Textile Structure
how cloth is put together
Concept

Cloth thickness — where it appears

How far a fabric stands off a flat surface, set in Peirce's geometry by the two crimp heights adding to the sum of the thread diameters. It is the closure condition of Peirce's geometry written as a dimension, and it is the constraint that fails when a cloth's threads swell.

Named by 49 essays across 9 fields — each of them below, with the objects they name alongside it.

Two sections, one yarn. The same yarn given a circular cross-section and a racetrack one of equal area, both jammed. The spacing the two models allow is nearly the same; the cover and the cloth thickness they predict are not.

A tow is not a yarn

Every geometric model on this site assumes a thread with a diameter. A reinforcement tow has a width and a thickness instead, and the two are eight to one — which moves the jammed sett, the crimp, the cover and the cloth's thickness at once, in four different directions, from one change.

applied · Sett Two sections, one yarn. The same yarn given a circular cross-section and a racetrack one of equal area, both jammed. The spacing the two models allow is nearly the same; the cover and the cloth thickness they predict are not.

Peirce against the racetrack, measured

Two models of a yarn's cross-section, given the same yarn and the same closure condition, agree on how densely the cloth can be set and disagree by nearly half on how thick it is. Which quantity is being asked about decides whether the choice of model matters at all.

setting · Sett The step between a figure and its ground. Three figures on a plain ground, all in one sheeting's threads at its own setts. A thread presses on the thread it crosses only where it turns, and it turns at its interlacings — so the pressing a region receives per unit area is the contact force at one turn times the turns per unit area, and the second factor is a property of the matrix exactly. A five-end satin turns two fifths as often as a plain weave, is pressed two fifths as hard, flattens less, and stands 55 µm proud of it. What the rows cannot show is that both regions are given a plain weave's weave angle: a satin's crimp is genuinely smaller and its turns genuinely gentler, so the real step is larger than this, by an amount not computed here.

A figured cloth has a step in its surface

A damask is one cloth in one set of threads at one sett, and it is not flat. A thread presses on the thread it crosses only where it turns, so a region that turns less often is pressed less often, flattens less and stands thicker — and the step is a ratio of interlacing rates, read off the matrix with no yarn property in it.

pattern · Blocks A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.40 N per crossing the sections flatten to aspect ratios of 1.75 and 1.84, the cloth thins to 0.263 mm, and the warp runs flat for 0.109 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

A flattened thread is a record of a force

A yarn in cloth is not round, and this site has modelled the flattening for as long as this collection has run with the amount of it left as a number somebody chose. Give the section a stiffness and ask what the cloth prefers, and the answer is a circle — at every stiffness, for every balanced cloth in the table. Flattening does not happen by itself; it happens because something pressed.

mechanics · Compression A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.42 N per crossing the sections flatten to aspect ratios of 1.79 and 1.88, the cloth thins to 0.260 mm, and the warp runs flat for 0.113 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

Peirce and Kemp are one cloth at two moments

This site has run two thread sections side by side since the setting field was built — a circle and a flattened racetrack — and said honestly that they disagree and that the disagreement is the point. They are not rival descriptions of the same fabric. They are descriptions of the same fabric before and after something pressed it, and the difference between them is a pressure that can now be named.

setting · Sett How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

fancy · Integrity The bracket that closes and the bracket that does not. A 25 tex cotton yarn at a packing factor of 0.6. Its bending rigidity lies between 0.00117 N·mm² — the sum of its fibres', with them free to slide — and 0.478, a solid rod of its own diameter: a factor of 408, and both ends are derivations. Its resistance to being squashed out of round has an upper bound of the same kind, 369 N/mm² for a solid section, and no lower bound at all, because fibres free to slide resist a change of shape with nothing. That is why the aspect ratio of a flattened yarn has been a free parameter here since the setting field was built: a quantity bounded below by zero cannot be estimated from its bounds, and has to be measured.

