The bias cut and the selvedge

Cut a panel with its length along the warp and it behaves as the warp does: firm, barely extensible, and inclined to hang in the direction the threads run. Cut the same panel at forty-five degrees and it behaves as a mechanism: it stretches, it clings, it hangs in a smooth cone rather than in folds. That is the whole reason the bias cut exists and it is not in question here.

What is in question is the price. Everybody in the trade knows that bias cutting wastes cloth; the figure usually attached to it is half, and the reason usually given is the diagonal. Both halves of that deserve computing, and they come apart in an interesting way.

One panel: exactly twice, and worse for every other shape #

Start with the arithmetic the slogan is really about, which concerns a single panel and is exact.

A w × h rectangle turned through forty-five degrees needs an axis-aligned box of side (w + h)/√2 in both directions. So the box has area (w + h)²/2, and its ratio to the panel is

(w + h)² ÷ 2wh

which by the arithmetic–geometric mean inequality is at least two, with equality exactly when w = h.

So a bias-cut square costs precisely twice its own area — no yarn, no sett, no cloth width, no free parameter anywhere in it — and that is the best case. A 400 × 900 mm skirt panel costs 2.35 times its area: 57 per cent of the enclosing square thrown away rather than 50.

Forty-five degrees is also the worst angle, which is asserted by search rather than assumed: for every rectangle the bounding box’s area is largest at 45°, so the true bias is the most expensive orientation there is. The slogan is, for one panel, an understatement.

But a cutting room does not cut one panel #

The trouble with that arithmetic is that a bounding box is a fiction. Nobody cuts a square of cloth around a panel and discards the corners; a marker is laid with panels interlocked, and identical panels at a single angle tile the plane exactly.

That is not an approximation either. Congruent rectangles tile the plane at any orientation whatever, so in the interior of a wide bolt a bias layout wastes precisely nothing: every scrap between two rotated panels is another rotated panel. The angle costs nothing where the cloth is not interrupted.

What interrupts it is the selvedge. A panel straddling the edge of the bolt is scrap, so the loss is a band along each edge — and the width of that band is set by the panel’s own size rather than by the bolt’s. Which gives the real scaling, and it is not the one the slogan implies: the penalty goes as the panel’s size over the bolt’s width, and falls as the cloth gets wider.

The minimum bolt, which is where the diagonal does appear #

There is one place the bounding box is exactly the right object, and it is a threshold rather than a rate.

A panel at forty-five degrees cannot fit on a bolt narrower than its own box: (w + h)/√2. For the 400 × 900 panel that is 919 millimetres, so a 900 mm bolt yields nothing at all — not a poor layout but an empty one — and the tally above shows exactly that, an 80-point penalty because the bias column is zero.

That is the useful form of the diagonal in this trade. It does not set the waste; it sets the minimum width, and the minimum width is why bias-cut garments are made from wide cloth and why a narrow silk was historically cut on the straight whether the designer wanted it or not.

What the straight cut is actually being compared against #

The comparison needs its other half computed too, and the other half is not 100 per cent.

A straight layout has its own waste, and it is a sawtooth: a 400 mm panel on a 1400 mm bolt fits three across with 200 mm left over, so the yield is 86 per cent; on a 1800 mm bolt it fits four with 200 left, which is 89; on a 2400 it fits six exactly, which is 100. The straight cut’s efficiency jumps up and down as the bolt width passes each multiple of the panel width, and a buyer choosing cloth by its width is choosing a position on that sawtooth.

So a bias penalty quoted against “the straight cut” is quoted against a number that is itself between 65 and 100 per cent depending on an arithmetic coincidence. The tally above prints both columns for that reason: the interesting quantity is the difference, and the difference is smaller than either the slogan or a single measurement suggests.

The slogan and the mechanism #

Putting it together: the trade’s “bias cutting wastes half the cloth” is a statement about one panel in a bounding box, where the true figure is at least a half and usually more. It is not a statement about a marker, where the waste is a selvedge effect that falls with the bolt’s width and can be well under a third.

That shape — a slogan that quotes the mechanism and forgets what bounds it — is one this site has met before. The standard pass found the bias’s extension quoted as half as long again, when the mechanism’s absolute ceiling is 41.4 per cent and real cloth jams at 27. Here the same trade’s account of the bias’s cost is a bound on the wrong object. Both survive because they sound like measurements.

The honest version is two sentences and it is more useful than the slogan. A bias panel needs a bolt at least (w + h)/√2 wide, and below that width there is no layout at all. Above it, the penalty against a straight layout is an edge effect of order the panel’s size over the bolt’s width, and it is smaller than the slogan on any cloth a modern mill delivers.

