What holds a thread in a seam

Pull hard on a seam in a satin blouse and something gives, but it is usually not the thread the seam was sewn with. The stitches hold, the sewing thread holds, and the fabric’s own threads slide out of the weave beside the stitching — leaving a row of holes and a gap where the seam used to lie flat.

The trade calls that seam slippage, tests for it, specifies allowances against it, and knows perfectly well that satins and loose twills are the fabrics that suffer. What it explains it with is float length, which is right and is a proxy: the quantity that actually resists slippage is friction at the crossings, and the weave enters through how many crossings there are.

The capstan, again, and the count is the weave #

The mechanism is the one the fancy weaves brought to this site: Euler’s capstan equation, in which a thread passing over another with a wrap angle θ under friction μ has its tension multiplied by e^(μθ) between one end and the other.

Two quantities feed it, and both are computed rather than assumed.

The wrap at a crossing is twice the weave angle, which Peirce’s geometry solves from the sett and the diameter. At 24 threads per centimetre in a 0.2 mm yarn that is 33.8° each side, so 68° of wrap at every crossing.

The number of crossings inside the allowance is the weave’s own interlacing count. A plain weave interlaces at every intersection, so a 10 mm allowance at that sett gives 24 crossings. A 2/2 twill interlaces at half of them: 12. A five-end satin, at two fifths: 9.5. An eight-end satin, at a quarter: 6.

Multiply the wrap by the count and put it in the exponent, and the grip is:

A plain weave holds its own threads more than five hundred times harder than an eight-end satin, at the same sett, in the same yarn, at the same friction. That is the trade’s rule of thumb with a size on it, and the size is exponential rather than proportional because friction accumulates multiplicatively.

Where the exponential stops, and it does stop #

An exponential in the crossing count cannot be the whole story, because it would say that a seam allowance of a few centimetres grips a thread with a force no thread could survive. Which is exactly what happens, and the boundary is the useful part.

A thread being pulled has two ways to go. It can slide, against the friction the capstan is accumulating. Or it can break, at its own strength. Whichever needs less force happens, so there is a crossing count past which the thread breaks rather than sliding — and past that count, extra allowance buys nothing at all.

The count is ln(strength ratio) ÷ (μ · 2θ), which at μ = 0.3 and 68° of wrap is 13 crossings for a hundredfold strength margin. So:

• a plain weave reaches it in 5.5 mm of allowance, and the other 4.5 mm of a 10 mm allowance is holding nothing;
• the three other weaves in the table never reach it in 10 mm at all, and their threads slide.

That is the sharp form of the trade’s practice. A satin is specified with a wider seam allowance not because a wider allowance grips proportionally harder, but because a satin is on the sliding side of the crossover and needs enough crossings to get across it. A plain weave is over the line inside half a centimetre.

Why the float is the trade’s proxy, and what it is a proxy for #

The trade explains slippage by float length and this arithmetic explains it by crossing count. They are the same statement, and seeing why is worth a paragraph.

Interlacings per intersection is the fraction of intersections where a thread changes face, and a float is a run between two of them. A weave with long floats has few interlacings, by definition: the two numbers are reciprocal, and this site has computed both since its first essays.

So “long floats slip” and “few crossings grip weakly” are one sentence. What the capstan adds is the functional form — exponential, not linear — which is what turns a rule of thumb into a specification. A designer told that a satin slips more will add a millimetre; a designer told that the grip falls by a factor of 500 will change the allowance, the stitch type, or the fabric.

The sett is the other half, and it moves the answer twice #

The crossing count depends on the sett as well as on the weave, and the sett enters the grip twice over in the same direction.

A closer sett puts more crossings into a given allowance — that is arithmetic. It also raises the weave angle, because a more crowded cloth has to bend its threads harder around each other, so each crossing wraps further. Both effects raise the grip, and their product is why slippage is a problem in loosely set fabrics and effectively absent in closely set ones.

What the standard measures, and why it is a gap #

Seam slippage has a test, and the test’s shape is worth knowing because it explains why the trade thinks in allowances rather than in forces.