The stiffness with no lower bound

A yarn's bending rigidity lies between two derivable ends and the ratio is the fibre count — wide, but closed. Its resistance to being squashed out of round has an upper bound of the same kind and a lower bound of exactly nothing, because fibres free to slide resist a change of shape with nothing at all. That is why nobody could ever compute the aspect ratio of a flattened thread.

mechanics · Compression From a calender's line load to a force at one crossing. A sheeting through a nip loaded at 30 N per millimetre of bowl width, with the cloth in contact over 5.0 mm. The pressure is the first divided by the second, 6.00 N/mm², and the force at one crossing is that pressure times the area a crossing owns — the product of the two thread spacings, 0.1374 mm². So the crossing carries 0.824 N, the sections flatten to 2.42 and 2.63, and the cloth thins from 0.388 mm to 0.218 mm. What the drawing cannot show is that two cloths through the same nip are not given the same treatment: the area a crossing owns varies fivefold across this site's table, and it is a factor in the force.

A calender spends the compression for good

Calendering was described on this site as moving a cloth from one thread-section model to another, which was right and had no number in it because the amount of the move was a free parameter. It is a pressure now — and the same nip setting turns out to give two cloths quite different treatments, because the force at a crossing is the pressure times the area a crossing owns.

finishing · Lustre finish How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

mechanics · Compression The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 221.0 µm from the highest point to the lowest, the root-mean-square roughness is 74.9 µm, and 30% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 190.8 µm and the weft at 182.8 µm, a step of 8.0 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far.

A cloth has an outside

Every quantity in this collection is a property of the inside of a fabric — a crimp, a cover, a hole, a fibre count. None of them says where the cloth stops. The outside is a height field the draft computes, its crowns stand at two different levels, and which of the two is higher decides what the cloth touches the world with.

cloth · Surface The bearing curves of 4 weaves in one cloth. How much of the plan is within a given depth of the highest point, for plain, 2/2 twill, satin 5, satin 8 — all in sheeting, all at the same sett, the same counts and the same thickness. They differ only in their drafts. At a hundredth of the cloth's thickness the last of them is touching 9 times the area of the first, and the gap widens as the depth shrinks, because the curves do not merely differ by a factor — they have different exponents. A crown that is a line opens as the square root of the depth and a crown that is a point opens in proportion to it.

The curve that says what a cloth touches with

Take a cloth's surface and ask what fraction of the plan lies within a given depth of its highest point. The answer is one curve, it answers every question of the form what does this touch, and its behaviour at the top is decided by a single bit of the draft — whether the longest float is one crossing or more than one.

cloth · Surface The tightest fold a 20 tex cotton yarn can be given. A cloth folded as sharply as it can be folded. The two yarn crowns on the inside of the fold cannot pass through one another, so the fold's radius is the yarn's own — 0.084 mm for a 20 tex cotton yarn — and there is no measurement of an iron anywhere in the argument. At that radius a fibre free to slide is strained 7.14%, which is √(packing × fibre tex ÷ yarn tex), and the whole yarn bending as a rod would be strained exactly one hundred per cent. Cotton's measured breaking extension is 6.0% to 10.0%, so the free bound does not survive and the locked one cannot. What the drawing cannot show is the fibres inside the yarn, which is exactly what the argument is about — the picture is the same either way and the strain is fourteen times different.

A crease is a fold the crimp cannot supply

A fold needs its outer face longer than its inner, and a woven cloth's way of supplying a length is to move crimp. That runs out at a radius of millimetres, and a pressed crease is tenths of one — so the fold is handed to the fibres. How hard it strains them turns on a question this collection has been unable to settle for two fields, and a crease settles it by refusing.

finishing · Crease What it costs to touch a cloth. The pressure needed to bring a stated fraction of the plan into contact, for plain, 2/2 twill, satin 8 in sheeting. Reaching two per cent of the plan takes 2.83 kPa on a plain and 0.05 on a satin 8, a factor of 55. The stiffness in this figure is fitted and is labelled as such. A yarn's resistance to being squashed out of round has no lower bound at all — a bundle of fibres free to slide is a fluid in cross-section — so no bracket exists to compute this from, and what is used is the value this collection fitted to measured fabric thickness. Every curve moves together across its published range, which is why the ratio between weaves survives and the absolute values are quoted with the fit named.

How much of a cloth is touching

Press a fabric against a flat plate with the weight of a hand and ask what fraction of it is actually in contact. The bearing curve answers, and the answer is about four per cent — of which the great majority is not the cloth's surface at all, but the hairs standing off it, which nothing in this arithmetic can see.

cloth · Contact How open a muslin is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this muslin it is 37.9 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 36.1° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 5.36 per cent open — 7.1 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none.

