven thread’s plane is the page and a knitted loop’s is a line across it, so a single picture of both is a picture of one of them.

The table is the honest form of the comparison, and a table is a poor picture.

What this rung is not claiming

Not that the two models are one model. They are computed by different methods for a reason: one thread has slack and the other has none.

Not that the angle is a spectrum. Two clusters and an empty middle.

And not that a knitted fabric’s mechanics reduces to it. The angle decides where a force points. What decides how big the force is, whether the fabric holds together, how far it stretches and what it weighs are four other questions with four other answers, and this collection answers each of them somewhere else.

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. 6 What a third fabric would look like. A two-by-two rib crosses at half its sinker loops rather than all of them, so its angle is between the jersey’s and the one-by-one rib’s — and the whole family sits on one curve with the gap as its parameter. That is the sense in which both halves of this collection meet here.

Where the collection’s two halves actually meet

It is worth listing what the two halves already share, because the angle is the fourth item rather than the first.

They share a contact force at every crossing, computed by different routes and compared directly. They share a thickness built from thread diameters. They share a friction account, in which what is held is a force and what holds it is a coefficient times a normal. And now they share a plane and an angle.

What they do not share is the method of solving for the thread’s shape, and they never will: a woven thread has no free run to solve and a knitted one is nine tenths free run. That difference is prior to everything above and is not a difference of degree.

Which rungs this stands on

That a woven thread’s path is a plane curve, which is Peirce’s construction and is what makes a thread diagram a section.

That a knitted loop’s is too, which is the rotation identity at a loop is a plane curve in another plane and is exact rather than approximate.

And that the contact force lies in the thread’s own plane, which follows from the first two by the same rotation and is checked against a finite difference of the energy at the force that holds a knit open.

Three results, none of them new here, and the angle between two planes is what they had not been asked together.

Where the ladder goes next

The angle is largest inside a fabric that has two beds, and a rib’s own course alternates between two very different values of it: a rib climbs a gap.

And the fabric at the shallow end of the range is the one whose thickness is hardest to measure, because so little is holding it up: what a thickness gauge reads on a knit.

Where this goes next

The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica this site does not have. Every crossing is a force A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself. 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. How much of a thread is spent going round the one it crosses. The share of a warp end's length that lies inside the wrap — the arc of radius half the combined diameter, which is as close as two centre lines can get — for every cloth in this collection's table, with a jersey at the foot for comparison. It runs from 7% on an open scrim to 54% on a sheeting, and what is left over is a straight run with no shape to solve. A knitted loop's figure is zero: its peak curvature never reaches the wrap's, so it touches at points and is free in between. That is the whole reason the same solver refuses a shirting and converges on a jersey, and it is a statement about the two fabrics rather than about the arithmetic. A woven thread has no room to bend Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.

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.

AnisotropyCloth thicknessContact forceContact pressureCrimpElasticaInterlacingLoopPeirce's geometry