A specimen is seamed, loaded across the seam, and the gap that opens beside the stitching is measured: the specification is a load at which a stated opening — six millimetres is a common figure — has appeared. It is not a strength test. Nothing has broken at the end of it; the threads have moved.

That is exactly the quantity the capstan governs. A thread slides when the tension exceeds what the friction can hold, and the gap is how far it has slid, so the test measures the onset of sliding rather than any failure of material. Which is why a fabric can pass a tensile-strength specification comfortably and fail a slippage specification badly, and why the two are always listed separately.

It also explains why the remedy is usually not a stronger thread. A slipping seam is a fabric problem: the answers are a wider allowance, a different seam type that clamps more of the cloth, or a finish that raises the friction — which is what a resin or a silicone anti-slip finish on a lining fabric is for. Raising μ raises the grip exponentially, so a modest change in the finish moves the failure load a long way.

The construction that solves it without closing the cloth #

There is a third answer, and it is one this site has already drawn for a different reason.

A leno crosses its warp ends around each other between picks instead of merely passing over and under. The crossing wraps the pick through π by construction — a half turn, rather than the twice-the-weave-angle a plain weave manages — and the compound-cloths field found the consequence: no plain weave, at any sett and any friction, grips its pick as hard as one leno crossing. The friction coefficient cancels out of that comparison, which is what makes it a result rather than a number.

Read into this rung, that says a leno resists slippage by construction rather than by crowding. An open gauze whose threads cannot slide is exactly the thing a loosely set fabric cannot otherwise be, and it is why leno is used for scrims, bandages, geogrids and anything else that must be open and stay put.

So the field has three ways to hold a thread in a cloth, and they are the three this site keeps finding: crowd it (a close sett, more crossings, a steeper wrap), wrap it harder (a leno crossing, a W-fastened tuft), or raise the friction (a finish, a coating, a texture). The first is geometry, the second is construction, and the third is chemistry — and only the first two are computable here.

What was counted, and how #

The wrap angle is Peirce’s, solved rather than assumed, and its solution is checked against the model’s own equations to a residual below 10⁻⁹. The crossing count is the interlacing count from cloth.js, computed from the matrix that drew the figure. The friction coefficient is measured, is stated at every use, and is the reason nothing here is quoted as a property of a weave.

Two assertions carry the argument. Within a figure, the crossings and the grip are asserted to fall together as the float lengthens — which would catch an interlacing count read the wrong way round. Across figures, the ordering of the four weaves is asserted at every friction coefficient in the reported range, because that is the claim: the sequence survives the whole range of μ, and only the sizes move.

The crossover count is arithmetic on a stated strength ratio and the effective allowance follows from it. That ratio — a hundred — is an input rather than a measurement, and it is the one number in this essay a reader should treat as illustrative: what it changes is where the crossover sits, and what it cannot change is that there is one.

The allowance a weave needs, which is the specification the arithmetic gives #

The crossover count is the useful output, because it converts directly into the one number a garment specification actually carries. A seam allowance does its job when it contains enough crossings for the thread to break rather than slide, and the crossings arrive at a rate set by the weave.

The count needed is fixed — thirteen crossings, at 0.3 friction and a hundredfold strength margin, whatever the cloth. The rate at which an allowance supplies them is the sett times the weave’s interlacing fraction. So the allowance a weave needs is the plain weave’s divided by that fraction:

Read that against practice and it lands squarely on it. Ten millimetres is the ordinary garment allowance and it is enough for a plain weave twice over and not quite enough for a 2/2 twill. Fifteen is the figure specified for satins and slippery linings, and it is enough for a five-end satin and short for an eight-end one — which is why an eight-end satin is the cloth the slippage standard was effectively written about.

Three things about that table are worth stating separately, because each is the kind of thing a rule of thumb cannot carry.

It is a reciprocal, not a linear scale. Halving the interlacings doubles the allowance, so the cost of a long-floated cloth in seam allowance is proportional to its float and not to some fraction of it. A designer moving from a plain weave to an eight-end satin needs four times the allowance and will be told to add a couple of millimetres.

