Colour and weave

Houndstooth is a two-and-two twill. Not a variant of one, not a special construction — the same interlacement as half the jackets ever made, with the ends and picks threaded four dark and four light.

That is worth sitting with, because it means the pattern a person sees in a cloth may have almost nothing to do with its structure.

What the eye sees #

At every intersection, the visible colour is the colour of whichever thread is on the face. Filled square: the warp’s colour. Empty square: the weft’s.

So the visible face is a function of two independent things — the draft, and the colour orders of warp and weft — and neither can be recovered from the appearance alone. Two quite different weaves in different colour orders can look identical, and the same weave in two colour orders can look like two different fabrics.

The computation is trivial and the consequence is not. It means that “what pattern is this cloth” and “what weave is this cloth” are different questions with different answers, and confusing them is the commonest error in reading a fabric.

Houndstooth, worked #

Take a 2/2 twill and thread four dark ends, four light, repeating. Do the same in the weft.

Where a dark warp end is on the face, the cloth shows dark. Where a light weft pick is on the face, it shows light. In the regions where dark warp meets dark weft, everything is dark whatever the weave does; where light meets light, everything is light. The interesting places are where dark meets light, and there the weave decides — and because the twill runs diagonally, the boundary between the dark and light blocks is broken into the characteristic points.

That is the whole of it. The pointed shape people recognise is the twill’s diagonal cutting across a colour boundary, and it is generated rather than designed.

Change the colour order to two-and-two and a quite different pattern appears from the identical weave. Change it to one-and-one and something different again — the classic sharp check of a “hound’s tooth” reduced to a fine pepper-and-salt.

Why this is a two-colour problem #

Once colours are threaded in, the visible face is a genuine two-coloured periodic plane pattern in the ordinary sense — unlike the draft itself, where the two states are physically different rather than interchangeable.

So the machinery applies without caveats: the visible pattern belongs to one of the two-colour plane groups, its symmetry can be computed, and a designer arranging colour orders is doing exactly what an ornamentalist does.

The repeat of the visible pattern is the least common multiple of the weave repeat and the colour repeat, which is why a four-end weave with an eight-end colour order gives an eight-end pattern, and why colour orders are usually chosen to relate simply to the weave.

Two orders, one weave #

Working an example through makes the mechanism concrete, and the useful example is the one where a small change in colour order gives a large change in appearance.

Take a 2/2 twill. Thread four dark ends and four light, and do the same in the weft, and houndstooth appears. Thread two and two instead and the pattern is a small even check with no points at all. The interlacement has not changed by a single square.

The reason is a relationship between two periods. The weave repeats every four ends and the colour every eight in the first case and every four in the second. When the two periods coincide, the colour blocks and the twill’s diagonal line up the same way in every repeat and a regular check results. When they do not, the diagonal meets the colour boundary at a different place in each repeat and the boundary is broken up.

So the visible repeat is the least common multiple of the two, and the character of the pattern depends on how the two periods relate rather than on either alone. That is why colour orders are chosen with the weave’s repeat in mind, and why a designer’s colour-and-weave sample book is organised by ratio rather than by colour.

Why plain weave is best at it #

Colour-and-weave effects are strongest in plain weave, which is not obvious and has a good reason.

The visible colour changes only where the face changes, so the number of opportunities for the colour pattern to express itself is the interlacing count. Plain weave interlaces at every intersection and therefore has the finest possible grid on which a colour pattern can act.

A satin interlaces once in five or eight, so most of its surface is warp whatever colour the weft is, and a colour order in the weft is largely invisible. Satins are made in solid colours for exactly this reason, and figured satins get their pattern from changing the weave rather than the colour.

That gives a neat inversion of the site’s usual theme. Almost everything about a fabric is decided by its structure — but what it looks like can be decided almost entirely by the colour order, and the structure’s role is to provide a grid fine enough for the colours to act on.

Log cabin, which is the cleanest demonstration #

If only one colour-and-weave effect were to be kept, it should be this one, because it isolates the point completely.

Weave plain cloth. Thread the warp one dark, one light, repeating — and after some number of ends, shift the phase so that the alternation starts on the other colour. Do the same in the weft.

What appears is a pattern of blocks that read alternately as vertical stripes and horizontal stripes. Every block is woven identically, in the same weave, with the same two colours in the same one-and-one order. The only difference between a vertical block and a horizontal one is whether the warp and weft alternations are in phase or out of it.

A person shown a log cabin cloth and asked what the weave is will describe two different weaves, and there is only one. That is the whole argument of this essay in a single fabric, which is why it has been a standard teaching example in weaving schools for a very long time.

