Crimp — where it appears
Named by 55 essays across 9 fields — each of them below, with the objects they name alongside it.
The crimp is the price of being cloth
A woven reinforcement is bought for stiffness along its fibres, and the usual explanation of what it gives up — fibre content, lost to the crimp — has the sign wrong. At a given thickness a woven fabric holds slightly more fibre than two flat plies of the same tows. What the crimp costs is stiffness, and the float length is the knob.
What comes off the loom is not the cloth
Every number on this site so far describes a fabric in one particular condition — held under tension, stretched to the reed width, never wetted — and none of them says so. That condition lasts until the cloth is taken off the machine, which is the first thing that happens to it.
Crimp, and why cloth narrows when it is pulled
A thread in cloth is longer than the cloth it crosses. Pull the fabric one way and that extra length is taken out of one direction and put into the other, so the cloth gets narrower without a single fibre stretching.
Relaxation is the crimp coming back
A cloth that shrinks in the wash has not lost any thread. The thread is exactly as long as it was; more of its length is now spent going up and down rather than along, and the difference is the shrinkage, computable to the last figure from the woven geometry alone.
Why the warp shrinks more
A woven cloth almost always loses more length than width, and the reason is not in the cloth. It is in the machine — the warp is held under tension for the whole of weaving and the weft is held for a fraction of a second — so the two systems arrive at the finishing works with different amounts of crimp missing.
A cloth cannot shrink past its own crimp
However hard a loom held the warp, there is a limit to what relaxation can take back, and it is not a fitted constant or a measured one. It is the crimp itself, read as c over one plus c, and it is the only result in this field with nothing empirical in it at all.
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.
Pre-shrinking is a subtraction done in advance
A compressive-shrinkage machine takes four per cent out of a cloth before anybody buys it, and the label then quotes what is left. The two numbers are fractions of different lengths, so they cannot be subtracted — and the difference between doing that correctly and incorrectly is most of the number.
The cloth gains weight by losing size
Mass per unit area is the number a fabric is bought by, and it rises by eleven per cent when a cloth relaxes — with nothing added, nothing removed and no thread changed. The quantity is a ratio, and finishing moves its denominator.
A cloth extends by moving its crimp
Everybody says the extension available along the warp is the warp's own crimp. On six of eight ordinary cloths it is not — a close sheeting has 14.61 per cent of warp crimp and reaches 4.03 per cent, because the limit lives in the weft.
A membrane is cut smaller than it is
A tensioned fabric roof is cut to a pattern smaller than the shape it will take, because stressing it makes it grow. At a fixed cloth thickness a prestress can only interchange crimp — one direction grows and the other shrinks — so everything that makes both directions grow is the crossings flattening, which is the one quantity here this site cannot compute.
What crimp interchange actually conserves
Pull a cloth lengthways and it narrows, because the crimp moves from one system to the other. Inextensibility says that much and no more — it is one equation short of an answer, and the second equation everybody uses is an assumption with a name.
Floats and abrasion
A satin is said to wear badly. It does not wear quickly — its flat face spreads the rubbing over more thread than a plain weave's crowns do. What it does is fail badly, and those are different quantities moving in opposite directions.
Terry needs two beams
A towel's pile is not woven longer. It is fed longer — from a second warp beam let off several times faster than the first — so the loop height follows from a ratio and a pick spacing, and the float length that decides a corduroy does not appear in it anywhere.
Every crossing is a force
A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.
Pulled both ways, only one can give
A cloth at constant thread length has one degree of freedom, so its reachable states are a curve rather than a region. Equal extension in both directions meets that curve at exactly one point — the state the cloth is already in — so the amount available is nought.
A cloth's Poisson ratio is not a material's
The ratio of a cloth's contraction to its extension has a name everywhere else in mechanics, and it breaks every rule the name comes with — above one half on all eight cloths measured, doubling across a four per cent span, and not reciprocal between the two directions.
What a fabric weighs
Every fabric is sold by its weight in grams per square metre, and the number is a sum of four products in which no term appears alone. One equation, four unknowns: a hundred and fifty grams describes an open coarse cloth and a close fine one, and the cover factors differ by a factor of nearly three.
Two coefficients, not one
Every friction on this site is a single number, and the account of why a cloth relaxes better when it is agitated depends entirely on there being two. Separating static from kinetic changes what a resting state is: a cloth at rest is held by one coefficient and a cloth being shaken by the other, and the band it can be left in narrows — by less than the ratio, because a cloth's restoring force is not linear in its extension.
A honeycomb gets its cells in the wash
The obvious mechanism is take-up on the loom, and the arithmetic says it is wrong: every end of a diamond passes through the long floats and the tight ones alike. What is left is finishing, and a cell is a region that wanted to shrink less than the cloth around it.
