A crease cannot cross a seam

Every fold in this ladder has been a fold of one cloth. A garment’s creases are not: the crease down a trouser leg crosses the side seam, the crease in a shirt sleeve crosses the underarm seam, and a pressed hem is a fold of three layers at once.

A stack folds differently from a cloth, and the difference is a length.

The outer layer has further to go #

Take n layers of thickness t round a fold of angle θ. The layers are concentric, so the outermost runs at a radius greater than the innermost by (n − 1)t, and an arc at a radius greater by s is longer by sθ.

slip = (n − 1) · t · θ

There is no fold radius in that. The radius cancels — a tight fold and a gentle one of the same angle demand exactly the same slip — which is the first surprising thing and is why the problem cannot be solved by pressing more gently.

For four layers of a quarter-millimetre cloth through a half turn:

0.78 mm of separation × π = 2.45 millimetres.

Two and a half millimetres is not a small length in a garment. It is most of a stitch spacing, it is several times a seam allowance’s own tolerance, and it has to appear from somewhere between the fold’s two ends.

Where it comes from in a loose stack, and where it does not in a seam #

A loose stack of cloth folded over — a pile of towels, a bolt turned back on itself — supplies the length by sliding. The layers move over one another at the fold, the outer one draws in a little from each side, and the fold closes.

That is why the edges of a folded stack of cloth do not line up. Everybody has seen it and the arithmetic above is the reason: the outer layer has been pulled 2.45 millimetres round the fold and its free end is that much short.

A seam is a stack that has been stopped from sliding. What holds a thread in a seam is friction at its crossings; what holds a layer is the stitch line, and it holds far harder. The stitch line pins the layers together every few millimetres, so the slip a fold needs has to be found between two stitches — and between two stitches there is three millimetres of cloth, in which two and a half millimetres of relative movement is being asked for.

At four layers the fold wants more than four fifths of the stitch pitch and at six it wants more than all of it. Neither is available: cloth between two stitches does not stretch by that much, and the stitches themselves do not move.

So the fold opens out, which is what a trouser leg does #

What happens instead is the only thing that can. The fold cannot close to the radius the press wants, so it opens — the crease runs into the seam, flattens across it, and reappears on the other side.

That is exactly what a pressed trouser crease looks like. Follow it down the leg and at the inside seam it is not a sharp line; it is a soft arc a centimetre or two wide, and the crease resumes past it.

And the arithmetic says how wide. A fold that cannot slip can still turn, provided the turn is spread over enough length that the two layers’ path difference is supplied by their own extension rather than by sliding. A cloth strains a per cent or so before it is damaged; over a fold spread across a length L the outer layer needs a strain of slip over L, so L must be at least slip over the strain the cloth can give.

At 2.45 mm of slip and a per cent of usable extension that is 245 millimetres, which is absurd — the crease would have to open over the whole leg. So the cloth does not supply it by extension either, and what actually happens is a compromise: the layers slip a little between stitches, the stitches themselves distort, the cloth strains a little, and the fold takes whatever radius those three between them allow.

Which is why the arc is a centimetre or two and not either extreme. Three mechanisms, each supplying a fraction, and none of them the one a single cloth uses.

Which explains three things a garment does #

The mechanism is one line and it accounts for a set of observations that are usually given separately.

A hem is pressed and a seamed hem is not flat. A seam stands proud and wears first for a related reason and a different one. A hem is two or three layers folded; the fold needs its slip; and where the hem crosses a vertical seam the stack becomes four or six layers and the hem’s own edge lifts. Every garment does this and it is usually called bulk.

A seam allowance is pressed open rather than to one side, where it can be. Pressing open makes the fold two layers rather than four at every point, which by the table above quarters the slip demanded. That is the standard instruction in every making-up guide and the reason given is bulk; the arithmetic says the reason is the fold.

And a topstitched seam is stiffer than a plain one by more than its thread accounts for. Topstitching adds a second line of pins, so the length available for slip between constraints halves and the demand doubles. The stitch itself weakens the cloth it passes through is the collection’s own account of what a stitch costs in strength; this is what it costs in flexibility, and the two are unrelated.

