About

This is a growing collection of illustrated essays about how cloth is put together. Each takes a single idea and draws it until the argument is visible — and every weave in every picture was generated from its own rule and then checked before it was allowed to appear.

Why the checking matters here

A weave is a binary matrix: warp up or warp down at every intersection, over a repeat that tiles the cloth. That is not a convenient encoding — it is the weave. So almost everything a weaver cares about is a decidable property of a small integer matrix, and there is no tolerance to choose and no residual to interpret.

The property that matters most is the one a draft cannot show. A grid of filled and empty squares can look entirely reasonable and describe fabric that falls into two independent layers, or that has warp ends lying loose on the surface which can be pulled straight out. Nothing about the drawing betrays it. Only the arithmetic does.

• The cloth is one cloth. At every intersection one thread passes above another. If the threads can be split into an upper set and a lower set such that at every crossing between them the upper thread is on top, the upper set lifts off. Read as a reachability condition that is a strong-connectivity test, and the number of separable layers is the number of components.
• Both directions. A fabric asserted to be one cloth must have one component; a double cloth asserted to be two must have two. Finding more layers than were claimed is the accident; finding fewer means an intended double cloth has been stitched together by mistake.
• Floats and interlacings are counted, never quoted. Measured cyclically, so a repeat that is only right in its interior does not pass.
• No yarn stretches. Every sheared or draped net is checked segment by segment, and a deformation that lengthened a thread throws.

A figure that fails those checks stops the build. That is not a formality: two real errors were caught this way before a single essay existed, and neither looked wrong on screen.

The bias, which is where the subject is taught worst

Cloth cut on the bias stretches half as long again and springs back, while the threads in it stretch by essentially nothing at all. The usual explanation — that the fabric is somehow elastic in that direction — is wrong, and the right one is more interesting.

A woven cloth is a pin-jointed net: two families of very nearly inextensible threads, crossing, free to rotate where they cross. That has a degree of freedom which has nothing to do with material stiffness. The cell angles change, the cell sides do not, and the sheet shears like a trellis fence closing. It is a mechanism, not a material — which is why the extension has a hard limit, reached when the threads jam against their neighbours at an angle that geometry decides.

The same fact, one step further, is why a flat cloth goes round a cylinder for nothing and cannot go round a sphere at all without shearing. Curvature has to be paid for in shear, and when the payment exceeds the locking angle the cloth wrinkles instead. That is why garments need darts.

Numbers the trade uses that do not mean what they seem

Thread count is the standing example. It counts threads and says nothing about how much of the cloth they cover, so it can be inflated by counting plies and it ranks fabrics in nearly the wrong order. The quantity that means what thread count is taken to mean is cover factor, and it is computable from yarn diameter and spacing.

Rather than ignore claims like that, the site states each fairly and then tests it. The rule is that a refutation must be a computation: it is not enough to say thread count is a poor measure, the figure has to compute cover for two fabrics of equal thread count and show them differing.

Where the models stop

Three places, and they fail differently.

The weave matrix is exact and says nothing about yarn. It decides floats, interlacings and integrity perfectly, and it has no notion of thickness, twist, friction or stiffness. Two cloths with the same draft can behave very differently.

The yarn geometry is a model, and there are several. Circular cross-sections and a fixed diameter are Peirce's 1937 assumptions; a real yarn flattens where it is gripped, and the racetrack and elliptical models relax that differently and give different numbers for the same cloth. Where a number appears here, the model that produced it is named.

The trellis has no stiffness and no friction. It says what is geometrically possible, not what a fabric will do. It cannot predict hand, or how a cloth hangs under its own weight, or the shape a wrinkle takes. Nothing here may be read as a mechanical prediction.

Vector, not raster

Every figure is generated SVG, recolours for the dark theme, and can be checked by the layout gates. Textiles are a texture-heavy subject and the temptation to photograph is strong; the rule is that raster may carry texture and never information a reader must read.

On being wrong

Corrections are welcome and will be made. A draft with the wrong caption looks exactly like a draft with the right one, which is the whole argument for checking.