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

Five 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 — and structure does not stop at the thread, since a chenille is a woven gauze cut into strips and a slub yarn is a periodic fault somebody chose. Where a fabric's structure has moved into its thread, the matrix goes quiet.

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. Where a mechanical answer is given here it comes from separate machinery, named where it is used, and never from the trellis.

The surface is a surface of yarn, and a spun yarn is not what touches anything. The outside of a cloth is computed here as a height field: the draft says which thread is on the face, Peirce's geometry says how high its centre line rides, and its own section adds a radius on top. Everything read off that field — what a cloth touches with, what a presser foot reports, where a rubbing lands, what reflects — is a statement about the yarn's outside. A spun yarn has fibre ends standing off it, held by nothing, and those are what a finger, a neighbouring cloth or a friction test meets first. Every contact number here is therefore a lower bound on the area and an upper bound on the pressure, and on a raised or brushed fabric it is describing a surface that is buried.

And which of a cloth's two systems is in contact is decided by a number nobody weighs. The warp's outside stands at half its crimp height plus half its diameter, the weft's likewise, and Peirce's closure condition pins their sum and not their division. The division is the crimp ratio, which this collection has argued at length is a bookkeeping convention rather than a measurement — and it turns out to decide which thread system meets the world, which is to say which one wears out. Every result about a cloth's surface here is conditional on it, the conditionality is stated where it bites, and the step it produces is a few micrometres on an ordinary cloth.

The forces are brackets rather than values. A spun yarn's bending rigidity lies between a free bundle of fibres and a coherent rod, and the two differ by the fibre count; its transverse stiffness has an upper bound and a lower bound of exactly nought, so it can only be read out of a measured fabric. Every result that depends on where in a bracket a yarn sits is reported at both ends, and the ones that do not depend on it at all are the ones worth having.

One of those brackets has since closed and one has not, and the difference is instructive. The bending bracket turns on whether a yarn's fibres can slide when it is bent, which is a comparison between the curvature the twist can hold coherent and the curvature the cloth imposes — and the second beats the first by more than an order of magnitude at every twist, so a thread in cloth is at the free end and the collection's habit of using it is a result rather than a convention. The strength bracket turns on whether a fibre migrates between the core and the surface of its own yarn, which is a property of how the yarn was spun and is not deducible from a construction. A wide bracket that closes is worth more than a narrow one that does not.

The thread is an assembly too

For most of this collection a yarn was a rod with a diameter, and the diameter was a mass divided by a length. It is not a rod. A yarn is a countable number of fibres — the yarn's count divided by one fibre's — and that single number decides its diameter in fibre diameters, the width of its stiffness bracket, and the floor beneath its evenness, the last two running in opposite directions. So there is no yarn that is both even and knowably stiff, and a finer yarn is necessarily a more irregular one.

Opening the thread adds exactly one fitted quantity to the collection and it is named wherever it appears. Fibres touch at points rather than along their length, and what fraction of the nominal contact carries pressure is a contact efficiency that nothing here computes. It scales every gripped length inversely and cannot change the shape of any curve, so every claim built on it is a claim about ordering — a longer staple wants less twist, a coarser fibre wants more — and each of those is asserted across a tenfold range of the fit rather than at the fitted value.

There is no clock anywhere in it. The collection does now say what a fabric fails to give back, and it says it as a geometry: a woven cloth reaches a strain first by moving crimp, which costs its threads nothing and returns in full, and only afterwards by stretching them. So a recovery curve has a corner in it, and the corner is a number about the sett rather than about the fibre. What is still absent is time. A fibre's recovery is quoted as the fraction returned immediately, the delayed part is carried for no fibre, and every statement here about how a fabric settles is a count of events — a wash, a cycle, a named relaxation state — rather than a duration. A garment shrinks most in its first wash and least in its tenth because its frictional barriers are spread, and this collection can say how widely without being able to say how long.

The loop is a shape, and it is not the same kind of object as a weave

The weave matrix decides a woven cloth because a woven thread has nowhere to go: between two crossings it is straight to within four parts in a thousand of its length, and every one of its bends is inside the wrap. A knitted loop has half its thread spare over the same span. That is not a difference of degree, and it is why the two halves of this collection are answered by different machinery — a woven cloth is a combinatorial object with a geometry bolted on, and a knitted one is a curve that has to be solved for.

So the loop here is a rod: the path a yarn takes between two interlacings is the one that minimises its bending energy at fixed length between fixed ends, which is the elastica, and the end forces fall out of the same solve as the multipliers on the endpoint conditions. Nothing about the shape is drawn or assumed. The same solver, handed a shirting, refuses — the slack a woven thread has to work with is four orders below what an unconstrained curve needs — and the refusal is the sharpest statement of the difference the collection can make.

Where that model stops, and it stops in five places.

The curve is planar, and the plane is not the fabric's. This was written here as an omission — the loop leaves its plane, the bending that costs is three per cent — and both halves turned out to be wrong. A half period leaves and arrives along the course direction, so the whole problem is invariant under rotation about that direction, and the solved curve is the flat one turned twelve degrees out of the fabric. That is exact rather than approximate. And the cost is negative: a climb lengthens the straight line the thread has to span, so it leaves less slack to be spent on curvature, and the loop's bending energy falls by 2.7 per cent rather than rising by three. What the third coordinate adds is a quantity the flat model had nowhere to put — a contact force through the fabric's thickness, 7.8 millinewtons a stitch, which is what holds a jersey's two faces two yarn diameters apart.

Torsion is still absent, and the curl radius is still not computed. What has changed is what is owed. A fabric whose neutral surface bisects its loops carries no curling moment at all, exactly, by a symmetry the model has and a real fabric does not — so the whole of stockinette's curl is an eccentricity of about five per cent of a yarn diameter, nine micrometres on a yarn a sixth of a millimetre across. The moment about the other axis cannot be computed here at all: turning a free thread's end tangents out of the fabric makes its flat configuration unstable, and the solver finds a shape that has left the fabric altogether. That refusal is checked rather than described, because both signs of curvature find the same wrong shape and their difference comes back at zero — which is the answer the right shape gives too.

The threads are allowed to pass through one another. Nothing in the energy knows that two courses are solid, so the extension ceiling the model reports — a little over three hundred per cent, where the yarn in a loop finally runs out of ways to be anywhere but straight — sits roughly three times beyond the strain at which a real jersey jams against itself. It is a ceiling on the yarn, not a prediction about a fabric, and it is used only as one.

The relaxed knit is not where the model puts it, and that is a result rather than an excuse. Differentiating the loop's energy along the fabric gives forces of tens of newtons a metre pointing downhill in both directions at once, with no interior minimum anywhere — and the three relaxation states everybody measures hold more bending energy at each successive stage, not less. A model contradicted in sign is wrong about the mechanism, and the mechanism it is missing is set: the yarn's stress-free shape is the loop, not the straight line the elastica measures against. What no measurement available here can do is say how much of the loop is set, because the fraction cancels out of every observable the collection has.

Friction is what holds a relaxed knit, and it is not enough in one direction. Because a loop's height is the course spacing plus one yarn diameter, the force lengthening the wales is identically the force pressing the yarns together at their interlacings. The contact force cancels, the balance reduces to a condition on the friction coefficient alone, and the condition is that it must exceed a half. No fibre in this collection reaches it; textile coefficients run from 0.2 to 0.4. Across the courses the same balance has a margin of nearly five. The exactness is the point — it is a statement no construction can escape by being tighter or looser, and it says the shape of a relaxed knit is held by something the model does not contain.

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.