Textile Structure
how cloth is put together
Concept

Loop length — where it appears

The length of yarn in one knitted stitch, which is the single quantity a relaxed plain knit's dimensions are all functions of. Everything else about a relaxed knit follows from it through dimensionless constants, so it is the master variable of the whole subject.

Named by 43 essays across 5 fields — each of them below, with the objects they name alongside it.

The knitted loop. One thread, bent into a course of loops, each of them drawn through the loop below. Nothing here is straight, which is why a knit extends in every direction while a woven cloth extends only on the bias.

The loop

A knit is one thread bent into loops, each drawn through the loop below. Nothing in it is straight, which is why it stretches in every direction while a woven cloth stretches only at an angle.

knits · Knit Yarn per needle, by structure. Wale spacings of yarn per needle position, at one loop length. A knitted loop is 4.30 of them — Munden's constant, measured rather than derived — a float across one needle is exactly one, and a tuck is taken as a stated multiple of a loop. Everything else follows by counting, so the percentages beside the bars are not estimates.

What a tuck costs

A knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop. That much is a count. What it does to the fabric's size is not, and this essay is careful about which is which.

knits · Stitch notation single float as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here.

The float in a knit

Five rungs of this anchor have taken the float to be a length of thread on the surface with nothing holding it down. A knit has one too, and it behaves the same way in the light and the opposite way in the hand — because a woven float lengthens its thread and a knitted one shortens its fabric.

knits · Float A fashioned edge at 1 wales in 2 courses. A knitted panel narrowing by 1 wale every 2 courses, drawn at the fabric's own aspect: a wale is 1.2791 times as wide as a course is tall, which is Munden's ratio of the two published constants. The edge therefore stands at 32.60 degrees from the wale, and that angle is the same in every yarn, at every gauge and at every loop length. Marks show where the 8 transfers fall.

A fashioned edge has a quantised angle

A knitted panel is shaped by transferring loops, so its edge steps by whole wales at whole courses and its angle is the arctangent of a fraction. The available angles turn out to be the same for every plain knit there has ever been — in any yarn, at any gauge, at any loop length — because the constant they scale by cancels the loop out. There are eighteen of them, and 16.67° between the last two.

knits · Shaping 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.

finishing · Knit geometry How far a tube can be tapered by its loop. The same 10 wales by 10 courses of plain knit at the two ends of the usable loop range for a 20 tex yarn — 3.44 mm at the loose end and 2.80 mm at the tight one — drawn at a common scale in centimetres. On 240 needles the circumference falls from 192.0 cm to 156.0 cm, a taper of 18.8 per cent, and the fabric becomes 1.51 times as dense.

A tube can only be shaped by its loop

On a circular machine the needle count is the cylinder, so a seamless tube's circumference is its wale count times its wale spacing — and the wale spacing is the loop length over one constant. The loop is the only free quantity, the yarn bounds it at both ends, and what is left is a taper of 18.8 per cent bought at the price of a fabric half again as dense.

knits · Shaping The loop that costs nothing to extend. A plain knitted loop at rest and extended by 35 per cent, with the arcs marked. The arcs' radius is the diameter of the yarn the loop wraps, 0.167 mm, and it is set by contact rather than by the fabric's dimensions — so extending the fabric lengthens the legs and bends nothing further. The bending energy is 0.0176 N·mm at both, and the model therefore asks no force at all for an extension a woven cloth would refuse. What the drawing cannot show is what a real knit's first few per cent do cost, which is friction and yarn flattening and is not a bending property.

A knit is soft because it bends

Ask the same energy question of a woven cloth and a knitted one and the answers are not different by a factor — they are different in kind. A woven cloth's bending energy changes the moment it is extended. A knitted loop's does not change at all, exactly, over the whole of its extension, because its arcs are held to a radius by contact rather than by the fabric's dimensions.

knits · Knit The weight against the constant nobody has measured. Areal weight against the stitch-density constant, for four two-bed structures at 20 tex on a 0.35 cm loop with a tuck taken as 1.15 of one. Every line is exactly straight through the origin, because the weight is tex times yarn per repeat times k_s divided by the area of the repeat and there is no fitted constant anywhere in that division. The three vertical rules are the only measured values this site has — Munden's published k_s for plain single jersey in three relaxation states — and none of them is the right value for any structure drawn here. Where each line should be read is the whole of what is missing.

