How far a cut edge frays

The applied field’s habit is to turn a description into a specification: not what does this cloth do but will this cloth do, which is always a pair of inequalities on quantities the earlier fields compute. The seams ladder has three rungs of it, and all three rest on one length.

A seam slips before it breaks put the number in its sharpest form. A thread pulled across a seam line either comes out of the allowance or breaks in it, whichever needs less force, and the boundary between the two is the crossover length. Compare it to the allowance a garment actually has — ten millimetres is ordinary — and the answer for each cloth is a verdict rather than a curve.

That rung found the ordinary allowance divides this site’s table rather than settling it: four cloths break and four slip. It is a good result and it was computed with an arithmetic that was four times too generous.

What moved #

Under the old arithmetic the crossovers ran from 4.3 mm to 73.7 mm across the table, and against a ten-millimetre allowance that is a split.

Under the capstan they run from 1.9 mm to 19.1 mm, and only the cheesecloth is above ten. Seven of the eight cloths now break rather than slip in an ordinary allowance, where before four did.

The verdict has therefore moved from this is a real design question, decided cloth by cloth to an ordinary allowance is comfortable for almost anything a garment is made from, which is a different engineering statement and matches the trade better. Seam slippage in practice is a problem of loose, slippery, satin-faced and filament fabrics — not of ordinary shirtings and sheetings, which is what the old table implied.

The individual numbers are worth setting out, because the pattern in them is not uniform. The sheeting goes from 7.4 mm to 2.0, the poplin from 7.3 to 2.2, the muslin from 10.3 to 3.2, the batiste from 4.3 to 1.9, the voile from 8.7 to 3.5, the filter cloth from 17.4 to 3.8, the duck from 30.7 to 5.4, and the cheesecloth from 73.7 to 19.1. The cloths that were furthest above the allowance are cut hardest — the duck by five and a half — because their coarse strong yarns make the argument of the logarithm large, and a logarithm is least generous where its argument is biggest.

The one that still slips is the one that should #

The cheesecloth’s crossover is 19 mm after correction: an open scrim in a coarse strong yarn, set at ten ends per centimetre, with almost nothing gripping the pick and a great deal of thread to break. It slips in any allowance a garment would use, and it is not a garment cloth.

That is the useful shape of a specification result. The boundary has moved to where it separates the cloths anybody would worry about from the cloths nobody would, which is what a boundary in the right place looks like, and the old one did not.

What a fray width actually is #

The same number, read the other way.

A cut edge frays until the threads still in the cloth are gripped over enough length to break rather than slide out. So a raw edge loses its threads to a depth of one crossover and then stops, and the depth is the crossover length.

Two millimetres for a plain cotton sheeting. Five for the same yarns in a five-end satin. Eight for an eight-end satin. Those are the widths a raw edge of each actually loses, and the old arithmetic said 7.4, 18.6 and 29.8 — a centimetre, two centimetres and three, which nobody has ever seen on a plain cotton.

A model that predicts a plain cotton frays a centimetre is refuted by any raw edge in a wardrobe, and it was on this site for a long time.

Which is the general problem with a comparative collection #

Nothing internal caught it. Every ordering in the seams ladder was right, every ratio between cloths was right, the friction sweep was right, and the exact inverse proportionality of μ and L was right and survives the correction untouched. The collection is built on comparisons and every comparison passed.

The error showed up only where a number was quoted against the world — a millimetre against a cut edge — and this site quotes very few. That is worth naming as a standing risk rather than as a one-off: a body of work built on internal consistency can be uniformly wrong by a factor and pass all of its own checks.

Why the correction hit the coarse cloths hardest #

The correction factor runs from 0.178 to 0.436 across the table, and the pattern in it is not noise — it says which cloths the old arithmetic was worst about, and the answer is the ones furthest from the boundary.