The grain line is a constraint on rotation, and it is the expensive one #

The angle above was a free choice. In a cutting room it is not, and the constraint that removes the freedom is worth separating from the bias question entirely.

A pattern piece carries a grain line: an arrow saying which way the warp must run. It is there because the fabric is directional in every property this site computes — the warp and the weft have different crimps, different extensions, different drape and different shrinkage — so two otherwise identical panels cut at right angles to each other are two different pieces of cloth and will not behave alike in a finished garment.

The grain line therefore forbids the one move that would make nesting easy: rotating a piece by ninety degrees to fill a gap. And on a napped, shaded or one-way printed cloth it forbids the 180° turn as well, which the enumeration above allows freely — so a real marker on a velvet or a corduroy is laid with every piece pointing the same way, and its yield is worse than any figure in this essay.

That is where the cutting room’s cloth cost actually lives. Not in the diagonal, which tiles; in the pieces that may not be turned, which do not. And the reason the constraint cannot be relaxed is a fabric fact rather than a habit: a raised nap reflects light differently along and against itself, and two panels of a corduroy sofa cut opposite ways are visibly two colours.

The minimum width sets the rate as well as the threshold #

The essay separates two roles for the bounding box: it does not set the waste, it sets the minimum bolt width. The separation is too clean, because the same number turns out to set the rate too.

A panel at forty-five degrees extends (w + h)/√2 perpendicular to the selvedge — which is exactly the minimum width, call it B. So a panel is lost if its centre falls within B/2 of either edge, and the two bands together are B wide. Since the centres are one per panel area, the area lost per unit length of bolt is exactly the band width, and

bias yield = 1 − B ÷ W.

One line, no enumeration, and it reproduces the tally. For the 400 × 900 panel, B is 919 millimetres, so a 3.2-metre bolt yields 71 per cent against a straight layout’s 100 — a penalty of 28.7 points, where the search found 28. At the minimum width the formula gives zero yield, which is the threshold arriving as the same statement.

So the diagonal sets both. It fixes the narrowest cloth on which anything fits, and above that it fixes how fast the penalty falls — as one over the bolt’s width, with the same constant.

That also says how far the falling goes, and the answer is discouraging. Ninety per cent yield needs a bolt ten times the minimum, which for this panel is nine metres; seventy-five per cent needs 3.7. So the penalty falls with width and does not fall far enough on any cloth anybody weaves, and the trade’s pessimism about bias cutting is partly re-earned — the mechanism is wrong and the magnitude is not so far off.

Which is why a bias garment is made of many small panels #

The formula’s dependence is on w + h rather than on the area, and that has a design consequence the essay’s own arithmetic cannot reach.

Cutting one panel in two halves the penalty by a third. Two panels of 400 × 450 have a bounding width of 601 millimetres against the single panel’s 919, so on a 1.4-metre bolt the yield goes from 34 per cent to 57. The cloth used has not changed; the boundary the pieces have to avoid has.

That is the arithmetic behind a piece of practice that is usually explained by fit. A bias-cut garment is built from many narrow shaped panels seamed together — Vionnet’s dresses are the standing example, and they are made of gores and inserts rather than of large pieces. The usual account is that the shaping is what the technique needs, and it is; the cloth cost says the same thing independently.

And it inverts the intuition about seams. In a straight-cut garment, adding a seam adds labour and buys nothing in cloth, since rectangles tile a strip anyway. In a bias-cut one, adding a seam buys yield in proportion to how much it reduces w + h — so the seam pays for itself in cloth, and on an expensive silk it may pay for its own labour twice over.

The rule is that a bias marker is priced by its largest piece’s diagonal, not by the total area of the pieces. A single large panel on a narrow cloth is the expensive case whatever else the marker contains, and breaking it up is the only lever that works on a bolt of fixed width.

A bolt is a strip, and every strip has two edges #

The general shape of this result is worth naming, because it recurs wherever cloth is cut.

The interior tiles and the boundary does not. Any layout of congruent pieces at one orientation loses material only where the region’s edge cuts a piece in two, so the loss is a perimeter quantity and the yield is an area quantity — which means the penalty falls as the region grows in the direction that is not fixed. A bolt is a strip: one dimension is a few metres and the other is a few hundred, so the boundary term is dominated by the two selvedges and the run length hardly matters at all.