The constants do not compose

The yarn in a knitted structure is a sum of what each element takes, exactly, over structures nobody has measured. The size the structure relaxes to is not a sum of anything — and the whole gap between the two is one number per structure, which would cost forty-five fabrics each to obtain.

finishing · Knit geometry A knit's restoring force, and the column that does not move. A plain knit of 20 tex cotton at a loop length of 3.5 mm, over the extension range its own geometry admits. The bending energy stored in one loop is the same number at every extension — the loop's arcs are held to a radius by the thread they wrap rather than by the fabric's dimensions, so extending the fabric does not bend anything more. The frictional resistance at the interlocks is not zero: it is μ times the force pressing there, times 2.34 interlocks per millimetre of width. So a knit's resistance to extension is dissipative rather than elastic, which is why it does not spring back and why its dimensions depend on how much it has been agitated. What the rows cannot show is the interlock force itself, which this site does not have for a knit and which is recorded as missing.

What stops a knit extending

A knitted loop's bending energy does not change as the fabric extends — exactly, over the whole range its geometry admits. Something resists, and it is not stiffness. It is friction at the interlocks, which is dissipative rather than elastic, and that single fact accounts for why a knit does not spring back, why a softener changes its dimensions and why the constants its size is quoted with contain no yarn property at all.

knits · Knit A loop's cell, dry and wetted. One stitch of a 20 tex cotton jersey at a 3.50 mm loop, drawn inside the rectangle of one wale by one course that its own dimensions give. The thread is drawn at its own width, and it already fills 1.129 of the cell dry — more than the whole of it, which is what an opaque jersey looks like from above. Wetting takes it to 1.355. So the reason a knit does not build a swelling pressure is not that it has room; it is that its dimensions are a loop length times a constant with no yarn diameter in them, so there is no closure condition to fail. What the drawing cannot show is the third dimension: the legs lie over one another rather than overlapping in the plane, which is exactly why an occupancy above one is possible.

A loop has no closure condition

A woven cloth can run out of room: its two systems must supply its whole thickness between them, and past a certain swelling they cannot. A knit has no such equation, so no critical swelling and no pressure. The obvious explanation — that a knit is open and has somewhere to put the swelling — is false, and the arithmetic refuses it.

knits · Knit Munden's states against the fibre swelling. What a plain knit does between its relaxation states, beside the swelling of the fibre it is made of. Going from dry-relaxed to wet-relaxed a jersey loses 5.66% along its courses and 2.44% across its wales, and going on to fully relaxed it loses 9.1% and 7.0%. The fibre swells 20%. Two things rule the swelling out as the cause: it is several times the whole change, and the change is markedly anisotropic while a swelling enters both of a loop's dimensions through the same loop length. What the bars cannot show is what does cause it, which is friction — water lets the loops move to where the yarn's own bending had been trying to put them.

A knit's change of state is not its swelling

A jersey is smaller wet-relaxed than dry-relaxed, by 5.7 per cent along its courses and 2.4 across its wales. Water is obviously involved, so the swelling is the obvious cause. Two things rule it out, and both are properties of the constants rather than measurements of a fabric.

knits · Knit A plain knit's two relaxation steps. Munden's three relaxation states are usually given as three sets of constants. Read as a path they are two steps, and the two compose to the whole exactly — which is a real check, because the three sets were measured independently. The first step is the larger in the course direction and the smaller across the wales, and the second is 0.64 of the first lengthwise. That is the shape of a laundering series and it is the same mechanism: a fully relaxed state is reached by tumbling rather than by waiting, so what the standard specifies is a quantity of agitation and not a duration. What the bars cannot show is the loop length, which cancels out of all four numbers because every dimension of a knit is a loop length times a dimensionless constant.