The factor is ln(1 + z)/z, and a logarithm divided by its own argument falls as the argument grows. So the correction is severe exactly where z is large, and z is the thread’s breaking load times the wrap angle over the contact force — large for a coarse strong yarn in a cloth that does not grip it hard.

The duck is that cloth. Sixty tex yarn at an open sett, so a great deal of thread to break and comparatively little pressing on it, and z is correspondingly enormous. The batiste is the opposite: a fine weak yarn in a close cloth, so z is small and the exponential has barely departed from its own tangent.

That means the old table’s spread was itself an artefact. Under the sum the crossovers ran 4.3 to 73.7, a spread of seventeen; under the capstan they run 1.9 to 19.1, a spread of ten. The correction did not shift a table sideways, it compressed one — and it compressed it by cutting the top hardest, which is where every verdict that mattered was.

There is a general form of that worth carrying. A linear model applied to an exponential process is worst where the process has run furthest, so the errors are concentrated in the cases the model was extrapolating into. A model checked at the middle of its range and used at the end of it is being checked where it is right and used where it is not — and the seams ladder had exactly that shape, because the cloths near an ordinary allowance were the ones the arithmetic was least wrong about.

Which is a second reason the error survived so long: the cloths whose answers were nearly right were the ones anybody would have checked.

A sheeting’s crossover was wrong by a factor of 3.7 and a batiste’s by 2.3; the duck’s, which nobody looks at, was wrong by 5.7.

What a seam allowance is chosen for #

Worth stating, because the correction changes which of the two reasons binds.

An allowance has to be wide enough that the seam does not pull out, and narrow enough that it does not bulk, fray in wear, or waste cloth. Those are the two inequalities and a specification is the interval between them.

Under the old arithmetic the first inequality was binding for half the table: an ordinary allowance was marginal, and the designer’s problem was slippage. Under this one it binds for one cloth in eight, and the allowance is set by bulk, fraying in wear and marker efficiency instead — which is what garment engineering actually optimises and what the pattern-cutting literature discusses.

The tuft, which does not rescale #

The third hat the number wears is a tuft’s anchorage, and it behaves differently from the other two because one side of the comparison is fixed by the construction rather than by the cloth.

A tuft is bound under one pick, or two, or three, depending on the fastening, and that bound length is a property of the weave and a matter of millimetres. A seam allowance and a fray width are both lengths somebody chooses or measures; a tuft’s is neither. So when every crossover shortens by a factor of four, a seam allowance stays where the pattern cutter put it and the tuft’s bound length stays where the loom put it, and the two comparisons move differently.

For the tuft the correction runs the right way: a shorter crossover means a tuft of a given bound length is more likely to break than to slide, which means better anchorage. The rung that priced a tuft in newtons found that no fastening reaches a carpet specification by friction alone — a domestic specification short by about three, a contract one by nearly six — and the correction narrows those gaps without closing them.

It narrows them because the pull-out force at a fixed bound length is larger under a capstan than under a sum: the exponential is above its own tangent everywhere. So the two readings are consistent and are worth keeping straight. At a fixed length the capstan holds harder; at a fixed force it needs less length. The first is what a tuft experiences and the second is what a seam allowance is chosen against.

What was counted, and how #

The crossover is computed for every cloth in the table at one weave and one friction, and the correction factor recorded beside it. The verdicts against a ten-millimetre allowance are a comparison and nothing more.

The crossovers themselves rest on three inputs and each is what it is. The breaking load comes from the site’s own yarn tensile model, which takes a fibre tenacity, a translation efficiency and a twist obliquity, and is quoted with its own uncertainty. The contact force comes from the weave angle Peirce’s geometry solves for at the construction, times a stated thread tension. The friction coefficient is a range — 0.2 to 0.4 for cotton on cotton — and never a value.

That last one is the reason no crossover here is quoted to more than two figures. Every length in this rung is proportional to 1/μ exactly, so the friction range alone puts a factor of two on it before anything else is considered, and the correction’s own factor varies by two and a half across the table.

