A cord's height has a ceiling and its width has none

A warp rib is bought for its cord. The two rungs below took the construction apart as a matrix and as a bundle of threads, and neither of them has a shape in it — the cord is the one thing a reader of the cloth actually sees, and neither rung can say how big it is.

It is a wave, and it has two dimensions.

The height is the weft’s crimp, and it has a ceiling #

The cord’s crests are the pick groups, so the cord’s height is how far a pick group stands above the mid-plane — which is the weft’s crimp amplitude, and Peirce’s geometry gives it.

That geometry has one condition at its centre and the condition is the ceiling: the two threads’ amplitudes must add to the sum of their diameters, h₁ + h₂ = d₁ + d₂. The two threads are wrapped round each other and between them they occupy exactly the cloth’s own thickness.

So a cord can never stand higher than the two yarn diameters together. At a quarter-millimetre warp and a quarter-millimetre weft that is 500 µm, and no rib, at any doubling, at any sett, exceeds it.

What a rib does is claim a larger share of that ceiling. The warp bends once every k picks rather than at every one, so it lies straighter, its own amplitude falls and the weft’s rises to make up the sum:

A plain weave splits the ceiling evenly; a rib gives the weft the larger share. That is the whole of what the doubling does to the height, and it is a diminishing return: each further doubling takes a smaller bite of what is left.

The pitch has no ceiling at all #

The cord’s width is the other dimension and there is nothing bounding it.

A cord is one group of picks and its pitch is k pick spacings — and k is a design choice with no geometric limit in it. At a doubling of two the pitch is 0.75 mm, at four it is 1.25, at six 1.75, and at twenty it would be nearly six.

So the two dimensions grow at completely different rates. The height rises by 29 per cent between a doubling of two and one of six, and the pitch rises by 133 per cent.

The consequence is the finding and it is the opposite of the intent: a bolder rib is a flatter one.

A designer who wants a more pronounced cord and reaches for a larger doubling gets a cord that is taller, wider, and less pronounced — because what the eye reads as pronounced is the slope of the surface, and the slope is the aspect.

Which is why a repp is made with two warps #

The trade’s answer to this has been in use for two centuries and the arithmetic above says why it works.

A repp — the upholstery cloth, not the light shirting — is a warp rib made with two warps: a fine one and a coarse one, alternating, both weaving the same rib. The coarse ends do the covering and the fine ones do the binding.

That moves the ceiling. The height cap is d₁ + d₂, and it is the warp’s diameter that enters as d₁ — so a coarser warp raises the ceiling directly. Doubling the warp’s diameter from 250 to 500 µm takes the cap from 500 to 750 µm, which is a fifty per cent taller cord with no change to the doubling and therefore no change to the pitch.

So the two levers are genuinely separate and only one of them is in the weave. The doubling sets the pitch and moves the height slightly; the yarn sets the ceiling and therefore the height. A designer wanting a bold cord should change the yarn, and a designer wanting a coarse one should change the doubling.

That is a different recommendation from the one the flattening result alone would give, and it is the useful one: not “do not use a large doubling” but “a large doubling is the wrong lever for boldness”.

The warp goes flat, which is the cost nobody quotes #

The height’s arithmetic has a mirror image and it is the part a weaver pays for.

The ceiling is shared, so every micrometre the cord gains is a micrometre the warp gives up. Between a doubling of two and one of six the warp’s own amplitude falls from 196 µm to 109, and its crimp falls with it — from 8.7 per cent to 2.5.

A warp with less crimp has less to give. A cloth extends by moving its crimp, so the warp direction’s extension is very nearly its crimp, and a 6/6 rib extends warpwise by a third of what a 2/2 does before the fibres take the load.

That is a real handling consequence and it points the same way as the flattening: a heavily doubled rib is a stiffer, less forgiving cloth in the direction the cord runs across. It is also why a rib cannot be set square — the two systems have parted company in crimp as well as in sett, and both partings widen with the doubling.