The angle matters and the radius does not #

The absence of the radius from the expression is worth a section, because it inverts the obvious remedy.

A single cloth’s fold is limited by its radius: the crimp runs out at a radius of millimetres and past that the fold is handed to the fibres, so a gentler fold is a kinder one and everything in the first rung is a statement about radius.

A stack’s fold is limited by its angle. The slip is (n − 1)tθ and R does not appear, so pressing a seam more gently does not reduce the demand at all — it reduces the strain in each layer and leaves the length difference exactly where it was.

So the two folds have different remedies. A single cloth’s crease is eased by a larger radius; a stack’s is eased only by fewer layers, a thinner cloth or a smaller angle, and of the three only the first is under a maker’s control.

That is why the standard advice is about layer count — grade the seam allowances, trim the corners, press open rather than to one side — and never about pressing more gently. The trade has the remedy right and the reason it gives is bulk rather than arc length.

What a cloth’s own thickness does to the whole table #

The slip is proportional to the thickness, so the table above is a table about one cloth and the range across cloths is large.

This collection’s own thickness figures run from 0.16 mm for a voile to 0.40 for a duck — a factor of two and a half — and the slip scales with them exactly. So a four-layer seam through a half turn demands 1.5 mm in a voile and 3.8 in a duck, against the same three-millimetre stitch pitch.

The duck’s demand exceeds its stitch pitch and the voile’s does not, which is a difference in kind from a difference in cloth, and it is the arithmetic behind something every maker knows: a heavy cloth cannot be pressed into a sharp fold at a seam and a light one can.

And it says what the remedies are worth in each. Grading the allowances takes a four-layer seam to three, which cuts the slip by a third; that takes a voile from 1.5 to 1.0 and a duck from 3.8 to 2.6, so the duck is still over its own stitch pitch and the voile was never near it. A remedy that fixes a light cloth does not fix a heavy one, and the reason is that the two are on opposite sides of a threshold rather than at different points of one scale.

That threshold is the useful thing to name. A seam can be pressed sharp when its slip is below the stitch pitch, which is (n − 1)tθ < p — three quantities a maker controls and one they do not.

And the same expression covers a fold that is not a crease #

The expression has no crease in it, so it covers every fold of a stack, and two of them are worth naming because they are not thought of as the same problem.

A cuff turned back is a fold of the sleeve and the cuff together, through a half turn, at four to six layers. Its slip is the table’s, and the reason a turned cuff never lies quite flat is the reason a crease will not cross a seam.

And a garment on a hanger folds at its own shoulder seam, which is a two-layer fold at a gentle angle and demands almost nothing — which is why it is the one place a garment folds without complaint.

So the family runs from the shoulder, where the slip is a fraction of a millimetre and nobody notices, to a hemmed and seamed corner at six or eight layers, where it exceeds anything the construction can give and the corner is simply thick.

Every one of those is one expression at a different (n, t, θ), and the observation that a garment’s difficult places are exactly its thick ones is not a coincidence about bulk — it is the arc-length demand rising with every layer added.

What was counted, and how #

The slip is geometry and is exact. Concentric arcs at separation s through angle θ differ in length by sθ, which is the definition of a radian and needs no model. What is supplied is the separation, which is (n − 1) layer thicknesses.

The thickness is a measured cloth’s. A quarter-millimetre is an ordinary shirting or trousering; the collection’s own thickness figures for named cloths run from 0.16 to 0.40, so the table’s numbers scale by a factor of two and a half across the range.

The comparison against the stitch pitch is a comparison and not a model. Three millimetres is an ordinary lockstitch pitch and the ratio is what is being reported; the claim is that the slip is a large fraction of it, not that the cloth between two stitches can supply any particular amount.

The trend is asserted over four stack depths, because a claim about a four-layer seam is a claim about a seam somebody chose. And the demand at four layers is asserted to exceed half the stitch pitch — a bound rather than a value, because a value would be a fact about one thickness and one pitch.

What the fold does to the seam rather than the other way round #

Everything above asks what a seam does to a fold. The question runs the other way as well and the answer is a fault with a name.