One minus the cover is a cloth with no thickness

The covering rule says a cloth's openness is one minus its cover factor, and this collection derived it and has used it ever since. It is the answer for a light directly behind the cloth. Move the light and a line of sight has to clear the hole at the top of the fabric and the same hole one thickness below, so the openness falls, and it reaches nothing at thirty-six degrees. Averaged over the whole sky a muslin is a seventh as open as the rule says.

setting · Cover Eight fibres through one cloth, and the bracket that hides them. A muslin is 19 per cent fibre and the rest air, so its thermal conductivity is a two-phase mixture, and how the fibre and the air are arranged is not something any amount of knowing the fractions settles. What is settled is Wiener's pair of bounds: heat across a series arrangement sees the harmonic mean and along a parallel one sees the volume average, and every real arrangement lies between. Each bar here is one fibre's whole band, from the series bound at its lowest published conductivity to the parallel bound at its highest. Not one of the eight clears any other, so this model cannot tell wool from nylon in a fabric — and the same cloth at twice the thickness has twice the resistance, exactly. Air's own conductivity is marked, and every band sits within a fifth of it.

Warmth is a thickness of air

A fabric is a tenth fibre and the rest air, so its thermal conductivity is a mixture of the two — and the pair of bounds that any mixture must lie between comes out narrower than the difference between wool and nylon. The model cannot tell one fibre from another in a cloth. What it can tell, exactly and with no bracket at all, is that twice the thickness is twice the warmth.

mechanics · Insulation The presser foot sinks 1.8 µm into a 2/2 twill. A thickness gauge presses a flat foot onto the cloth at 1 kPa and reads the gap. It does not read the geometric thickness. The foot sinks until the area it is touching can carry the load, and that is 2.57% of the plan at a depth of 1.8 µm — so a 2/2 twill in sheeting whose outside stands 381.6 µm apart measures 379.8 µm. How far the foot sinks is a property of the draft, because the bearing area near the top is, and a weave with plateaux stops the foot in a fraction of the distance a plain weave lets it travel. The transverse stiffness used here is fitted to measured fabric thickness rather than predicted: across its published range the reading moves between 377.0 µm and 380.6 µm.

A thickness gauge reads the draft

A presser foot does not stop at the top of a cloth. It sinks until the area it is touching can carry the load, and how far that is depends on the shape of the bearing curve near the top — which is a property of the weave. So there is a weave term inside a measurement nobody thinks of as a weave measurement, and it is worth about one per cent.

cloth · Contact An operation multiplies a spread by its own log-slope. A transformation does not leave a population's spread alone: if y goes as the kth power of x then a small spread in x becomes k times that spread in y, exactly in the limit and nearly so at the CVs a yarn has. So an operation with an exponent below one narrows the population it acts on — a thickness that goes as the square root of a load comes out at half the spread it went in with — and one with an exponent of four widens it fourfold. This is the same derivative that decided every bias in this ladder, read for its magnitude rather than for its curvature, and it is why a finish can be a variance-reducing operation without anyone having chosen it for that. The straight line through the origin is the whole of the rule; the departure from it at the right-hand end is the second-order term arriving, which is where the linearisation stops being one.

A finish spends a spread before it spends a mean

Every operation on a cloth multiplies the variation it inherits by its own log-slope, so an operation with an exponent below one makes the cloth more even and one above it makes the cloth less even. Calendering, which is bought for evenness, has an exponent of 1.4.

finishing · Lustre finish The crossings under a 25 mm² presser foot. A 25 mm² foot on a muslin covers 12 ends and 11 picks, which is 132 crossings — and a first guess treats those as 132 chances of finding a thick place. They are not independent chances. Every crossing along one end shares that end's diameter, so the largest crossing is the largest end plus the largest pick, and the number of tries is 23: the threads. Each cell here is shaded by its own two diameters, and the darkest is at the meeting of the darkest row and the darkest column, which is what that identity looks like. The gauge rests on it and reads 0.431 mm, against 0.342 for the cloth's mean crossing — 26% over.

