Textile Structure
how cloth is put together
Pattern and colour

The diagonal is not a thread

A twill's diagonal is the most looked-at feature in weaving and the most misdescribed. No thread in the cloth runs diagonally. What runs diagonally is a pattern of which thread is on top, which is a different sort of object.
13 min read 6 figures Measured, not claimedA weave is a matrix

Look at a pair of jeans in good light and there is an unmistakable diagonal running up the cloth. It is the first thing anybody notices about denim, it is what the word twill means to most people, and the natural inference from it is wrong.

No thread in that cloth runs diagonally. The warp runs straight up and the weft runs straight across, exactly as in a plain weave. There is no third thread system and nothing is laid on the bias.

The 2/2 twillThe 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4
Fig. 1 A two-and-two twill over three repeats. The columns are warp and the rows are weft, both perfectly orthogonal. What runs diagonally is the boundary between the filled and empty regions — a pattern of which thread is on the face, not a thread.

What the diagonal actually is

A twill is a sequence of ups and downs, shifted by one end for each successive pick. That shift is the entire mechanism.

Pick one has warp up over ends one and two. Pick two has it up over two and three. Pick three, three and four. Follow the region where warp is on the face and it moves one end across per pick — which draws a diagonal on the point paper and on the cloth.

The thing moving is the region, not any material. It is the same sort of object as the bright band that runs along a row of theatre lights switched on in sequence: something is travelling, and it is not any of the bulbs.

Why the mistake is so durable

Three reasons, and they compound.

The visual evidence is genuine. There really is a diagonal, and it really is a structural feature rather than a printed one. The instinct that it means something about the construction is correct; only the inference about threads is wrong.

The word “bias” is nearby. Cloth cut at forty-five degrees behaves quite differently, and the vocabulary of diagonal directions in fabric is already occupied by something that genuinely is about direction. The two get conflated.

Twill really does behave directionally. A twill cloth has a right and a wrong way up, panels cut from it must run the same way, and it wears differently along the diagonal. So the diagonal has real consequences, which makes a wrong explanation of it feel confirmed.

Following one thread

The quickest way to convince anybody is to make them follow a single thread, which a draft permits and a photograph does not.

Take a column of the draft — one warp end — and read down it. It is up, up, down, down, up, up, down, down: a simple alternation with a period of four. Take the next column and it is the same sequence shifted by one pick. Every column is the same sequence at a different phase.

Where the floats are in the 2/2 twillEvery longest run of warp on the face, marked. A float is smooth because nothing interrupts it, which is the same reason it snags: there is a length of thread lying on the surface with nothing holding it down.2/2 twillwarp floats of 2weft floats of 216 markedruns measured round the repeatfloat 2
Fig. 2 The floats in a two-and-two twill, marked where they run. Every mark is vertical, because every warp float runs along a warp thread and warp threads run straight up. The diagonal in the picture is the arrangement of those vertical marks, not their direction.

That is the demonstration. The float marks run straight up the cloth. Their starting positions step across by one per pick, and the eye joins the starts into a line — but the marks themselves, which are actual lengths of actual thread, are vertical.

A reader who has followed that once will not make the mistake again, which is the argument for showing it rather than asserting it.

The angle is not forty-five degrees

A detail that gives the game away, once noticed.

If the diagonal were a thread it would run at whatever angle the geometry put it at. Since it is a shift, its angle depends on the ratio of the thread spacings: one end across per pick means the angle is set by how far apart the ends are compared with the picks.

In a cloth with equal ends and picks per inch, the diagonal is at forty-five degrees. In a warp-dense cloth like denim — many more ends than picks — the ends are closer together, so one end across is a shorter horizontal step and the diagonal is steeper than forty-five degrees. In a weft-dense cloth it is shallower.

Mills use this deliberately. A steep twill is a warp-dense cloth, sometimes with a shift of two ends per pick to steepen it further; gabardine is the standard example, at around sixty-three degrees. A reclining twill is the reverse.

So the twill angle is not a property of the weave at all. It is a property of the weave and the sett together, and the same draft gives different angles in different cloths.

Right hand and left hand

The shift can go either way, and the choice is not cosmetic.

A Z-twill runs from lower left to upper right; an S-twill the other way, named for the middle stroke of the letters. Denim is traditionally Z; some mills use S deliberately for a different surface.