The reed is not the sett
A reed holds the warp at a pitch, and the cloth that leaves it is narrower — by exactly the weft's crimp, with nothing fitted and nothing approximated. On a close balanced sheeting that is 12.75 per cent, on an open scrim 1.87, and a weaver who allowed one figure for both would be wrong by a factor of nearly seven.
What water does to a thread
A cotton fibre in water is a fifth wider and a hundredth longer. If it grew equally in both directions a wet cloth would simply be a bigger cloth and nothing structural would follow; because it does not, every ratio of a diameter to a spacing in a cloth moves, and they all move the same way.
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.
A cloth gives back less than it took
Everything this collection computes about a deforming fabric is reversible, and no fabric is. The repair is not a new material property: a woven cloth has two routes to a strain, one of them costs its threads nothing and comes back in full, and where the first route runs out is a number about the sett with no fibre in it at all.
The crimp ratio is not a measurement
Peirce's geometry is two thread systems, four unknowns and three equations. It cannot say how the crimp divides between warp and weft, so every cloth solved so far has been drawn at a ratio somebody chose. Give the threads a stiffness and the missing equation arrives — and for six of the eight cloths in the table it arrives with no material constant in it at all.
The blow that sets the pick
The take-up gear decides how far the cloth moves between picks and says nothing about the blow that puts each pick where it goes. That blow is a force, and a virtual-work argument gives it in one line — with the length of the beat-up zone cancelling out of the answer exactly, which is the part worth having.
The locus gets a force
This site has drawn the set of states a cloth can reach without stretching any yarn, and has never been able to say which of them it is in or what it would cost to move. Both questions are one derivative of a bending energy — and the answer explains the flat start every fabric's load–extension curve has, which is not slack yarn but a symmetry.
The most a cloth can give back
The extension a cloth can find without stretching a thread comes from crimp, so setting a cloth closer ought to give it more. It does, up to a point, and then takes it away again — and the point is a cover factor of 0.4078 at a budget of 7.2857 per cent, identical to six figures for every yarn count from five tex to a hundred.
Wicking is slower along a crimped thread
A front travelling up a warp end travels the thread's path, which is longer than the cloth by exactly the crimp. So the wicking coefficient measured on the fabric is the yarn's own divided by (1 + c)² — 14.3 per cent lost at eight per cent crimp, whatever the liquid.
A cloth relaxes until its threads stop pushing
The finishing field treats the relaxed state as a place a cloth arrives at. With an energy along its own locus and a friction at its crossings it is not a place but a band — and which point of the band a piece of cloth stops at depends on which side it came from, which is why washing it twice gives two answers.
What holds a tuft in, in newtons
The pile ladder computed a tuft's anchorage as a capstan ratio and said, correctly, that a ratio was all it could offer. A ratio multiplies a tension and there was no tension anywhere on this site. There is one now — and the answer, put beside what a carpet is actually specified at, falls short by a factor of three.
Why agitation helps a cloth relax
Every standard relaxation procedure agitates: tumble it, wash it, steam it, work it. The explanation given is that agitation lets the fabric find its own dimensions, which is true and is not a mechanism. The mechanism is that a sliding contact resists less than a stuck one — and putting a number on it shows the effect is real, is smaller than the obvious arithmetic suggests, and does not account for what a relaxation procedure achieves.
The half of the beat-up that is all zone
The elastic half of the beat-up force is exact and the length of the beat-up zone cancels out of it, which is this site's own result and disagrees with every practical account of weaving. The frictional half is nothing but zone — and requiring the total to match the force a loom is actually built to apply puts the number of picks still sliding at about one.
The swelling a cloth cannot take
Three of the eight cloths in this collection's table have no wet state at all. A thread of fixed length cannot wrap a partner that has grown by a fifth, so the closure condition fails and the geometry has nothing to offer. What happens instead costs megapascals, and the alternative route is not merely dearer but unavailable — the thread would break first.
Two shrinkages, one tape measure
A cloth that comes out of a wash smaller has done two different things and the tape cannot tell them apart. One is geometric, recoverable and finished in minutes; the other is frictional, permanent and needs agitation. This collection can now compute both, and they have different signatures.
A wet cloth is set closer than it was woven
Cover is a diameter over a spacing. Wetting moves the diameter and does not move the spacing, so every cover factor on this site is multiplied by exactly the swelling ratio and every jamming sett divided by it — which is an identity, and the one statement in this ladder a reader can check by hand.
A cloth has one budget for two directions
A woven cloth looks as though it carries two independent reserves of free extension, one along the warp and one across the weft. It carries one. There is a single curve of states and a single position on it, so every hundredth spent one way is refunded the other — which means a pre-tensioned cloth has not used its recovery up, it has moved it.
