Bias-cut cloth

Why does a woven fabric stretch generously on the diagonal while barely giving along its own threads?

Take a square of ordinary woven cloth and pull it along its threads. It hardly moves. Turn it forty-five degrees in your hands and pull again, and the same square gives a startling amount — it lengthens, it narrows, and it goes fluid where a moment ago it was crisp.

Nothing was added and no thread was cut. The fibres are identical in both trials; they cannot have become more elastic because you rotated your grip. So the give is coming from somewhere other than the fibres. Where?

Look at what a woven cloth is. Two sets of straight yarns crossing at right angles: warp running the length of the roll, weft running across. They are not bonded at the crossings; they simply pass over and under one another, held by friction and the pressure of the weave.

Now pull along the warp. The load goes straight down the yarn, and a yarn is a bundle of fibres already twisted tight — cotton, linen and silk extend only a small percentage before breaking. There is a little apparent give first, but it is not extension: the yarns undulate over and under their crossmates, and a pull straightens that waviness. Once the crimp is gone, so is the give.

Pull on the diagonal and the situation is wholly different, because no yarn lies along the pull at all. Both sets are at forty-five degrees to it. What the load can do is not stretch them but swing them: the two sets rotate toward the pull, and the rectangular grid shears into a rhombus. Nothing has to lengthen. The crossings merely change angle, resisted by friction and by yarns crowding each other — forces far smaller than it takes to stretch a fibre.

How much give does the geometry allow? This can be worked out exactly, and it is worth doing because the answer explains more than the stretch. Take one cell of the weave as a square of side one; its diagonal is the square root of two, about 1.414. Now shear it into a rhombus of the same side, closing the yarn angle from ninety degrees to sixty. A rhombus of side one with interior angle theta has diagonals of two times the cosine of half theta and two times the sine of half theta. At ninety degrees both come to 1.414, as they must. At sixty degrees they are 1.732 and 1.000.

So the diagonal you are pulling along has grown by about twenty-two percent while the diagonal across it has shrunk by about twenty-nine percent — and not one yarn has changed length by a thousandth.

That predicts something a fibre-stretching account never would: the bias give must arrive bundled with a large narrowing, roughly in the proportion the geometry sets. It does. A bias-cut skirt lengthens and pulls in as it hangs, which is why it moulds to a body without darts — the shaping is the cloth agreeing to shear.

The geometry also predicts a limit. Keep shearing and the yarns crowd until they cannot rotate further — the weave jams — and beyond that any further pull must extend fibres, so the cloth goes abruptly stiff. Bias cloth is very compliant up to jamming and then nearly rigid, a signature quite unlike rubber.

What would falsify it? Cut identical strips at zero, twenty-two and forty-five degrees, hang the same weight from each, and measure length and width: the mechanism requires the width loss to scale with the extension, not to lag behind it. Second, immobilise the crossings — a heavily starched, bonded or very densely set cloth — and the bias give should largely disappear though the fibres are unchanged. It does. Third, a knitted fabric, built from interlocking loops rather than a crossed grid, should stretch in every direction rather than only on the diagonal. It does, which puts the effect in the grid rather than in textiles generally.

The first is that "true bias" — exactly forty-five degrees, where both yarn sets are equally inclined to the pull — is a definition rather than a discovery. Off that angle the behaviour is not merely weaker but asymmetric: one yarn set takes more of the load, so the cloth twists as it stretches rather than simply narrowing, which is why a garment cut a few degrees off true hangs crooked however well it is sewn.

The second is that warp and weft are not equivalent, though the geometry treats them as a symmetric grid: warp yarns are tensioned throughout weaving and are usually stronger and less crimped, so a cloth normally gives slightly more across than along. The square cell models the bias mechanism well enough, but it is an idealisation.

The third concerns what is mechanical here and what is taste. The mechanism depends on no viewer, and works identically for a sailmaker, who has the opposite interest and cuts panels so loads run along the threads precisely to keep bias shear out. What is conventional is the aesthetic use: the fashion for bias-cut evening dress that made the technique famous, commonly credited to Madeleine Vionnet's work between the wars, is a period preference that has come and gone since. Attributions of that kind vary between sources and should be held loosely; the geometry does not.

• Why sailmakers and tyre makers spend so much effort keeping bias shear out, and how a ply is oriented to do it.

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