A balanced twill need not have two equal faces

Every weaver’s first distinction between twills is the one the notation states: 2/2 is balanced, 3/1 is warp-faced. A balanced twill has as much warp as weft on its face, and the natural inference is that its two sides are therefore alike. A 3/1 cloth is warpy on one side and weft-faced on the other, so its back is plainly a different fabric. A 2/2 twill looks the same either way, and every balanced twill a beginner meets does too.

That inference runs out at eight ends. The 2/1/2/3 twill is balanced, four cells over and four under on every pick. Its face shows warp floats of two and two. Its back shows warp floats of three and one. They are two cloths that happen to share a ratio.

To say how often that happens, the motions of a cloth have to be got straight, and that turns out to correct something the count of twills took for granted.

Three motions, and which one does what #

A twill is one pick’s pattern, a cyclic word of over and under, moved along one end for each following pick. Three things can be done to a finished piece, and each does something different to the word.

Turn it end for end. Spin the piece half a turn on the table without lifting it. The first pick becomes the last and the first end the last, so the word reads backwards. The diagonal keeps its hand, though, since a half turn in the plane changes no handedness, and warp stays warp and the same face stays up. Nothing has happened to the cloth, and yet its word has been reversed, so a word and its reverse are the same fabric.

Mirror it. This is the only motion that is not physical. It keeps the word and changes the sign of the shift, which is the difference between a Z twill and an S twill. The word does not see it at all.

Turn it over. Every cell changes face, since what was over is seen from behind as under; and the ends are read from the other side, so the word reads backwards; and the diagonal changes hand. So the back of word $w$ is the reverse of its complement, with the shift’s sign changed.

Only the last can change what a viewer of a face is looking at. Reversibility is the question of what that third motion does, once the first has been taken out.

Thirty-four, twenty-eight, seventeen, and twenty-one #

Counting with these three motions in hand gives a set of numbers, and the one the earlier essay published is not among them.

On eight ends there are 34 cyclic words of over and under, leaving out the two that never change face. Taking out end for end merges six pairs and leaves 28 cloths. Taking out the turn-over as well leaves 17, when a warp-faced cloth and its weft-faced back count as one.

How many twills a repeat admits reported twenty-one, and described its second operation as the mirror image. Enumerating its operations confirms its number and changes its description. Reading a run list backwards is not the mirror. A run list alternates over and under, so reversing it swaps which runs are over: 3/1 reversed is 1/3, which is the same twill’s back. So the operation that earlier count quotiented by is the turn-over, and the end-for-end turn was not among its operations. That is why twenty-one sits between 28 and 17: it merges each cloth with its back, and it keeps a word and its reverse apart, which are one piece of cloth given a half turn.

The twenty-one is a correct count of what it says it counts. It is not the number of different fabrics on eight ends, which is 28, or 17 if a fabric’s two sides are held to be one. The sentence in the earlier essay that identifies reversal with the diagonal’s direction is wrong. The direction of a twill’s diagonal is not in its word, and it is multiplied in afterwards. Every other conclusion of that essay survives, since its argument that a catalogue must quotient by something was the right argument, applied with one label on the wrong operation.

Which balanced twills are the same on both sides #

A twill is balanced when its word has as many overs as unders. It is reversible when its back, with the end-for-end turn taken out, is the same word: its complement appears among its rotations or among the rotations of its reverse. Reversibility is stronger, as a complement with a different number of overs could never match, and how much stronger is an enumeration.

No odd repeat has a balanced twill at all: an odd number of cells cannot be split evenly. At four ends both balanced twills are reversible, and at six all three are. At eight the first two exceptions appear, and they are each other’s backs: 2/1/2/3 and 3/2/1/2. Then the share falls: ten of sixteen at ten ends, twenty of fifty at twelve, thirty-seven of 133 at fourteen, seventy-four of 440 at sixteen, and 143 of 1,387 at eighteen, about one in ten.

