Textile Structure
how cloth is put together
Generator

The flattening a knitted fabric asks for, against its own tightness

The flattening a knitted fabric asks for, against its own tightness
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.

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.

24 essays call touch-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

Two courses as centre lines, and their closest approach. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, as centre lines, with the closest approach marked. 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. What cloth is

The closest approach is not the crossing

Two wavy curves that touch at a point are not necessarily closest at that point. Whether they are depends on one thing: whether they run alongside one another or cross. That distinction decides which of this collection's two fabrics fits together and which does not.

 A yarn is pressed on part of its length and free on the rest. The distance from each point of one course to the nearest point of the course below, for a 20 tex cotton jersey at a 3.5 mm loop, along two wales. It runs from 0.780 diameters at the worst to 3.81 at the freest, and 20% of the length is inside one diameter of its neighbour. The line at one diameter is where a round yarn would begin to overlap. What the profile says is that a single flattening ratio is an average: the section a yarn takes changes along its own length, which every racetrack section this collection has drawn assumes it does not. What cloth is

Where a yarn is thinnest

A yarn in a fabric is pressed where it crosses and free where it does not, so its section changes along its own length. Every flattening this collection has ever quoted is a single number for a profile that runs from four fifths of a diameter to nearly four.

 A yarn is pressed on part of its length and free on the rest. The distance from each point of one course to the nearest point of the course below, for a 20 tex cotton jersey at a 3.5 mm loop, along two wales. It runs from 0.780 diameters at the worst to 3.81 at the freest, and 20% of the length is inside one diameter of its neighbour. The line at one diameter is where a round yarn would begin to overlap. What the profile says is that a single flattening ratio is an average: the section a yarn takes changes along its own length, which every racetrack section this collection has drawn assumes it does not. What cloth is

A fabric is a population of contacts

Only a fifth of a knitted fabric's yarn is inside a diameter of its neighbour. So a fabric's friction lives in a fifth of its length, and every calculation this collection makes about withdrawal, slippage and fraying has assumed it lives everywhere.

 A woven cloth asked the same question, and the answer is nearly one. The closest approach two crossing threads make, for four cloths from an open voile to a dense duck, in units of the separation they have where they touch. A value of one means the closest approach is exactly at the crossing and the cloth fits together; anything below one is an overlap. The values run from 1.000 to 0.955, so the worst overlap in the table is 4.5% of a contact separation — against 22% for a knitted fabric. The overlap rises with the crimp, which is what identifies the mechanism: the vertical gain from moving away from a crossing is the crimp, and a cloth that barely crimps has nothing to gain by moving. Weaves

A woven cloth asked the same question

A knitted fabric's two adjacent courses occupy the same space by a fifth of a diameter. A woven cloth's two systems overlap by nothing at all in an open cloth and by four and a half per cent in a dense one — and the difference is that they cross rather than run alongside.

 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. Knits and other structures

The fabric that does not fit

Every solve in this collection minimises an energy over a centre line, and a centre line has no thickness. Nobody had checked whether the fabric that comes out of it can be built. It cannot: two adjacent courses of the relaxed jersey approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere.

 The section the fabric asks for, beside the one the model drew. A 20 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.167 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 78% of it, which is the closest the fabric's own adjacent courses come to one another — 0.130 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett. Knits and other structures

The flattening nobody fitted

A yarn in cloth is not round, everybody knows it, and nothing has ever predicted how flat. This collection's own knitted geometry turns out to require a flattening of four fifths — from a solve that knew nothing about flattening, made no allowance for it, and would have been written the same way if the idea had never occurred to anybody.

 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. After the loom

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.

 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. Knits and other structures

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.

 A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot. Cloth doing a job

A knot is nothing but contact

A knot has no fastening in it. Nothing is glued, hooked, sewn or threaded through a hole: a thread is bent round itself until the friction where it presses on itself is more than the load. That makes a knot the purest contact problem in the subject, and the place to look first for what contact does.

 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. After the loom

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.

 A knot, and the tension falling through it. A 20 tex cotton thread wrapped through 1 turn at a bend radius of 1 yarn diameters, with a coefficient of friction of 0.30. The wrap is drawn as a spiral because a thread taken round a pin comes back beside itself rather than onto itself. The marks round it are the fraction of the entry tension still there: 100%, 69%, 47%, 32%, 22%, 15%. The bend has already spent 3.6% of strain at the outside of the thread before any of that happens, against a breaking strain of 6.6%. So the largest total is at the entry, before the knot has done any gripping at all — which is where a knot in a real yarn is observed to break, and why a knot's efficiency is a property of its first bend rather than of the knot. Cloth doing a job

Where a knot breaks

It breaks at the entry, before the knot has done any gripping at all. Two quantities run along a knot's path and only one of them rises; the other falls from the first millimetre; and their sum is largest where the thread arrives.

