Why clothes need darts

Wrap a piece of cloth round a cylinder and it fits perfectly, with no wrinkles and no effort. Wrap the same cloth round a ball and it will not lie down: there is too much of it somewhere, and pushing the excess away only moves it.

That is not a fact about cloth. It is a fact about surfaces, and it was settled by Gauss in 1827.

Developable and not #

A surface is developable if it can be flattened into a plane without stretching. A cylinder is: cut it along a line and unroll it. A cone is. A flat sheet obviously is.

A sphere is not, and neither is a saddle, and neither is nearly any surface that curves in two directions at once. Gauss’s Theorema Egregium says the reason: a surface carries an intrinsic quantity — Gaussian curvature — that is unchanged by any bending that does not stretch. A plane has zero everywhere. A sphere does not. So no amount of bending turns one into the other.

A woven cloth is very nearly inextensible along its threads, so it is very nearly restricted to deformations that preserve length. It goes round a cylinder for nothing, and it cannot go round a sphere without doing something else.

What the cloth does instead #

The something else is shear, and it is the bias mechanism doing the work.

A woven sheet is a pin-jointed net. It cannot stretch, but it can change the angle at every crossing, and that gives it exactly one way to accommodate curvature: the cells shear. On a surface with positive curvature the cells close; on a saddle they open.

The zero in that figure is worth pausing on. It is not a modelling assumption; it is what the algorithm produces when it lays an inextensible net over a developable surface, and any error in the placement would show up immediately as shear that should not exist.

The fishnet construction #

The way to compute a drape is old and it is not an optimisation.

Choose two yarn paths across the surface, crossing at a point — the only choice in the whole procedure, and it corresponds to a person deciding where to lay the cloth first. Then every other crossing is forced: given three corners of a cell already placed, the fourth is the point on the surface at one pitch from two of them.

Nothing is optimised, nothing is iterated, and nothing is chosen after the start. That is why two people draping the same cloth from the same starting cross get the same answer, and why a drape can be predicted rather than measured.

Every node in the figures here is placed that way, and every segment is checked afterwards to be exactly one pitch. A drape in which a thread had stretched would throw.

The shear grows outward #

The amount of shear is not uniform. It is smallest where the cloth was first laid and grows with distance from that point, because the accumulated curvature enclosed grows with the area covered.

That gives the practical rule everybody who has draped anything already knows: place the cloth where it needs to be right, and let the trouble go outward. A garment is draped from the shoulder or the bust point; a composite preform is laid from the deepest part of the mould.

How much shear a sphere costs #

The requirement can be computed, and seeing it vary makes the mechanism concrete.

Halve the radius and the shear rises sharply: about twenty-two degrees at radius six, and forty-eight at radius four for the same patch of cloth. The requirement grows faster than the curvature, because it accumulates over the area covered as well as depending on how sharp the surface is.

That is the practical experience of draping in a sentence. A shallow curve costs nothing, a moderate one costs some, and a tight one costs more than any woven cloth has — which is why a fitted bust, a shoulder or a heel is always managed with seams and shaping rather than by asking the fabric to do it.

Where the budget runs out #

Shear is available only up to the locking angle, and past that the threads are touching and the mechanism has stopped.

So a drape has a budget. If the shear required to cover a region is within the locking angle everywhere, the cloth lies down. If it is not, something has to give — and the cloth buckles out of plane, which is a wrinkle.

That is the whole engineering content of draping, and it explains the two standard remedies.

Cut some out. A dart removes a wedge of cloth and sews the edges together, which introduces exactly the curvature the surface needed and removes the excess the shear could not absorb. Gores and panel seams do the same thing distributed rather than concentrated.

Use the bias. Cutting a piece so its threads run at forty-five degrees to the direction of greatest curvature puts the shear where the cloth has most of it available. That is why bias-cut garments conform without darts, and why they need so much more cloth.

A saddle does the opposite #

Negative curvature is the mirror image and it is worth seeing because it makes the mechanism unmistakable.

On a sphere the cloth has too much material and the cells close to absorb it. On a saddle it has too little and the cells open. Both are the same mechanism responding to the sign of the curvature, and both are asserted in these figures rather than assumed: a drape whose shear ran the wrong way for its surface would throw.

The saddle, and why it is not symmetric #

Positive and negative curvature both cost shear, and they are not equivalent in practice.