It is also a good demonstration of a more general point about patterns: a visible feature can belong to a superposition rather than to either of its parts, and no amount of looking at the result separates them. The same is true of a twill’s diagonal, which belongs to the shift and not to any thread.

The classic effects #

Four are worth naming because they are everywhere and they are all the same mechanism.

Hound’s tooth — 2/2 twill, four and four. The one everybody knows.

Shepherd’s check — 2/2 twill, four and four in two colours with the twill running the same way, giving a simple check rather than the pointed tooth; the difference is which twill direction and how the colour blocks align.

Pepper and salt — plain weave, one and one. No pattern at all at a distance, a fine speckle close to, and a colour that is neither of the two threads.

Log cabin — plain weave, one and one in blocks that reverse, so that alternating blocks appear as vertical and horizontal stripes although every block is identically woven. It is the purest demonstration available that the appearance is not the structure.

Why the effect is computable #

Everything in this essay follows from one line, and it is worth writing it out because it makes the whole subject arithmetic.

The visible colour at pick $i$ and end $j$ is

$$C(i,j) = \begin{cases} W_{\text{warp}}[j \bmod P] & \text{if the draft has warp up}, \\ W_{\text{weft}}[i \bmod Q] & \text{otherwise,} \end{cases}$$

where $P$ and $Q$ are the two colour repeats. Three independent inputs — the draft, the warp order, the weft order — and one output.

That is how every figure on this page is generated. Nothing is drawn, nothing is designed, and the pattern that appears is not chosen by anybody; it is what the three inputs produce.

The computability has a practical use beyond figures. A mill designing a colour-and-weave range does not need to weave samples to see what the combinations look like: the appearance follows from the three inputs, and the sample is confirmation rather than discovery. That has been true since long before computers, since the calculation is one a person can do on point paper with two coloured pencils.

What this means for identifying a cloth #

The practical consequence is a caution.

Reading a weave from a photograph is unreliable, because the visible pattern is a superposition of two things and a photograph shows only their product. Identifying the weave requires either a solid-coloured area, or a magnifier and patience, or unravelling a corner.

That is why weaving analysis is done with a pick glass and a needle rather than by eye, and why fabric identification in a museum or a laboratory involves taking a cloth apart. The alternative is guessing from an appearance that was designed to be interesting rather than legible.

It is also, incidentally, why the diagonal of a twill is so persistently misunderstood. The visible diagonal is a feature of the pattern rather than of the threads, and a colour order can strengthen it, weaken it, or hide it entirely.

Where the effect is used #

Three places, and they are more different than they look.

Tailoring cloth. Almost the whole vocabulary of men’s suiting patterns — houndstooth, shepherd’s check, glen check, birdseye, nailhead — is colour-and-weave on a small number of twills. The patterns are woven rather than printed, so they survive wear and cannot be cut off-register, and they are cheap because nothing about the weaving is unusual.

Handweaving. Colour-and-weave is the standard way a handweaver gets a complicated-looking cloth from a simple threading, since changing the colour order costs nothing and changing the weave costs shafts. Log cabin and its relatives are beginners’ projects for exactly that reason.

Camouflage and dazzle. A pattern that is woven rather than printed cannot be worn off, and the mechanism scales down to thread level in a way a print does not. Fabrics designed to break up an outline at a distance exploit the optical mixing that colour-and-weave produces.

The common thread is that the effect gives pattern for free. The loom is doing nothing it was not already doing; only the order the threads went in has changed, and that is decided at warping rather than at weaving.

How many patterns one weave and one colour order give #

The phase result is demonstrated above on a plain weave and left as a demonstration. It is a count, and the count says how large a colour-and-weave sample book has to be.

The two free parameters are where the warp’s colour sequence starts and where the weft’s does. With a four-and-four order in both directions that is eight choices each, so

sixty-four combinations from one weave, one colour order and one pair of yarns.

They are not sixty-four different cloths. Shifting the whole fabric by four ends and four picks returns a two-and-two twill to itself — its repeat is four — and shifts both colour phases by four, which with an eight-end colour order exchanges dark for light in both systems. So the sixty-four fall into thirty-two pairs, each pair being one pattern and its own colour negative.

Thirty-two houndstooth-family patterns, from a draft that never changes and two cones of yarn. That is a fair description of a suiting range, and it explains why such ranges are so large for so little manufacturing variety: the mill is selling phases.

The plain weave case is the same count at its smallest. A one-and-one order gives two alignments each way, four combinations, falling into two pairs — stripes and checkerboard, each with its negative — which is exactly the pair the essay demonstrates and is the whole of what that construction can produce.