The sett moves the whole column. At eighteen threads per centimetre rather than twenty-four, every allowance in the table grows by a third — and grows again because the weave angle falls, so the wrap at each crossing is shallower and more crossings are needed. A loosely set satin is the compound worst case and is exactly the cloth the trade warns about.

And friction is the cheapest lever on the list. The count needed goes as one over μ, so raising the friction coefficient from 0.3 to 0.4 cuts every allowance by a quarter. An anti-slip finish is therefore worth about two and a half millimetres of allowance on a five-end satin, bought without changing the cut — which is why such finishes exist on lining fabrics, where the allowance is constrained by the garment rather than by the cloth.

The general shape is one this collection keeps meeting. A specification carries a single number where the arithmetic has a reciprocal in it, so the number is right for the case it was calibrated on and wrong by a factor for everything else. Ten millimetres was calibrated on plain and twill shirting cloth, and it has been applied to satins ever since.

What the picture cannot show #

The figures here draw a thread past a row of crossings with a factor at each, and the one thing they cannot draw is the quantity being computed.

The grip is a product of factors, so it is a number with no place on the drawing: nothing in the picture is 4,600 times anything else. The bar beside the section is a logarithm for that reason, and a reader who reads it as a length is reading it wrongly — which is exactly the sort of thing a caption has to say out loud.

Nor can the figure show the thread sliding. It draws a static state, and the whole argument is about a competition between two ways of failing. The mark on the bar stands in for the thread’s own strength, and where the bar passes it the mode changes; that is an annotation rather than a depiction, and it is the honest limit of a section drawing.

What the drawing does carry is the count. Twenty-four crossings against six is visible at a glance, the interlacing count that produced it is computed from the matrix, and the reader can see that the satin’s thread is held in fewer places without having to believe an exponent.

Where the model stops #

A seam is not a single thread being pulled. It is a row of stitches loading a whole set of threads at once, with the load shared unevenly and the threads nearest the needle holes taking most of it. The capstan gives the anchorage of one thread and the seam’s behaviour is a collective failure.

μ is not a constant of cloth. Yarn-on-yarn friction depends on the fibre, the finish, the twist, the moisture and the crossing angle, and reported values for cotton on cotton run from 0.2 to 0.4. Every size in this essay is at a stated μ. What survives the whole range is the ordering, and that is the claim the assertions are on.

The stitching itself changes the cloth. A needle passing through crowds and sometimes severs threads — the subject of the next rung — and the thread it leaves behind clamps the fabric locally, which raises the friction in exactly the region this arithmetic is about. Neither effect is here.

And nothing here is a seam strength. The step from a grip ratio to a force needs the tension the seam actually carries, the number of threads engaged, and a statistical account of which one fails first. What is computed is a ratio and a crossover, which is where the geometry ends.

One number in the table deserves a second look, because it is the one a reader is most likely to disbelieve. A factor of 4,600 sounds absurd for a fabric holding one of its own threads — and it is not a force, it is a ratio: the tension at one end of the buried thread divided by the tension at the other. Ratios of that size are ordinary in the capstan’s world, which is why three turns of rope round a bollard hold a ship. The reason it does not mean the thread is unbreakable is precisely the crossover above: past thirteen crossings the ratio is larger than the thread’s own strength margin, and the thread stops being the thing that moves.

Who found it, and when #

The capstan equation is Euler’s, from 1762, and textile mechanics has used it for the reason a knot holds since long before anybody wrote it down for a seam. Seam slippage as a specified property is twentieth-century garment engineering, with standard tests measuring the load at which a stated gap opens beside a seam.

What appears not to be written down anywhere is the exponent. The trade’s account is float length, the standard’s account is a measured load, and the arithmetic connecting them — wrap angle times crossing count in an exponent, with the crossing count being the interlacing count the weave already has — sits in between and is nobody’s. This site is fond of that position: the two ends were known and the middle was a computation nobody needed until they wanted to know why the factor was so large.

Where the ladder goes next #

The seam that holds the cloth is sewn with a needle, and the needle has to get through. Whether it finds room between two threads or drives through one is decided by the same clear gap the filter cloths were specified by — so a seam has an optimum stitch density rather than a maximum, and the next rung computes where the two limits cross.