The direction interacts with the twist direction of the yarn. If the twill runs the same way as the yarn’s twist, the surface fibres lie along the diagonal and the twill line is subdued; if it runs against, the fibres cross it and the line is pronounced. That is a genuine structural interaction between two different scales, and it is why the same draft in the same sett can look sharply lined in one yarn and softly in another.

Nothing about that is available from the matrix, which knows the shift direction and nothing about yarn twist.

The same mechanism at other angles

Because the diagonal is a shift, changing the shift changes the angle, and mills use several.

A shift of one end per pick gives the ordinary twill. A shift of two doubles the horizontal step and gives a shallower line, or a steeper one depending on which axis it is measured against. A shift that reverses at intervals gives herringbone, in which the diagonal zigzags — and nothing has changed except the sequence of shifts.

The 2/1 twillThe 2/1 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.2/1 twilllongest float 212 interlacings per repeat1 separable clothrepeat 3 × 3generated, then counted3×3
Fig. 3 A two-and-one twill, the smallest twill there is: three ends, three picks, and a diagonal produced by the same shift as any other. It is the structure of a great deal of lining and pocketing cloth.

A broken twill shifts without mirroring, which removes the continuous diagonal while keeping the float structure. Some denim is woven this way precisely to avoid the pronounced twill line, and the cloth is otherwise identical.

The list is worth having because it shows the diagonal is a parameter rather than a property. It can be steepened, reversed, broken or removed without changing what the threads are doing in any material sense — and a feature that can be removed while the structure stays put was never a feature of the structure.

What is structure and what is not

The general question this essay is an instance of: which visible features of a cloth are telling a reader about its construction?

The diagonal is structural — it is a real consequence of the shift, present in any colour.

A colour-and-weave pattern is not. Houndstooth is the same twill as a plain jacket cloth, and its pattern comes from the threading order. The visible face is a function of two independent inputs, and appearance alone cannot separate them.

Lustre is structural — it comes from float length, and it is available in any fibre.

A printed pattern is obviously not, and prints are frequently mistaken for weaves in photographs.

A raised or brushed surface is not, or rather it has destroyed the structural evidence: flannel is a plain or twill cloth whose surface has been deliberately disturbed, and its handle has almost nothing to do with its floats.

The practical rule that follows is the one weaving analysts use: do not read a weave from an appearance. Take a corner apart, or use a pick glass on a solid-coloured area, and count.

Wear, and the one place the diagonal is material

Having spent the essay insisting the diagonal is not a thread, it is worth conceding the case where it has a physical consequence beyond appearance.

A twill wears preferentially along the float direction. In denim, the warp floats are on the face, and the abrasion at a crease removes the indigo-dyed surface of those floats — revealing the undyed core of the yarn and the pale weft beneath. The resulting whiskering follows the crease rather than the diagonal, but the texture of the worn area follows the float structure.

Where the floats are in the 3/1 twillEvery longest run of warp on the face, marked. A float is smooth because nothing interrupts it, which is the same reason it snags: there is a length of thread lying on the surface with nothing holding it down.3/1 twillwarp floats of 3weft floats of 116 markedruns measured round the repeatfloat 3
Fig. 4 Denim’s floats: three warp on the face for every one weft. The abrasion that produces the pale creases of worn jeans acts on these lengths, and the way a pair of jeans fades is a direct record of where the warp floats lie.

There is also a genuine directional effect in cutting. A twill cloth has a right and a wrong way up, so all pattern pieces must run the same way or the diagonal reverses between panels and the garment looks wrong. That is a real constraint that costs cloth, and it exists because the diagonal is a genuine feature of the arrangement even though it is not a thread.

So the correction in this essay is narrow and worth stating precisely. The diagonal is real, it is structural, and it has consequences. What it is not is a direction that any material in the cloth runs in — and the bias, which genuinely is a direction, is a completely different thing that happens to be described with the same word in ordinary speech.

The same error elsewhere

The pattern of mistake — inferring a material direction from a visual direction — recurs, and it is worth recognising.

The moiré in two overlaid fabrics moves as they slide, and nothing in either fabric moves. The watered effect in moiré silk is a physical flattening rather than a weave. The apparent spiral on a barber’s pole climbs indefinitely while the paint goes round. In every case something is genuinely travelling and it is not the substance.

The general lesson is that a pattern can have a direction that no constituent has. That is not a subtlety about weaving; it is a fact about patterns, and cloth is simply the place where most people meet it without being told.