Which is why sample books are organised by ratio #

The essay notes that a designer’s book is arranged by the relation between the two periods rather than by colour, and the arithmetic says which relation.

The visible repeat is the least common multiple of the weave’s repeat n and the colour repeat P, in each direction independently. The pattern’s character is set by their greatest common divisor, because that is how often the colour boundary and the weave’s own phase come back into the same relationship.

For a four-end twill:

A colour repeat sharing a factor with the weave gives a compact pattern and one coprime with it gives a large one. Four and eight — the orders the trade actually uses — are the two that share the most with a four-end twill, so they give the smallest visible repeats and the most regular patterns.

And the coprime cases are the ones nobody uses. A five-end colour order on a four-end twill has a visible repeat of twenty ends and twenty picks, and every one of its twenty repeats meets the twill’s diagonal at a different phase — which produces a pattern with no discernible unit at ordinary scales, and reads as a texture rather than as a check.

So the trade’s preference for four and eight is not habit. It is the choice that keeps the visible repeat small, and a small repeat is what makes a check read as a check.

That also gives the design space a shape. A colour-and-weave range is a search over gcd and phase, and the two do different things: the gcd sets the scale and the phase sets the figure. A mill wanting a new pattern at the same scale changes the phase and gets thirty-one more; one wanting a different scale changes the ratio and gets a different family.

What the model does not include #

Two limits.

Colour is treated as discrete. Each thread is one colour and the face is one thread’s colour. Real yarn is a blend, a marl, a heather or a space-dye, and a fabric’s apparent colour at a distance is an optical mixture of many threads rather than a mosaic of individual ones.

No account is taken of scale. At a distance the eye mixes adjacent threads and a pepper-and-salt reads as a solid grey. Whether a given colour-and-weave effect reads as a pattern or as a colour depends entirely on the viewing distance and the thread fineness, and nothing here computes that.

What the weave still decides #

It would be easy to leave this essay thinking the structure does not matter, and that is the wrong conclusion.

The colour order decides the pattern. The weave still decides everything else: how the cloth handles, how it drapes, how densely it can be set, how it wears, whether it snags. A houndstooth in a twill and the same colour order in a satin would look related and behave nothing alike.

So the honest summary is a division of labour. The weave decides what the cloth is; the colour order decides what it looks like. They are close to independent, which is unusual and useful, and it is why a mill can offer one construction in a hundred patterns at no extra cost.

It also means that reading a fabric requires separating them, which is why weaving analysis is done with a pick glass on a solid area rather than from a photograph of the pattern.

Why the phase matters as much as the order #

One further thing the effect does, which is easy to demonstrate and hard to believe until it is.

Take plain weave and a one-and-one colour order — dark, light, dark, light in both directions. Threaded one way the surface comes out as stripes. Shift the warp order by a single end and the same weave in the same two colours comes out as a checkerboard.

Nothing about the interlacement has changed. Nothing about which yarns are used has changed. What changed is the phase between the warp’s colour sequence and the weft’s, and it produced two surfaces with nothing in common.

That is why colour-and-weave drafts specify the starting end as well as the order, and why a mill re-threading a warp has to get the phase right or produce a different cloth. It is also the sharpest available demonstration that the pattern is not in the weave: a property that can be changed by moving the starting point cannot be a property of the interlacement.

How much the surface gives away #

The effect described here has a sharper form, and it is worth stating because it turns a curiosity into a measurement.

At an intersection where the two crossing threads happen to be the same colour, the surface looks identical whichever thread is on top — so the weave leaves no trace there at all. Count those blind intersections and the loss of information follows immediately: with a one-and-one colour order over a four-by-four repeat, eight of the sixteen intersections are blind, so at most 256 surfaces exist and the 22,874 drafts collapse onto them at an average of eighty-nine apiece.

That is why the cloths in this tradition are named for their appearance. Houndstooth is an appearance, not a construction, and it could not have been named after its weave because several weaves give it.

Colour and weave as a two-colour problem works the enumeration through, including the result that no two-colour order blinds fewer than half the intersections and that turning the cloth over gives no further information at all.

Where the ladder goes next #

The symmetry framework this sits in is weaves as plane patterns, where the draft’s own two states are the complication.

The single most misread visible feature is the twill diagonal.

What the pictures here cannot show. These figures colour each intersection with the face thread’s colour, which is a mosaic rather than a fabric. Real yarn has hairiness, tonal variation and an apparent colour that changes with distance, and none of that is drawn.