A knit relaxes for as long as it is allowed to

Munden's three states are usually given as three sets of constants. Read as a path they are two steps, they compose exactly, and the second is not a smaller version of the first — the fabric shrinks twice as much along its courses as across its wales on the first step and rather less than half as much on the second.

knits · Knit When a knit has a hole between its loops, and when it has none. A loop's occupancy is its length times its diameter over the cell it sits in, and Munden's constants make that cell ℓ²/(k_c·k_w). The loop length cancels once and what is left is d·k_s/ℓ, where k_s is Munden's own stitch-density constant — so whether a knit has a hole between its loops depends on d/ℓ and on nothing else: no gauge, no count, no fabric dimension. The three curves are the three relaxed states of 20 tex cotton. Each crosses one at a loop length of 3.34, 3.63, 3.95 mm respectively, and a jersey is knitted at 2.63 to 3.44 mm — the shaded band. Every commercial jersey is therefore on the wrong side of the threshold in at least two of its three states: it closes its own holes as it relaxes, and its air goes through its threads rather than between them.

A knit has no hole to lose

Every argument in this ladder is planar: threads at a spacing, a rectangle between four of them, a channel down it. Applied to a jersey it returns nothing at all, and the nothing is the finding. A loop's occupancy is its diameter times Munden's own stitch-density constant over its loop length, with no gauge and no fabric dimension in it — and the whole commercial range of tightness is on the wrong side of the threshold.

knits · Knit geometry The same error twice: scattered, and in a period. Two bands of 96 ends, drawn at spacings that differ from the reed's by the same root-mean-square amount — 5% of a dent, which is 21 µm. The upper band's errors are independent; the lower band's repeat every 4 ends, which is what one shaft set forward or one dent of the reed too wide produces. The displacement is drawn 8 times over scale, because at true scale it is half a pixel and both bands are picket fences; the arithmetic below it is at the true amplitude. The upper band reads as an even cloth with a little texture in it and the lower one has stripes, and nothing about the eye is needed to say why: at 96 ends the periodic arrangement is 7.8 times stronger at its own frequency than the scattered one is at any frequency, and over the 256 ends a buyer takes in at once it is 12.8. The ratio grows as the square root of how much cloth is looked at, which is why a fault-finding sweep is done at a distance rather than close up.

A course is one thread and a warp is many

A woven fabric draws its warp from two thousand packages side by side, so a yarn's drift averages out across the width. A weft knit takes whole courses from one package, so the same drift becomes a band — and the standard remedy for that turns an invisible error into a visible one.

knits · Knit The stripe a knitting machine chooses. A circular machine takes its yarn from a fixed number of feeders arranged round the cylinder, and feeder k lays every F-th course. So any difference between packages — a shade, a count, an evenness — is reproduced in the fabric with a period of exactly F courses, and at 20 courses per centimetre that is a band every 48.0 mm on a 96-feeder machine. The machine chooses the period, not the yarn. The spacing is proportional to the feeder count and inversely proportional to the course density, as the turn of the cylinder requires whatever the bars show, and a one-feeder machine produces no band at all from the same packages.

Why a knit shows a thick place

A woven cloth has hundreds of separate warp ends and averages a yarn's faults among them. A knit has one thread and a machine that repeats — so a difference between two packages becomes a stripe, and the machine chooses its period.

knits · Variation Whether a cloth's hairs can reach one another. n_A λ² for each construction in this site's table — the hairs per square millimetre times the square of their own length, which is the pure number that asks whether a hair can touch its neighbour. It is a count times an area, so it has to be a pure number. Every one of them is under one, which means no ordinary woven cotton cloth has a hair layer at all: it has isolated whiskers on a bare surface. The dashed line is the threshold. The spread across the whole table is only 1.9-fold, because the density goes as the sett times the root of the count and those move in opposite directions as a cloth is made finer — so construction is almost powerless here, and everything that crosses this threshold does so by finishing rather than by weaving.

A knit gives up its fibres more easily

Knitwear pills and shirting does not, and the fibres are often the same fibres. The difference is a count of yarn per unit area and a pressure between threads, and both of them push a knit over a threshold that a woven cloth of the same yarn cannot reach.

knits · Knit 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.

knits · Elastica What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

What a knit gives when it is pulled

How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight.

knits · Knit A jersey gets taller before it gets shorter. How much a 20 tex cotton jersey shortens along its wales as it is pulled along its courses, with the course spacing at every extension chosen to minimise the loop's energy rather than assumed. Over the first 81% it is negative — the fabric gets 2.0% taller as it is pulled wider — and only then does it start to contract, reaching 88% at the geometric limit. A material with a negative Poisson ratio is a curiosity; a knit has one over part of its range for a reason with no material in it at all, which is that widening a wale at a fixed loop length first lets the loop's tightest bends open and only later starts taking height away from it.