Naming a twill

The notation is worth knowing because it is compact and it encodes exactly the two things that matter.

A twill written 2/2 has two ends up and two down in the sequence; 3/1 has three up and one down. The first number is the warp on the face, the second the warp underneath — so 3/1 is warp-faced and 1/3 is the same cloth from the other side.

Add a direction: Z for a diagonal running lower-left to upper-right, S for the other way. And add the shift where it is not one: a 2/2 twill, 2 step moves two ends per pick and gives a steeper line.

The 2/2 twillThe 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4
Fig. 5 A two-and-two twill: two filled, two empty, shifted one end per pick. The whole of the notation is visible in the first row and the first shift, and everything else is repetition.

Four characters therefore specify the structure completely, which is why weaving specifications are so terse. What they do not specify is the angle, because the angle depends on the sett as well as the shift — and a specification that gave the angle instead of the shift would be describing a cloth rather than a weave.

That distinction between the weave and the cloth runs through this whole site. A weave is a rule; a cloth is that rule in a particular yarn at a particular sett with a particular finish, and most of the interesting numbers belong to the cloth rather than to the rule.

What the eye is good at

It would be unfair to leave the impression that looking is useless. The eye is extremely good at some of this and it is worth knowing which.

Finding a diagonal, judging its angle, spotting a repeat, seeing that two areas differ — all of these are things a person does instantly and a computation does laboriously.

What the eye cannot do is the things this site keeps computing: whether the cloth hangs together, how long the floats are round the repeat, how many interlacings there are, what the cover is. Those have no visual signature, or a signature that a plausible-looking wrong answer shares.

So the division of labour is clear enough. Look for the things that are visible, compute the things that are not, and be careful about the ones where looking gives a confident answer that happens to be wrong. The twill diagonal is the third kind.

Two threads, checked

The claim of this essay is a negative one — that no thread runs diagonally — and negative claims are the ones this site normally insists on computing rather than asserting. So it is worth saying how it is established.

A draft is a matrix. Its columns are warp ends and its rows are weft picks, by construction, and there is no third index. So the question “does any thread run diagonally” is not a question about the cloth at all: the representation has only two thread directions, and a diagonal thread is not expressible in it.

The 2/2 twillThe 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.2/2 twilllongest float 216 interlacings per repeat1 separable clothrepeat 4 × 4generated, then counted4×4
Fig. 6 Three repeats of the weave in question. Every filled square belongs to exactly one column and one row, and there is nowhere in the representation for a third direction to live.

What the figure shows is that the diagonal is a feature of the arrangement of filled squares, and the float map shows that every actual length of thread on the face runs vertically. Between them that is as close to a demonstration as a structural question gets.

The reason the mistake survives anyway is that it is not a mistake about the cloth. It is a mistake about what kind of thing a pattern is, and no amount of looking at a cloth corrects it — which is why it belongs on a site that keeps insisting the visible and the structural are different questions.

Where the ladder goes next

The weaves themselves are plain, twill and satin, and the other great source of misread appearance is colour and weave.

The framework in which a direction is a symmetry statement is weaves as plane patterns.

What the pictures here cannot show. A draft drawn on point paper shows the shift and therefore the diagonal, at an angle set by the square cells of the paper rather than by any cloth. The real angle depends on the sett, and no drawing on this page is at a real cloth’s proportions.

Where this goes next

plainfloat 1 · 8 interlacingsfirmness 1.002/2 twillfloat 2 · 16 interlacingsfirmness 0.505-end satinfloat 4 · 20 interlacingsfirmness 0.40a filled square means the warp is on the facecounted from the matrices, not from the pictures Plain, twill and satin Three rules, and everything else in weaving is a variation on them. What separates the three is not appearance but one trade — how often a thread changes face against how far it runs when it does not. only the order the coloured threads are put in changesthe weave2/2 twillfour dark, four light▪▪▪▪▫▫▫▫two dark, two light▪▪▫▫the face colour computed, not designed2/2 twill Colour and weave Thread the same weave with a different order of coloured ends and a pattern appears that is nowhere in the draft. Houndstooth is an ordinary two-and-two twill, and nothing about its interlacement is unusual at all. weaveleave the faces aloneexchange themplain1616basket16162/2 twill16163/1 twill16none5-end satin20noneevery candidate operation tried against the repeat Weaves as plane patterns A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.