A seersucker is made at the loom
Every other relief weave in this collection gets its shape after the loom, from a difference of crimp between two regions of a few per cent. A seersucker's surplus is thirty per cent and is put in as the cloth is woven — an order of magnitude more, which the square root turns into a factor of four in depth and no more than that.
Wetting moves a cloth to another locus
A cloth's constant-thread-length locus is built at a fixed thickness. Swelling changes the thickness, so a wetted cloth is not somewhere else on its own locus — it is on a different one, and the distance between the two least-energy states is the shrinkage. Five of the eight cloths here have a wet state, and one of them gets bigger.
How sharply a weave lets a cloth fold
A fold's length difference is paid for out of crimp, and crimp is not spread evenly along a thread — it is made at the interlacings and nowhere else. So what a fold has to spend is not the weave's average crimp but whatever is inside the few picks the fold crosses, and in an eight-end satin half the warp ends have nothing there at all.
A crepe is a yarn that will not lie still
A crepe cloth's pebbled surface is the yarn's own torque acting on a cloth that cannot resist it. The instability that makes a slack yarn snarl says how large the pebbles should be, and the answer is a millimetre — which is what a crepe looks like.
Two layers need two beams
A layer weaving a metre of cloth eats one plus its crimp metres of warp, and a beam delivers one rate. A plain face over a five-end satin back differs by twelve percentage points of crimp, which is twelve metres of warp over a hundred-metre piece and one pick spacing of slack after four millimetres of weaving. The difference has nowhere to go and does not settle — so the only double cloth that can share a beam is two layers of the same weave at the same sett.
Twist decides where in the bracket
A yarn's bending rigidity can only be bracketed, and the bracket is three hundred wide. What decides where a yarn sits in it is whether its fibres can slide — and for a thread bent by its own crimp in a cloth, the answer is not close.
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.
A fancy yarn has its crimp in the wrong thread
A bouclé is made by feeding an effect thread eighty per cent faster than the core it wraps, so the yarn contains more thread than it is long — which is crimp by this collection's own definition. A cloth's crimp is in the thread that carries the load, so removing it is the cloth's first extension. A bouclé's surplus is in a thread that carries nothing, so pulling the yarn stretches the core at once and the loops never straighten. Thirty-nine per cent of the yarn's mass is on the load path, and the other sixty-one is decoration.
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 woven cloth is not linked at all
Every thread in every woven cloth passes over its neighbours and comes back. None of them passes through. So the linking number of any two threads in any weave is zero, at any crimp, permanently — and almost everything a cloth does that a knitted fabric does not follows from that one number being nought.
A thread between two crossings is an elastica
Every crossing force Peirce's thread path can give comes from a tension, and a cloth on a table has none. What presses its threads together there is their own unwillingness to be bent — and recovering that needs a shape nobody had, because a path assembled from an arc and a straight line has a bending moment that jumps.
A woven thread has no room to bend
Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.
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.
A cord's height has a ceiling and its width has none
A warp rib's cord is a wave, and its two dimensions come from two different places. The height is the weft's own crimp amplitude, and Peirce's closure condition caps it at the two yarn diameters together — 500 µm for a quarter-millimetre yarn, of which a 2/2 rib reaches 304 and a 6/6 rib 391. The pitch is the doubling times the pick spacing and has no cap at all. So a bolder rib is taller and wider, and wider faster: the aspect falls from 0.41 to 0.22.
A damask is the only figure that costs its beam nothing
Figure and ground consume warp at different rates, and the difference accumulates down the length of the figure. An eight-end satin figure on a plain ground puts its two regions fourteen per cent apart, which on a loom absorbing a millimetre of slack bounds the figure at eight and a half millimetres. A damask's two regions are the same satin used two ways, so complementation leaves both float lengths where they were — and their crimps are identical rather than close, to the last bit of a double.
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.
A figured warp needs a beam for every share of its figure
Figure and ground take up warp at different rates, so a figured cloth on one beam is bounded in how long its figure may run, and a second beam is the obvious escape. It escapes only for ends that live wholly in one region. An end that crosses the figure for part of the repeat consumes warp at its own rate, and two ends can share a beam only if they spend the same share of the repeat in the figure and never drift a slack apart inside it. So a round figure twelve blocks across needs four beams, ninety-six blocks across needs twenty-nine, and any crimp difference at all — a third of a per cent will do — costs every one of them over a piece.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Peirce's geometrySettJammingCrimp interchangeRelaxationShrinkageSpecificationFrictionContact forceCover factorLoom stateBending rigidity