And it compounds with the beam. A warp at 2.5 per cent crimp and a weft at 2.5 per cent are the same number here because the geometry is symmetric in the two, but a repp woven with two warps has them at different crimps, and two warps at different crimps need two beams — which is exactly the construction the boldness argument recommends. So the recommendation has a price attached and the price is a beam.

Reading the doubling off a finished cloth #

The arithmetic runs the other way too, and it gives a small piece of forensics.

A finished rib shows two things anybody can measure with a rule and a glass: the cord pitch in millimetres and the pick density. Their product is the doubling, exactly — k = pitch × picks per unit length — because the pitch is k pick spacings by definition.

So a cloth’s doubling is recoverable without unweaving anything, which is not true of most constructions. And the height is recoverable too, from a thickness gauge: the cord’s height is the weft’s amplitude, the two amplitudes sum to the cloth’s own thickness, so measuring the thickness and the pick density between them gives the warp’s share.

Which makes the rib one of the few weaves whose full geometry can be read off the outside. Most of what this collection computes needs the draft; here the draft is recoverable from the cloth, because the construction has exactly one parameter and it is visible.

That is worth a sentence because it is the practical use of the whole essay. A buyer holding two repps can tell which has the bolder cord by measuring the aspect, and can tell whether the difference was bought with the doubling or with the yarn by measuring the pitch — and those are two different purchases with two different prices.

What the aspect is doing that the height is not #

The claim that the eye reads the slope rather than the height is worth being explicit about, because it is the step that turns a geometry into a design statement.

A surface’s visible relief under raking light is set by the angle its facets make with the light, and for a wave of height h and pitch p that angle goes as h/p. A wave twice as tall and four times as wide is half as steep and catches less light, however much bigger it is.

So a cord’s boldness under a directional light falls as the doubling rises, even though the cord itself is growing. Under diffuse light the height is what matters and the ordering reverses — a 6/6 rib really is thicker to the hand and reads bolder in a photograph taken flat.

That is a genuine dependence on lighting and it is the sort of thing the trade knows without stating: a repp is specified for upholstery, which is seen under room light at an angle, and the doublings used are small — two and three, rarely four. A figure shows by its shine rather than by its step is the collection’s own version of the same distinction, and it points the same way.

What a weft rib does, which is the same thing turned #

Everything above is a warp rib — the picks grouped, the cord running across the piece. A weft rib groups the ends instead and the whole argument transposes, with one asymmetry that does not.

The geometry transposes exactly. Peirce’s closure condition is symmetric in the two systems, so a weft rib’s cord height is the warp’s amplitude, capped at the same d₁ + d₂, and its pitch is k end spacings. Every number above holds with the two directions exchanged.

The setting does not transpose. A rib’s two densest setts stand at (k + 1) to 2k, and the rung this ladder starts from shows that the grouped system is the one that may be set densely. So a warp rib’s pitch is set by a weft sett that rises with the doubling, and a weft rib’s pitch is set by a warp sett — and warp setts are chosen for the beam and the reed rather than for the design.

And the loom does not transpose at all. Ends per centimetre is a reed choice and is coarse-grained: reeds come in stock dents per inch and the ends per dent is a small integer, so the achievable setts are a discrete set. Picks per centimetre is a change-wheel choice and is nearly continuous.

So a warp rib’s cord pitch can be tuned finely and a weft rib’s cannot, which is a purely mechanical asymmetry between two constructions that are geometrically identical. It is also why warp ribs are far commoner: the cord that runs across the cloth is the one whose scale a weaver can actually adjust.

What was counted, and how #

The geometry is Peirce’s, unchanged, and the two amplitudes come from the same solver every crimp on this site uses. What is supplied here is the bending pitch: a warp end in a k-fold rib turns once every k picks, so the spacing Peirce is handed in that direction is k times the pick spacing, which is the one modelling step and is the same one the relief weaves use.