A thickness is a maximum, not a mean

A presser foot rests on whatever is highest beneath it, so the thickness of a fabric is an extreme value — and an extreme grows with how much cloth is asked. The standard specifies the foot's area because the foot's area is in the answer.

mechanics · Compression The surface of an 8-end shading. The height of the cloth's own surface at each tone of an 8-end shading, for the two chains, on a sheeting at 0.50 N in the end. A thread presses on the thread it crosses only where it turns, so a region that turns more often is pressed more often and finishes thinner — and firmness rises towards the midtone of a shading. The spread chain therefore sinks 84 µm between its ends and its midtone and the consecutive chain 43 µm, with 2 of its steps at exactly one thickness against the spread chain's 0. The tone scale is level by construction and the surface under it is not. What the plot cannot show is the light: a step in the surface reads as a line under a raking beam whatever the tone is doing, which is why a relief nobody specified is visible at all.

A tone ramp is a valley, and the satin digs it

A shading's tone is exact and its lustre is measured; its thickness is neither, and nobody specifies it. A firmer weave is pressed harder at every crossing and finishes thinner, so an eight-end shading sinks eighty-four micrometres between its ends and its midtone — about a third of the cloth's whole thickness, on every cloth tried. Build the same chain on a twill instead of a satin and the sag is exactly nothing.

pattern · Shading At 0.1 kilopascals the plate is standing on hair. A flat foot pressed onto sheeting at 0.1 kPa, over 3 mm of cloth. It stops 25 µm above the cloth's own crowns, because that is where the hairs it is bending can carry the load: of the 10 hairs drawn, 9 reach higher than the foot and are laid over under it, and the rest are untouched. The vertical scale is set by the approach and not by the layer — 5188 pixels to the millimetre off the cloth against 192 along it — because the foot's height above the crowns is tens of micrometres and the layer it stands in is more than a millimetre, so a picture at one scale shows the second and not the first. The thickness reported is 0.432 mm against 0.382 mm for the cloth itself, so 12% of the reading is a population and not a fabric. The hair layer carries up to 0.23 kPa before the foot reaches the crowns at all. A laid-over hair is drawn as two straight segments where a real one is an elastica: the corner is a convenience and the height it turns at is the measurement. What the picture cannot show is that a bent hair leans on its neighbours, which the arithmetic behind it does not know either.

A light touch never reaches the crowns

This collection found that a plain weave touches at points and every other cloth touches along lines, and that the difference is an exponent rather than a factor. It is a real result about a real surface, and at the pressures a fabric is actually touched at, nothing ever reaches that surface.

cloth · Contact A seam stands 763 µm proud of a cloth 382 µm thick. A 10 mm seam allowance of 3 plies in a 60 mm panel of one, in sheeting. The seam stands 763 µm above the body of the garment — which is 102 times the depth at which the body cloth first comes into contact with anything at all. So a flat surface rubbed across this garment touches only the seam, over 16.7% of the area drawn, until it has crushed a whole thickness of fabric. Everything this collection computes about where wear lands on a woven surface applies inside that 16.7%, and the other 83.3% is not being touched.

A seam stands proud and wears first

A seam allowance is three plies where the garment is one, so it stands three quarters of a millimetre above a cloth whose own surface has a few micrometres of contact in it. Anything flat rubbed across the garment touches the seam and nothing else — all of the wear on two or three per cent of the area, until a whole thickness of fabric has been crushed.

applied · Seams A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

A rib climbs a gap

A jersey's yarn crosses one diameter between interlacings because that is what a crossing of two threads is. A rib's crosses the whole distance between the two beds. Nothing else in the model changes, and that one length is the whole mechanical difference between the fabrics.

knits · Two-bed 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.

What a thickness gauge reads on a knit

The structure says two yarn diameters and the gauge says more, and the gap is not an error in either. A gauge lands on the highest crowns, through a canopy of protruding fibre, under a load that has already begun to compress both — and it does that on a surface that is nothing but crowns.

cloth · Contact A calender multiplies the highlight by 32, and all of it is width. A 2/2 twill in sheeting pressed at increasing force, with the specular area recomputed at each state from the site's own compression model. It rises from 0.91% of the plan to 28.9%, a factor of 32, while the cloth thins from 381.6 µm to 186.1 µm. The gain is not in the length of the crowns: that moves by 5 per cent. It is in their width, which moves by 31.8 times, because pressing puts a flat top on the section and a flat top has one normal rather than a fan of them. The finish does not polish the thread. It changes the dimension of the highlight, from a line to a band, and the arithmetic says so by refusing to put any of the gain in the other factor.