The verdicts are reported as a count — how many cloths fall each side — rather than as a table of margins, because a margin computed from a length known to a factor of two is not a margin.

What the verdict is worth to somebody choosing an allowance #

The useful output of an applied rung is a rule somebody can act on, and this one is short enough to state.

Ten millimetres holds anything ordinary. Seven of the eight cloths in the table break rather than slip in an allowance a garment already has, and the exception is an open scrim.

The weave matters more than the cloth. A plain sheeting needs two millimetres of grip and an eight-end satin of the same threads at the same setts needs eight. So a satin-faced fabric is a different problem from a plain one made of the same yarn, and a specification that names a cloth by its weight and count has not said which.

And the number scales with 1/μ exactly. A softener that halves the friction doubles every length here, and softeners are applied at the finishing stage, after the cloth has been specified. So a seam verdict computed on greige cloth is a verdict about a fabric that will not be sewn.

Those three together are the shape of the answer: the allowance is comfortable, the weave is what to watch, and the finish can undo the margin without anything on the specification changing. The last is the one most likely to be missed, because friction is the only input here that a mill can alter after the construction is fixed.

Where the model stops #

The allowance is treated as a length of grip and nothing else. A real seam has a line of stitching in it, the stitching compresses the cloth locally and raises the contact force there, and the thread being pulled crosses that line. So a real seam grips harder than a plain length of cloth of the same width, and every number here is conservative in the direction of slipping — which is the safe direction for a specification and the wrong one for an explanation.

A tensioned contact force, in a garment that is mostly not tensioned. The contact force used is a loaded cloth’s. A seam under load is loaded, so this is the right choice for the slippage question and the wrong one for the fraying question — a raw edge in a wardrobe is relaxed, and a relaxed cloth’s contact force is a smaller number, which makes a relaxed cloth fray further than these figures say.

No sewing thread anywhere. The stitch that weakens the seam is about the damage a needle does, and this rung is about the cloth’s grip; the two have not been put together and the interaction is real, because a perforated cloth grips differently along the perforation.

And one weave at a time. A seam crosses a cloth in one direction and the two systems have different crossovers, because they have different counts and different setts. Everything here is the weft’s.

The generalisation #

A specification boundary is only as good as the absolute number under it, and a body of work can get every ratio right and the absolute number wrong.

The seams ladder was built on comparisons — this cloth against that, this friction against that, this weave against that — and comparisons are what a collection like this one is good at and checks continuously. The one thing it does not check is the scale, because the scale never appears in a comparison.

What catches a scale error is contact with something outside the work, and there is usually exactly one such contact available: a fact everybody knows. A raw edge of cotton loses about two millimetres. A carpet tuft holds at a few newtons. A loom pushes with several hundred newtons per metre. Those numbers are worth more to a model than any number inside it, because they are the only ones the model cannot have been fitted to.

The corollary is a habit rather than a result: when a ladder is built entirely out of ratios, find the one place it touches the world and check there first.

Who found it, and when #

Seam slippage is one of the oldest measured properties of fabric and has a standard test with a stated opening and a stated load. The trade knows exactly which cloths slip and has known for a century.

The pull-out mechanics behind it — a capstan build-up along an embedded length, with a floor from the surrounding compression — are standard in fibre and composite mechanics and are measured directly there.

What is this site’s is the collision between the two: taking a crossover length computed from a sum of independent contacts, comparing it against a garment’s allowance, getting a verdict that split the table, and only afterwards finding that the same arithmetic said a plain cotton frays a centimetre.

Where the ladder goes next #

Sideways, the friction the whole rung is proportional to is two numbers rather than one, and the difference between them decides how a seam behaves under repeated small loads rather than one large one.

Along the seams ladder, the missing piece is the stitching line itself: a real seam’s grip is not a plain cloth’s, and nothing here models what a row of needle holes and a compressed thread does to the contact force along the line where it matters most.