A jersey gets taller before it gets shorter

Pull a knit along its courses and the first thing it does is grow along its wales — by two per cent, over the first eighty per cent of extension, before it turns round and contracts. The transverse response changes sign, and there is no material in the explanation at all.

knits · Knit 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.

How far a knit could go if its yarn were the limit

The yarn in a stitch allows three hundred and twenty per cent course-wise extension before the straight line between two interlacings reaches the thread spanning it. A jersey jams at about a hundred. The factor of three is the finding: what stops a knit stretching is not the loop running out of yarn.

knits · Knit Why a cuff is ribbed. The force a knit pulls back with, over the range a cuff is used across. It rises the whole way — 0.79 N per metre at 23% to 3.38 at 104% — and it is small throughout, which is the combination a cuff needs and almost nothing else supplies. A rib gets its extension by geometry, folding alternate wales to opposite faces so that its relaxed width is about half its opened one, and it gets its recovery from the loop reconfiguring. Neither is the yarn stretching, which is why a cuff made of a fibre with no elastic recovery at all still works.

A rib pulls back on a force the loop supplies

A cuff has to give a great deal at almost no load and come back reliably, and no ordinary material does both. A rib gets its extension from folding, which is geometry, and its return from the loop reconfiguring, which is now a computable force — under four newtons a metre over the whole range a cuff works across.

knits · Two-bed 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.

Two knits with one tightness factor are one knit

The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does.

finishing · Knit geometry What a raising machine can catch in a 2/2 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

What holds a nap in a knit

A fibre buried in a woven cloth breaks rather than slides once about thirty-four millimetres of it is held, and a cotton staple buries about fourteen. In a knit the same figure is over a metre — seventy-five times the burial available — so nothing is ever close, and a raised knit sheds for the whole of its life.

finishing · Nap What holds a thread in, as two factors. The two quantities whose product is the grip on a buried thread, for a woven poplin and a jersey of the same yarn. Each contact in the knit is lighter by 8.0 times, and the contacts are further apart by 5.6 — one per half loop length against one per thread spacing. They multiply rather than competing, so the grip per millimetre of buried thread is 31 times weaker in the knit, and the crossover length at which a thread breaks rather than slides moves with it: 11.6 mm in the cloth and 461 in the knit. The two factors are drawn separately because their product is two orders of magnitude and a bar chart of it would put the knit's bar below the width of a line.

The knitted pilling criterion gets its number

Knitwear pills and shirting does not, and the standing explanation here has been that a knit presents more exposed yarn under less pressure between its threads. The second half had no number. It has one now, and it is bigger than the argument needed: the grip on a buried fibre is thirty times weaker in a knit than in a woven cloth of the same yarn.

applied · Pilling 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.

How thick a knit is

Two centre lines pass at a diameter and each has a radius on either side, so a plain jersey is two yarn diameters thick with nothing fitted. It does not depend on the gauge, it lands inside the band of this collection's woven cloths, and it is lower than any gauge will read.

knits · Knit 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.

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

applied · Damage What the third dimension changes, and by how much. Every number the planar loop model produced, beside the same number with the climb in it, for a 20 tex cotton jersey at a 3.5 mm loop. Four of the five fall and none moves by as much as four per cent, which is the useful part of the answer: the planar model was not wrong about a jersey, it was a projection of the right curve. What it could not have at all is the quantity that is not on this list — the force through the fabric's thickness, 7.81 mN a stitch, which a model with no thickness has nowhere to put.

The constants say nothing about thickness

Munden's two constants give a knitted fabric's wale and course spacings from its loop length alone, and the tightness factor collapses every fabric's shape onto one curve. Neither reaches the third dimension: two knits that are one knit in plan are two different thicknesses.

finishing · Knit geometry 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.

mechanics · Elastica What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

A seam must give what the knit gives

A knitted seam fails because it is too short, not because it is too weak: the thread in it is nearly two thousand times stronger than the load it carries. What decides whether it survives is one line of geometry — the extension a seam can reach is twice the fabric's thickness times the stitches per unit length.

applied · Seams A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.