These counts take in words that repeat within themselves, so 2/2/2/2 appears at eight ends though it is the four-end twill written twice, and the eight-end plain weave is there as well. Of the six reversible words at eight ends, then, two are short twills already seen, and the new ones are 4/4, 3/1/1/3, 2/1/1/1/1/2 and 1/1/2/1/1/2. Leaving the repeats out would thin both columns and would not change the falling share.

So the conflation has been harmless for a reason. A trade whose ordinary twills stay at four and six ends has never met a counterexample, and the 2/2 in every textbook is at the short end of a table whose long end says the opposite.

Half a word fixes a reversible twill #

The fall has a plain cause. A reversible word is one whose complement is reachable by a rotation or by a rotation and reversal, and either route tells half the word what the other half must be.

In the first kind, a rotation by half the word, the second half is the complement of the first: 1011 followed by 0100, which is the 1/1/2/1/1/2 at eight ends. In the second, the word read backwards is its own complement: the right half undoes the left in reverse order, which is what 3/1/1/3 does. Both kinds together account for every reversible twill the enumeration found, and they overlap heavily at short repeats: at eight ends all six reversible words are the second kind and four are also the first. At twelve, seven are the first kind and nineteen the second, with six that are both, which is the twenty.

Either way, half the word is free and the rest is forced, so the number of reversible words grows as a power of two in half the repeat. The balanced ones have only the one constraint, and the count of words with exactly half their cells over grows as the binomial coefficient, which is close to two to the whole repeat. A share that is one exponential over another falls fast, and it falls from the first repeat that gives it room.

That is why balance cannot stand in for reversibility. A balance is one equation on a word, and a reversal is half a word’s worth of them. Most balanced twills satisfy the one equation and no others.

What differs between two balanced faces #

Both faces of any twill interlace at the same rate. Every change of face on a pick is a change of face seen from behind, so the interlacings, and through them the firmness, are the same on both sides of any cloth at all. So is its cover, its weight and how it frays. Every property that is about the threads holding each other cannot tell a face from a back.

What can tell them is whatever reads the face’s floats. A float on the face of the 2/1/2/3 is at most two, and on its back at most three. The float decides how far a thread runs unsupported on a surface, so the two faces differ in how they take abrasion, in where a snag can catch, and in how much of the surface is one straight thread for the light to read. A designer keeping to a float limit of two would accept this cloth from its face and reject it from its back.

And it can be made harder to see. At twelve ends a balanced twill can show the same floats on both faces, in a different order, and still not be reversible.

Six of the fifty balanced twills at twelve ends are like this, twenty of 133 at fourteen, eighty-two of 440 at sixteen, and 270 of 1,387 at eighteen. Comparing a face’s float lengths with a back’s is therefore not a test for reversibility. It is a necessary condition, and from twelve ends up not a sufficient one. The distinction between the two sides of those cloths is entirely in how the floats are ordered along the pick, which is a difference of texture and diagonal, with no number in the notation’s usual summary to show it.

No twill is reversible in a real yarn #

Everything so far has been about the draft, and the draft leaves out the fact with the largest consequence.

Turning a cloth over changes the diagonal’s hand. A Z twill’s back is an S twill, always, for every word, as the third motion shows. A yarn’s twist does not change. A helix is a right-handed or left-handed object when seen from any direction at all, so a Z-twist yarn is a Z-twist yarn from behind. So the argument about twill direction has a consequence nobody needs a matrix for: a yarn whose surface fibres lie across the diagonal on one face lies along it on the other.

So the 2/2 twill, the most reversible in the whole table, sets its warp against its diagonal one way on the face and the other way on the back, and the same for its weft. Its two sides can read differently in the light with an identical word, which is one reason a cloth’s right side is often found by looking rather than by reading its draft. Reversibility in a matrix is a property of the word. The thing a viewer calls a reversible cloth needs the word and a yarn, and the yarn’s hand breaks the symmetry for every twill. Only a yarn with no net hand can leave it alone: a cloth with alternate Z and S ends, or a filament yarn with no twist in it.