That has a consequence a marker maker will recognise. Lengthening the marker does not help a bias layout — the two selvedge bands grow with it — while widening the cloth does. The right response to a bias-cut style is to buy wider cloth, not to lay a longer marker, and the arithmetic says so with no appeal to experience.

It also explains a piece of trade practice that looks like superstition. Bias-cut garments are often cut in a tubular or folded layout, or from a cloth deliberately woven wider than the garment needs. Both are ways of removing a selvedge from the problem, and removing a selvedge is removing half of the waste.

What was counted, and how #

The placement is an enumeration rather than an argument. Panels are put on the lattice their own tiling defines, the lattice’s offset is swept over a grid of positions within one cell, and the best count wins. Nothing is fitted and nothing is estimated.

The check that makes it trustworthy is the straight case. At zero degrees the number of panels is the number of whole panels across times the number along — arithmetic anybody can do — and the routine is asserted to find it. A placement search that disagrees with a closed form in the case where a closed form exists is a placement search with a bug, and this is the only case where one exists.

Three further assertions sit on the bias column. The bounding-box ratio at forty-five degrees is asserted to be exactly two for a square and strictly more for anything else. Forty-five degrees is asserted to be the worst angle, by search over nine thousand of them. And the penalty is asserted to fall between the narrowest bolt and the widest, which is the claim the whole essay turns on and the one a reader would otherwise have to take on trust.

What the picture cannot show #

The layout figures draw a bolt with panels on it, and the honest limitation is that they draw the best placement the search found rather than the best that exists.

Offsets are swept on a grid, so a placement a fraction of a millimetre off the grid that would fit one more panel is not found. The counts are therefore lower bounds, and the penalties upper bounds — which is the safe direction for the argument being made, since the claim is that the penalty is smaller than the slogan says.

Nor can the figure show the grain. A panel drawn at 45° looks identical whether its warp runs up its length or across it, so the constraint that actually costs a cutting room its cloth is invisible in the drawing. The directional properties that make the grain line necessary — a different crimp, a different extension, a different shrinkage each way — are in other essays and cannot be put on this one’s picture.

Where the model stops #

Real panels are not rectangles. A skirt panel is a shaped piece with a curved hem and a flared side; a sleeve is not convex. Real markers interlock shapes whose complements are other shapes, and a nesting problem over irregular pieces is a hard combinatorial problem that this arithmetic does not touch. What survives is the mechanism: whatever the shapes, a single orientation tiles a plane region with a loss concentrated at the boundary.

A marker carries many different panels, not one repeated. That helps rather than hurts — small pieces fill the gaps large ones leave, which is most of a cutting room’s skill — so the figures here are pessimistic about a real marker and optimistic about nothing.

Grain is not only an angle. A pattern piece carries a grain line because the fabric’s behaviour is directional, and cloth also has a nap, a shade direction and sometimes a one-way print. Those forbid rotating a panel by 180°, which the enumeration here allows freely, and on a napped cloth the real constraint is much tighter than the geometry.

And nothing here prices the making. A bias-cut panel is harder to sew, needs its seams stabilised, hangs differently as it is handled, and takes longer at every operation. The cloth is often the smaller part of the cost, and a computation about yield is a computation about one line of a costing.

One number in this essay is worth isolating because it is the only exact one, and it is the one the slogan gets nearest to. A bias-cut square costs exactly twice its own area — the AM–GM bound, with equality only for a square — and it is the best case over all shapes at 45°. Everything else here is an enumeration over placements, a sawtooth in a bolt width, or a measured penalty. So the trade’s “half the cloth” is an exact statement about the one case nobody cuts, and an underestimate of the panel-in-a-box figure for every shape that is not square.

Who found it, and when #

The bias cut as a technique belongs to Madeleine Vionnet in the 1920s, and the cloth cost is part of the technique’s reputation from the beginning — bias dresses were expensive garments, made from wide silk, and the waste was real.

The arithmetic above suggests the reputation attached itself to the wrong quantity. What made a bias garment expensive was the minimum width it demanded, the labour it took, and the shaped panels it needed — not a diagonal throwing away half the cloth in the middle of a marker. Markers have been laid by hand for a century and by computer for fifty years, and the trade’s own nesting software finds the tiling the enumeration above finds, without anybody having to say why it works.

Where the ladder goes next #

The bias cut is one of the two things a cutting room does about the fact that cloth is flat. The other is the dart, and the total angle a dart has to remove is fixed by the shape rather than by the fabric — a hemisphere costs one full turn, whatever its size, and how much of that the cloth can supply by shearing instead is a property of its sett.