How dense a knitted fabric is

A fabric's areal weight is what the trade specifies and it says nothing about bulk. Divide it by a thickness and the answer is a density — 0.40 grams a cubic centimetre for a jersey, a quarter of the fibre it is made of — and that quarter, the share of the volume that is not air, is the number every other property follows.

setting · Weight A knit's warmth is its thickness, and a rib's thickness is a machine setting. Thermal resistance in tog for a 20 tex cotton at a 3.5 mm loop, for a plain jersey and for a one-by-one rib at four bed gaps. The bar is the lower bound and the mark beyond it is the upper — Wiener's two bounds on a mixture of fibre and air at the fabric's own fibre fraction, which is 26.6% for the jersey and falls to 8.9% for the widest rib. The jersey holds 0.088 tog and the rib at five diameters 0.334 — a factor of 3.8, bought entirely by opening the beds. Every one of these is a small fraction of the still-air layer that clings to a garment's outside, which is worth about 1.2 tog on its own.

A knit is warm because of where its yarn is not

Warmth is a thickness of still air, and until a knitted fabric had a thickness there was nothing to compute. It has one now, and the answer is that a rib's warmth is a machine setting: opening the beds from two diameters to five nearly trebles the fabric's resistance without changing a gram of yarn.

applied · Insulation What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

The modulus a knit has instead of one

A fabric's stiffness in extension is usually its yarn's modulus with the geometry taken out. A knit's is not: dimensionally it can only be a bending rigidity over a length cubed, and the same yarn laid straight and parallel is thirty thousand times stiffer than the fabric made of it.

mechanics · Knit What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

applied · Pressure How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

mechanics · Knit The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

A wet knit's yarn is flatter

A knitted fabric's own geometry demands a flattened yarn, and how flat depends on how much room the fabric leaves. A wet cotton yarn is a tenth thicker than a dry one at the same length, so a wet fabric leaves less room — and asks its yarn to be flatter by an amount the geometry gives.

finishing · State The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

A flattening that follows the tightness factor

Eighteen solved fabrics — five loop lengths, three relaxation states, three counts — and the flattening each one's geometry demands falls on a single curve against one dimensionless group. Nothing about the fibre or the count survives except through that group, which is what turns an arithmetical result into a structural requirement.

knits · Contact Two courses at the yarn's own width, and the place they overlap. The solved course of a 24.2 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.184 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.140 mm — 0.761 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 76% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

What wetting does to the bending limit

A rod cannot be bent to a radius below its own. A relaxed knitted loop sits at twice that limit, and a wet yarn is a tenth thicker in the same loop — so wetting moves a fabric a tenth of the way towards a bend it cannot physically take.

finishing · State 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.

mechanics · Elastica Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

A loop bends at twice its own radius

A rod of radius r cannot be bent to a centre-line radius below r without occupying its own space. A knitted loop's tightest bend is at 2.04 yarn radii — twice the hard limit, and falling as the fabric tightens. That is a ceiling on how tight a knit can be, from contact alone.

knits · Contact 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%.

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.

mechanics · Elastica The extension ceiling, with the yarn given a thickness. As a jersey is pulled along its courses the wale spacing grows and the course spacing has to fall, because the yarn between two interlacings is a fixed length. The upper curve is how small the course spacing may be before the yarn runs out; the lower is how small it may be before two adjacent courses occupy the same space. The geometric ceiling is 322% and the contact one 299% — 7% lower. That is the result and it is a negative one: a measured jersey extends by about a hundred per cent, so contact between courses is not what puts the computed ceiling three times beyond a real one. The candidate this ladder was written to test is ruled out.

Contact is not why a jersey stops

The model says a jersey can be pulled to three hundred and twenty per cent along its courses. Real ones stop at about a hundred. The recorded diagnosis was that nothing stops adjacent courses passing through one another — and giving the yarn a thickness closes seven per cent of a gap of two thirds.

knits · Elastica The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

The count that decides how flat

A knitted fabric's demanded flattening is a function of its tightness factor, and a tightness factor is the square root of a count over a loop length. So a coarser yarn at the same loop is flatter — which is a prediction about a spinner's choice that nobody has framed as one.

setting · Count The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

mechanics · Contact

Named alongside it

The objects these essays reach for when they reach for this one.

Tightness factorSpecificationElasticaStitch densityJammingWaleLoopMunden constantsYarn diameterBending rigidityCourseRelaxation

All concepts