On a sphere the cells close, so the constraint is the locking angle: the threads eventually touch and stop. On a saddle the cells open, and opening has no such hard limit — the threads move apart, and what stops it is the cloth becoming so open that it is a net rather than a fabric.

So a woven cloth accommodates negative curvature more easily than positive, up to the point where it becomes transparent. Anyone who has fitted a garment over a waist — which is a saddle, curving one way round the body and the other way up it — has met this: the fabric goes round without complaint and looks thin and strained where it has spread.

The remedy for excess positive curvature is a dart. The remedy for excess negative curvature is a gather or an ease, which adds cloth rather than removing it. Both are the same accounting done in opposite directions, and a garment usually needs both.

Where the practice came from #

The mathematics is Gauss’s and the practice is much older and entirely independent of it.

Tailoring solved this problem empirically over centuries. The transition from draped clothing — the toga, the sari, the himation, all of which avoid the problem by not requiring the cloth to fit — to fitted clothing in mediaeval Europe is exactly the point at which the shear budget started being managed deliberately, with set-in sleeves, darts and shaped panels. Every one of those is a way of getting a flat sheet round a doubly curved object.

Sailmaking arrived at the same place from the other end: a sail is a doubly curved surface made of flat cloth, and the panel layout of a sail is a solution to precisely this problem. So is the gore pattern of a hot-air balloon, and so is the leather panelling of a football.

The modern quantitative version came from composites, where the same question is a manufacturing constraint with a cost attached, and where the fishnet algorithm has been standard since the 1950s.

What a dart is, geometrically #

Sewing is not the language geometry uses, so it is worth saying what the operation actually is.

A dart removes a wedge from a flat piece and joins the cut edges. In doing so it removes some of the material and, more importantly, it changes the intrinsic geometry: the cone that results has a point of concentrated Gaussian curvature where the dart’s apex is, and is flat everywhere else.

So a dart is not a way of getting rid of excess cloth. It is a way of injecting curvature at a chosen point, and the excess cloth is what is left over when the curvature has been put where it is wanted. That is why the placement of a dart’s apex matters so much more than its width, and why moving a dart around a pattern piece — rotating it to a different seam — leaves the fit unchanged as long as the apex stays put.

Several darts, or a curved seam, distribute the same curvature over a line rather than concentrating it at a point, which is why panelled garments fit more smoothly than darted ones and cost more to make.

That is also why the simplest garments in every tradition are made of rectangles. A rectangle wrapped is a cylinder; a rectangle folded is a cone; and both are free. Everything that fits a body more closely than that is paying the curvature bill somewhere.

How wide a dart has to be #

A dart injects curvature at its apex, and that is the whole of what it does. Which means its width is not a matter of taste: it is fixed by how much curvature has to go in, and the conversion is one multiplication.

Removing a wedge of angle δ from a flat piece and closing it leaves a cone whose apex carries a total Gaussian curvature of exactly δ, and the angle a given form demands is a fixed total that a pattern cutter can look up. What that total does not say is how wide to cut, and the step from one to the other is that a dart of angle δ, measured w from its apex, opens to a gap of δ × w — with δ in radians, which is the only place radians are needed anywhere near a cutting table.

A bust approximated as a cap of thirty degrees’ half-angle demands 0.84 radians. At sixty millimetres from the apex that is a fifty-millimetre opening, which is far wider than any single bust dart in any real pattern.

The discrepancy is not an error. It is the measurement of what the cloth did for free, and it is the reason this essay’s two halves belong together: the shear absorbed the rest. Every degree the fabric shears is a degree of dart the cutter does not have to sew, so the dart supplies only what is left after the cloth has taken what it can — which gives the reason, in one line, for two pieces of trade knowledge usually stated as taste. A bias-cut piece needs less darting, because the shear is available where the curvature is; a firmly woven cloth needs more, because its threads jam sooner and it absorbs less. A gabardine and a loosely set crepe cut to one pattern do not fit the same body.

Why the apex matters more than the width #

The multiplication also explains why a fitting fault is so much more often a misplaced apex than a mis-measured width, and it puts a number on how much more.

Curvature is injected at the apex and nowhere else, so a dart whose point falls short of the body’s own curvature builds a cone whose tip is in the wrong place. The cloth is then flat where the body curves and curved where the body is flat, and the mismatch is the injected angle times the distance the apex missed by: an apex twenty millimetres short of a bust point that needs 0.84 radians leaves about seventeen millimetres of loose cloth standing away from the body.