A fold that cannot get its slip puts the outer layers in tension and the inner ones in compression, at the fold and along it. Over a crease pressed repeatedly into the same place, that is a cyclic load on the stitching at the fold — the thread is pulled by the layers trying to move past each other, every time the garment is folded and pressed.

So a seam that a crease crosses is loaded by the crease, and it is loaded in shear along the stitch line, which is exactly the direction a lockstitch is weakest in.

That is a mechanism for something the trade observes and attributes to wear: a trouser fails at the crease line where it crosses the inside seam, more often than anywhere else on the leg. The seam standing proud takes the abrasion is one reason and it is not the only one — the crease is also working the stitching there every time the garment is pressed.

Neither this collection nor anybody else has the number, because it needs a fatigue model for a thread under a cyclic shear whose amplitude is the frustrated slip. What can be said is the direction and the location, and the location is testable: the failure should be at the intersection of the crease and the seam rather than anywhere else along either.

Which fold in this ladder each of the four is #

Setting the ladder’s four folds beside one another puts this rung in its place, and the placing is by what limits each of them.

A crease in one cloth is limited by its radius: the crimp runs out at a radius of millimetres and past that the fibres take it.

A wrinkle in one cloth is limited by nothing sharp at all, which is why it settles no question — both branches of the bending bracket are survivable at its radius and the model has two answers.

A crease across a stack is limited by its angle, and the radius does not appear.

And a fold at a weave’s own scale is limited by where in the repeat it falls: a satin has places with no crimp to spend, so the same cloth folds differently depending where in its repeat the fold falls.

Four limits, four different quantities, and the only thing they share is the word. That is the reason the ladder needed four rungs rather than one, and it is worth stating because “a fold” reads as one phenomenon and is four with almost nothing in common.

The one thread through all of them is the accounting: a fold needs its outer face longer than its inner, and every rung is about where that length comes from. Crimp, fibre strain, interlayer slip, the weave’s own float — four sources, and each rung is the case where one of them is the binding one.

Where the model stops #

The layers are treated as inextensible and they are not. A cloth extends by moving its crimp, so each layer can give a per cent or so of length before anything is strained — and 2.45 mm over the few centimetres a fold occupies is far more than a per cent, which is why the fold opens rather than closing. Where exactly it settles needs the three mechanisms above solved together and this collection does not have that.

The stitch is treated as a rigid pin. A lockstitch has slack in it, the thread stretches, and a seam does slip a little at every stitch — so the constraint is softer than “no movement” and harder than “free”. How much softer is a question about thread tension and stitch balance.

The stack is treated as concentric. Real layers of cloth compress at the inside of a fold and spread at the outside, so the separation at the fold is less than (n − 1)t and the slip is correspondingly less. That correction is in the direction of making the fold easier and it is a compression this collection could compute and has not.

And nothing here is about what the crease does to the cloth. The fibres at a crease are strained by the ladder’s own bracket, and a seam’s fold is at a larger radius so its strain is smaller — so where a crease crosses a seam the cloth is less damaged and the crease is less permanent, which is the same fact from the wearer’s side and is not computed.

Who found it, and when #

That a stack of layers slips at a fold is elementary and is known to anybody who folds paper: the classic demonstration is that a folded sheaf’s edges do not line up, and the reason given is the same arc-length argument.

Applying it to a seam appears not to be done, and the reason is probably that the making-up literature is procedural. It says grade the allowances, press open, trim the bulk — all of which are the right actions — and explains them by thickness, which is the visible property. The arc-length reading says the same actions are right for a different reason and predicts one thing thickness does not: that the demand is independent of how sharply the fold is pressed.

That is the testable part, and it is the part worth carrying. A maker who believes the problem is bulk will press harder; a maker who believes it is slip will not, because pressing harder does not supply length.

Where the ladder goes next #

Four rungs have taken a fold from a single cloth to a stack, and every one has treated the fold as a shape held once. A crease is not held once: it is pressed, worn, sat on, washed and pressed again, and what survives that cycle is a question about how a set decays rather than about how it is made.