That makes the balanced-and-reversible cloth of the textbook one a draft can draw and a twisted yarn cannot quite weave. A plain weave escapes, as its diagonal has no hand to change: the word 1/1 is its own back and its own mirror.

The other two-faced cloths #

This is not the only place a fabric’s two sides have turned out unalike for a reason in its construction. A jersey has two surfaces because loop heads and loop legs present different crowns, and there the difference is built into the stitch. In a twill the difference is optional, and which twills opted out is what the enumeration counts. A broken or herringbone twill reverses its diagonal partway across, so that its two sides have the same mix of hands and the yarn-twist asymmetry is averaged, stripe by stripe. That is a second reason to weave one, beside the figure it draws.

Plain, twill and satin sets the three foundation weaves against one trade, a thread’s changes of face against its run. The back of a warp sateen is a weft sateen, and a sateen’s word has one over in it, so no balanced word lives there at all. So twills are the one foundation family where a designer has a real choice about the two faces, and it is a choice most of the family’s names do not record.

How it was counted #

A twill’s pick is every binary word of length $n$ that is not all one value, over or under, with step one between picks. The word is rotated for where it starts, reversed for the end-for-end turn, and complemented and reversed for the turn-over, and the words are grouped by the least string each of those groups produces. All of that is walked in full, at every even repeat to eighteen ends and every count of motions to twelve.

Required of it, not shown: that the group the earlier essay used reproduce its count at every repeat from four to twelve, including its twenty-one; that each added motion only merge classes; that no odd repeat have a balanced twill; that every reversible twill show one list of floats on both faces; that all balanced twills be reversible up to six ends and that 2/1/2/3 be a counterexample at eight.

Where the word stops describing the cloth #

The cloth is square and of one yarn here. A warp and a weft are rarely the same thread at the same sett, and the same draft in a warp-set cloth shows more warp on both faces in every way that matters to a viewer, so a balanced word is not a balanced surface. Turning the piece a quarter turn, which would swap warp and weft, is not counted as a motion for the same reason: a warp is not a weft in a real cloth.

A word is not a texture. The enumeration separates cloths and does not say how far apart two are. At sixteen ends, two non-reversible faces may differ in one float’s position and nobody may see it, and nothing here measures that.

The twill’s step is one. Words at other steps are other weaves. A satin is a step word, and the arithmetic of what its back does is a different calculation, already made in the essays on satins.

And the yarn’s hand is qualitative here. That one face’s fibres lie along the diagonal and the other’s across it is geometry; how much that changes what a viewer sees is a question about light that no matrix asks.

Who worked out which part #

That a 3/1 is warp-faced with a weft-faced back, that a 2/2 looks alike on both sides, and that twist and twill direction interact are all old trade knowledge. Counting necklaces and bracelets under a group is Burnside’s lemma, and binary necklaces are thoroughly tabulated in combinatorics.

What is added here is the sorting of a cloth’s real motions: which of them reverses a word, which changes a hand, which complements. Also the correction of how the earlier count described itself; the exact separation of balanced from reversible, with the first counterexample at eight ends and the first with equal float lists at twelve; and the observation that hand and twist make every twill unlike itself in a real yarn.

Still open: how much further the two faces of a reversible twill sit apart in the light #

The draft says a 2/2 twill’s two faces differ only in a hand, and the yarn says that hand matters. How much it matters is a question for the same reflection model that priced a satin’s shine. A twill’s line stands out when its yarn’s surface fibres run across it, and a calculation of that kind would take a fibre’s tilt on the crown of a float, set it against the line’s direction for each face, and give a contrast for each.

The prediction with a number in it would be the ratio of the two faces’ line contrast as a function of a yarn’s twist angle. It should run to one at no twist and rise from there. If that ratio is large at an ordinary twist factor, then the choice of a Z or an S twill in a singles cloth is doing most of what a right side is. If it stays small, the visible right side of a balanced cloth comes from finishing and not from the draft at all.