Seventeen millimetres is a visible bubble. Against it, a dart cut two millimetres too wide changes the injected angle by four per cent of nothing much and is invisible.

So the sensitivities are not comparable, and the practice follows: a fitter marks the apex, and the width is taken up afterwards. It also says why a large cup is harder to fit than a small one — the same twenty-millimetre error at a larger δ leaves proportionally more cloth loose, so a form with more curvature is less forgiving of the same mistake in exactly the ratio of its curvatures.

What the model does not know #

Three limits, and the first is the one that matters.

It has no bending stiffness. The model says where a cloth can lie, not where it will. A real fabric hangs under its own weight, resists bending, and settles into a shape decided by a balance the model does not contain. Wrinkle shape and fold spacing are entirely outside it.

It has no friction. Shearing takes force, and a real drape may not reach the geometrically available configuration because nobody pushed hard enough.

The drape is not unique in practice. The construction is deterministic given the starting cross, and a real cloth can be smoothed differently and settle somewhere else. What the model gives is one reachable configuration, not the only one.

Why the fit of a woven garment is a compromise #

The last consequence is one everybody has experienced and few connect to geometry.

A body is a doubly curved surface that changes shape when it moves. A woven cloth can be shaped to fit one configuration exactly, by putting the right curvature in the right places with darts and seams. It cannot fit two, because the shaping is sewn in and the body is not.

So a fitted woven garment fits the posture it was cut for and binds in every other. A jacket cut for a standing figure pulls across the back when the wearer reaches forward; trousers cut for standing strain at the knee when sitting. Tailoring manages this with ease — deliberate excess in particular places — which is a way of making the garment fit several configurations imperfectly rather than one perfectly.

A knitted fabric escapes the problem entirely, because it can change its own dimensions and take the curvature it needs at the moment it needs it. That is why sportswear is knitted, why knitted underwear replaced woven, and why a knitted garment can be made in a fraction of the pieces.

The trade is the familiar one. The woven garment holds its shape and does not follow the body; the knitted one follows the body and does not hold its shape. Both are the same compliance question that runs through the whole subject.

Sails, balloons and footballs #

The same problem is solved repeatedly outside clothing, and the solutions look alike once the geometry is visible.

A sail is a doubly curved surface that has to be made from flat cloth. The traditional answer is broadseaming: cutting the panel edges to slight curves so that joining them introduces exactly the curvature wanted. That is a dart distributed along a seam, and sailmakers were doing it for centuries before anybody described it as managing Gaussian curvature.

A hot-air balloon is a sphere made of gores — long tapered panels which are developable strips, joined so that the tapering supplies the curvature. A football is the same idea with a small number of large panels, and the classical thirty-two-panel arrangement is a truncated icosahedron precisely because it approximates a sphere with flat pieces.

The common structure is that a developable strip is free and curvature has to be paid for at the joins. Cut the surface into pieces each of which is nearly developable, and the bill is settled by the seams. A dart is the same operation done once and concentrated, and a panelled garment is the distributed version.

Measuring how far a cloth fails to stay flat #

The impossibility has a laboratory measurement attached, and its construction is more geometric than it looks.

A circular specimen laid over a smaller circular pedestal falls into folds, and the fraction of the annulus its shadow still covers is the drape coefficient. A perfectly stiff cloth does not drape and scores one; a perfectly limp one hangs vertically and scores nearly nothing.

What makes it computable is the hem. The draped edge is a closed curve of cloth and cloth does not stretch, so whatever shape it takes its length is the flat specimen’s circumference — and that one constraint fixes the amplitude of the folds given their number.

The finding is a negative one and it is worth having: the coefficient is measuring how far the hem has come in and almost nothing else. Two fabrics with the same coefficient can fall into five deep folds and eleven shallow ones and look quite different. Bending stiffness and the drape coefficient works it through.

Where the ladder goes next #

The limit that sets the budget is the locking angle, and the mechanism supplying the shear is the bias.

What the pictures here cannot show. Every drape here is a net of lines on a mathematical surface. A real cloth has weight, thickness, friction and bending stiffness, and it wrinkles into shapes that a kinematic construction has no way to predict. What these figures show is what is geometrically possible, and no more.