The setts come from stripe.js’s own ribSetting, so the pick spacing is the densest that weave may actually be set at rather than a number chosen to make the arithmetic come out. That matters here: a larger doubling permits a denser weft, so the pitch grows more slowly than k alone would suggest — 2 to 6 in doubling gives 0.75 to 1.75 mm rather than 0.75 to 2.25.

The ceiling is asserted rather than quoted. Every cord computed must stand lower than the two diameters together, which is the closure condition read as an inequality and is what would catch a solver returning a state that does not close.

And the three trends are asserted separately: the height rises, the pitch rises, the aspect falls. A version asserting only that the cord gets bigger would have been satisfied by its getting bolder, which is the opposite of the finding.

The hopsack has no cord at all, and the arithmetic says why #

Grouping both systems is the third member of the family and it is the one that produces no cord, which the geometry above explains rather than merely records.

A cord exists because one system’s crimp is larger than the other’s — the crests of the higher-amplitude system are the ridges. A hopsack groups both systems equally, so the two amplitudes are equal, exactly as they are in the plain weave it is derived from, and the surface has no preferred direction of relief.

What a hopsack has instead is a cell: a square of k by k threads, alternately warp-faced and weft-faced, which is a checkerboard rather than a wave. Its relief is the difference between a warp-faced patch and a weft-faced one, which for equal yarns is nothing at all.

So the three doublings give three surfaces and they are not three degrees of one thing:

• a warp rib, a wave across the cloth, with a height capped at the two diameters;
• a weft rib, the same wave along it;
• a hopsack, no wave, and a coarse flat checkerboard whose scale is k threads.

That is a cleaner separation than the drafts suggest. On point paper the three look like three members of a family, and six of nine matrix measures cannot even tell a hopsack from a twill; in the cloth two of them have a directional relief and one has none, for a reason that is one line of Peirce’s geometry.

And it says what a hopsack is bought for, which is not relief: it is cover at low weight, from the group’s own arithmetic, with the softness that comes with it. The rib buys relief and gives up isotropy; the hopsack buys cover and gives up stiffness. Two constructions from one operation, and they are bought for different things.

Where the model stops #

Peirce’s threads are circles. A yarn in a cloth flattens where it is gripped, and a cord is exactly where the gripping is hardest — the pick group is squeezed between two warp turns. Peirce against the racetrack is the collection’s own account of what flattening does to these numbers, and it lowers the height and raises the width, so the flattening result is if anything understated.

The pick group is treated as k separate picks at their own spacing and it is not quite: the picks of a group are beaten up together and lie touching, so a group’s own width is closer to k diameters than to k spacings. That narrows the pitch and would raise the aspect, in the opposite direction from the flattening — by how much is not computed and it is the largest single uncertainty here.

Nothing here is about finishing. A calender flattens the cord, a milling shrinks the cloth and raises it, and both change the aspect by amounts that depend on the machine. The section is a loom-state geometry.

And the light is asserted rather than modelled. That relief reads as h/p is a statement about facet angles that this collection has no reflectance model behind. What can be said is that the two lighting cases order the doublings oppositely, which is a qualitative claim that does not need one.

Who found it, and when #

The rib weaves are old and their setting is trade knowledge; the two-warp repp is a named construction with a literature of its own.

Peirce’s geometry is from 1937 and its closure condition is the first thing any account of it states. That the condition is a ceiling on a cord’s height does not appear to have been said, and the reason is probably that Peirce’s model is used to compute crimp and cover, which are what a mill measures, rather than relief, which nobody measures at all.

The flattening — that a larger doubling gives a taller and less pronounced cord — is this collection’s, and it is the kind of result worth having because it contradicts the obvious move. The lever that makes a rib coarser is not the lever that makes it bolder, and the two have been conflated for as long as the construction has been described by its doubling alone.

Where the ladder goes next #

Every rung so far has grouped the two systems by the same amount or by nothing. There is no reason they should agree: grouping the ends by two and the picks by three is a perfectly good cloth, it has a name in the trade, and its unit area, its setts and its cord are all different from anything the symmetric family produces.