A calender buys the width

Press a cloth and its lustre multiplies by twenty-four. None of that comes from the length of its crowns, which moves by six per cent; all of it comes from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refuses to put any of the gain in the other factor.

weaves · Shine 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.

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.

cloth · Contact Two exponents in a compression curve. How far a flat plate sinks into plain, 2/2 twill, satin 8 in sheeting, against the pressure it is applying, on logarithmic axes where a power law is a straight line. The measured slopes over the light end of the range are 0.50 for the plain, 0.67 for the 2/2 twill, 0.67 for the satin 8 — against two thirds predicted for any weave carrying a float and one half for a weave carrying none. The prediction is one line of algebra: pressure is a stiffness times a strain times a bearing fraction, the bearing fraction is a square root of depth for a plateau and linear in it for a point, so the pressure is the three-halves power in the first case and the square in the second. Nothing is fitted to produce it; the slopes are measured afterwards and compared. At the heavy end every curve bends, because the crowns have merged and the cloth has stopped being a surface and started being a solid — which is a different regime with a different law, and it belongs to the compaction of a fibre mass rather than to the geometry of an interlacement.

A cloth compresses along its own bearing curve

A fabric's pressure–thickness curve is always fitted with an empirical power law and the exponent is reported without explanation. It is not empirical. At light loads it is two thirds for any weave carrying a float and one half for a weave carrying none, and the two numbers come out of one line of algebra with nothing fitted in it.

mechanics · Compression 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.

A rib is quietest at two diameters

Open the beds of a rib and everything about it should get stronger. It does not. The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters and rises on both sides of it, because a crossing's climb and an interlacing's own climb cancel there.

knits · Two-bed A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

mechanics · Compression 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.

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.

cloth · Contact 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.

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.

cloth · Contact 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 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 state is a thickness too

This collection's rule is that a fabric dimension quoted without its relaxation state is not a measurement. A knitted fabric has three dimensions and only two of them obey the rule: its thickness is the same in every state, because the interlacing that sets it does not relax.

finishing · State 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 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.

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.

weaves · Contact 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 Two numbers that stay at zero however large the fabric gets. The linking number of two adjacent courses, and the writhe of one course per wale, for tubes of 6 to 20 wales. Both sit at zero and stay there: the largest departure anywhere on the plot is 1.5e+1, which is the sampling. A quantity that should grow with the fabric and does not is the cleanest kind of null result: the model has the geometry of knitting and none of its topology, and making the fabric bigger does not make the topology appear.

Five symptoms of one omission

Five things this collection recorded as unexplained, found in four different ladders over three years of work. They are the same defect seen from five directions, and the defect is one sentence written for good reasons with no visible cost at the time.

knits · Topology 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.

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.

knits · Contact 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.

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.

knits · Contact A voile hung at 2.5 times its window, seen in plan. A curtain of voile gathered to 2.5 times the width of its window, drawn in plan with the window above it and a line of sight crossing a flank. A length of cloth spans its own length times the cosine of its flank angle, so a fullness of 2.5 stands every flank at 66.4 degrees and a line of sight normal to the window meets the cloth at that incidence. This cloth's holes close completely at 47.3 degrees, which is a fullness of 1.48, so at 2.5 times every flank passes no line of sight at all and the whole of what comes through arrives at the crests. Flat the cloth is 49.3% open and hung it is 3.0%. What the plan cannot show is the cloth's own drape, which rounds every fold drawn here as a corner.

A curtain is gathered so that it is seen edge-on

A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

pattern · Opacity 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 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.

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.

finishing · State 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.

The force that holds a knit open

Resolving a knitted loop's contact force out of the fabric's plane leaves a fifth of it pointing through the thickness. That fifth is 7.8 millinewtons a stitch, fifteen kilopascals over the area a stitch occupies, and it is the whole reason a jersey has a thickness rather than a plan.

mechanics · Elastica A 4-layer stack round a 180° fold. 4 layers of 260 µm cloth taken round a 180 degree fold, drawn as concentric arcs at their own separation. The outer layer runs round a larger radius than the inner one, so it must be longer — by the angle times the separation, which is 2.45 mm here and 0.82 mm at every interface. In a loose stack that length is found by the layers sliding over one another at the fold. In a stitched seam they cannot: the stitches pin them every 3 mm, so the slip a fold needs is 82 per cent of the distance between two stitches and has nowhere to come from. What the arcs cannot show is what happens instead, which is that the fold opens out to a radius the stack can manage.