 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. Knits and other structures

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.

 The section the fabric asks for, beside the one the model drew. A 24.2 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.184 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 76% of it, which is the closest the fabric's own adjacent courses come to one another — 0.140 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett. After the loom

The diameter that does need a state

A fabric dimension quoted without its relaxation state is not a measurement. The rule was applied to the two plan dimensions and then to the thickness, and it was never applied to the one dimension that is not the fabric's at all — the yarn's.

 What a knot costs, and what decides it. Knot efficiency against the radius the thread is bent to, for 5 fibres at 20 tex, computed at the free end of the stiffness bracket — each fibre bending about its own axis, so the strain at the outside of the bend is half a fibre diameter over the radius. At one yarn diameter of radius, cotton keeps 46%, which is the rule of thumb that a knot halves a rope's strength. What separates the curves is not the knot and not the friction: it is the fibre's own breaking strain, which is how much the bend is allowed to spend before there is nothing left. Cloth doing a job

A knot halves a yarn and says why

The rule is quoted for every rope and every knot and derived nowhere. It comes out at forty-six per cent for a cotton — but only if the yarn's fibres bend individually. A yarn bending as a solid section has already spent five times its breaking strain before any load arrives, so it could not be knotted at all.

 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. Knits and other structures

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.

 What a knot costs, and what decides it. Knot efficiency against the radius the thread is bent to, for 5 fibres at 20 tex, computed at the free end of the stiffness bracket — each fibre bending about its own axis, so the strain at the outside of the bend is half a fibre diameter over the radius. At one yarn diameter of radius, cotton keeps 46%, which is the rule of thumb that a knot halves a rope's strength. What separates the curves is not the knot and not the friction: it is the fibre's own breaking strain, which is how much the bend is allowed to spend before there is nothing left. Cloth doing a job

Which yarns knot well

A knot's efficiency depends on three things and only one of them is the knot. The other two are the fibre's fineness and its breaking strain, and across this collection's own table those give efficiencies from six per cent to eighty-three at exactly the same bend.

 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. Knits and other structures

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

 The section the fabric asks for, beside the one the model drew. A 20 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.167 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 78% of it, which is the closest the fabric's own adjacent courses come to one another — 0.130 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett. Setting and geometry

What a sett is when the yarn is not round

A jamming condition says how close threads can be set, and it says it in terms of a diameter. A thread in a cloth does not have one diameter: it has a wide one and a narrow one, and which of them a jam is about depends on which way the threads are jamming.

 The section the fabric asks for, beside the one the model drew. A 20 tex cotton yarn in cross-section, at 900 times life size. The circle is what every solve in this collection has assumed: a diameter of 0.167 mm from the count and the packing factor. The ellipse has the same area and a minor axis of 78% of it, which is the closest the fabric's own adjacent courses come to one another — 0.130 mm. Nothing about that number was fitted to a fabric. It falls out of a solve that knew nothing about flattening, and it lands inside the range the trade reports for yarn in cloth, which this collection has carried as a free parameter ever since it first put a number on a sett. Setting and geometry

What a flattened yarn does to its cover

A cover factor is a sett times a diameter, and it decides how much of a cloth is thread and how much is hole. A flattened yarn is a third wider than a round one of the same area, so a cloth of flattened yarn covers more at the same sett — and every opacity, permeability and shade computed from a round diameter is wrong in one direction.

 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. Setting and geometry

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.

 What flattening costs, against what the loop's bending is worth. The energy of squashing a 20 tex cotton to the 78% the fabric's geometry demands, as a multiple of the whole bending energy of one stitch, at three lateral rigidities. The free bound is exactly zero: a bundle of fibres free to slide resists a change of shape at constant area not at all, so flattening is free and the bracket on it has no floor. The coherent bound is 36 times the loop's bending, so a yarn that could not rearrange could not be knitted into this fabric at all. The fabric flattens, so it is reading the free end — which is the fourth ordinary observation on this site to land at that end of the bracket. Mechanics and drape

Flattening is free and impossible

The fabric demands a flattening and the yarn has to supply it. At one end of this collection's oldest bracket the deformation costs exactly nothing; at the other it costs thirty-six times the whole bending energy of a stitch. The fabric flattens — which is the fourth everyday observation in one phase to land at the same end.

 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. Mechanics and drape

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.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

A fabric reads its own bracket four ways

A yarn's stiffness is unknown to a factor of three hundred, and no laboratory measurement has closed it. Four unrelated everyday observations — a snarl, a knot, a flattened yarn and a cloth's own thickness — all say the same thing about which end of it a yarn sits at.

 Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one. Mechanics and drape

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

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