A crease cannot cross a seam

The outer layer of a folded stack has further to go than the inner, by the fold's angle times the stack's own thickness. A four-layer seam of quarter-millimetre cloth taken through a half turn needs its outer layer to be 2.45 millimetres longer than its inner — and a stitch line every three millimetres has pinned them. So the fold opens out where it crosses the seam, which is what a trouser crease visibly does, and the arithmetic gives the radius it opens to.

finishing · Crease The surface of every even six-end shading, against the twill's. The height of the cloth's surface at each tone of a six-end shading on a sheeting at 0.50 N, for every chain of the 64 even classes, drawn once per distinct shape — 5 of them — and for the cyclic square's twill read one step at a time. The twill is level to the micrometre and floats 5, 4, 3, 4, 5. Every even chain sinks to exactly 43.3 µm at its midtone; the shapes differ only at the second and fourth tones. What the plot cannot show is which of these a raking light would reveal, which depends on the finish.

An even shading cannot keep its surface level

Sixty-four six-end shadings hold every middle tone to a float of two, and the question left over was whether any of them keeps a tone ramp's surface level the way a twill read in order does. None does, and all of them sink by exactly the same depth — 43.3 micrometres on a sheeting, a sixth of the cloth. The reason is a counting argument a recording engineer would recognise: a limit on how long a thread may float is a limit on run length, and a run-length limit forces a floor on how often the thread changes face. A level ramp needs every tone at the extremes' rate, which needs a float of half the repeat at the midtone and more beside it — exactly the floats the twill in order has.

pattern · Shading How little eccentricity a curl needs. The radius a jersey would curl to, against how far the fabric's neutral surface fails to bisect its loops. The model's own loop is bisected exactly — its crest sits half a diameter behind the mid-surface and its trough half a diameter in front — so it carries no curling moment at all, and the curve here is what any departure from that would buy. The moment is proportional to the eccentricity, so one solve settles the whole curve. A radius of three millimetres, which is about what a fine jersey rolls to, needs 5.6% of a yarn diameter — 9.4 micrometres on a yarn 167 micrometres across. That is why curl is easy to see and hard to model: the whole of it lives inside a rounding error on the geometry. Only the course-wise edges are drawn, because bending about the other axis turns the thread's end tangents out of the fabric and a free thread given turned ends buckles clean out of it — 3.2 times the fabric's own thickness, for 37% off its energy. That configuration is not available to a thread with neighbours, so the second moment is refused rather than computed.

How little asymmetry a curl needs

A model with a thickness can finally be asked why stockinette rolls. It answers that it does not — its curling moment is exactly zero, by a symmetry — and the useful part is what that costs to break: five per cent of a yarn diameter buys the whole of the curl anybody has ever seen.

mechanics · Elastica 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 a knit gives up when it is pressed

The woven half of this collection has had a compression curve for several rungs — a thickness that falls under load, a bearing area that grows, a pressure at every point. The knitted half had a plan and no depth. It has a relaxed thickness and an initial slope now, and the two fabrics turn out to resist for different reasons.

mechanics · Compression What the count moves at 150 grams. Plain cotton cloths that all weigh 150 g/m², from 20 tex to 200 tex, with sett and crimp solved together. Thickness, which is two yarn diameters, rises 3.16 times across the line; the cover factor of each thread system falls 2.75 times; their product with one plus the crimp is the same number at every count, because the weight has fixed the volume of fibre and the count only decides whether it is laid out flat or stacked up.

A weight fixes the fibre and not the drape

Every plain cotton cloth of 150 grams a square metre contains the same fibre, and the count decides only how it is arranged. Across the counts that can make that weight, thickness rises threefold and cover falls in step, so their product holds still. The bending length does something stranger: at the bound a woven yarn actually sits near, it depends on neither the count nor the weight, only on the fibre.

setting · Weight

Named alongside it

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

Contact forceLoopSpecificationCompression energyYarn diameterElasticaContact pressureJammingTightness factorBearing curveInterlacingPacking factor

All concepts