How many twills a repeat admits

A twill is written as a sequence of runs: 2/2, 3/1, 2/1/1/2. Over that many, under that many, over that many again, until the sequence sums to the repeat. On eight ends there are sixty-four such sequences.

There are not sixty-four twills on eight ends. There are twenty-one. The other forty-three are the same cloths written down differently, and the two operations that identify them are worth naming precisely, because a catalogue that skips them is counting notations.

Where the sequences come from #

A twill sequence is a composition of the repeat — an ordered list of positive integers summing to it. The compositions of $n$ number $2^{n-1}$, by the standard argument: write $n$ as a row of units and choose, at each of the $n-1$ gaps between them, whether to cut.

Half of those compositions have an even number of parts and half an odd number. A twill’s runs alternate over, under, over, under, and the alternation has to close when the sequence wraps, so the number of parts must be even. That halves the count to $2^{n-2}$: sixteen on six ends, sixty-four on eight, a thousand and twenty-four on twelve.

That is the number of ways a twill on $n$ ends can be written. It is a count of strings, and nothing about it has yet touched cloth.

The two operations #

Two changes to the string leave the cloth alone.

Rotation by two. The sequence describes one pick and the draft is built by shifting it one end for each following pick. Starting the sequence at a different point in its own cycle produces the same cloth begun at a different pick, which is not a different fabric — a length of cloth has no first pick. The rotation must be by an even number of positions, because an odd rotation swaps the over-runs with the under-runs, and that is a different operation with a different meaning.

Reversal. Reading the sequence backwards gives the mirror image, which is the same cloth turned over or, equivalently, the same twill with its diagonal running the other way. Every quantity the matrix can produce is identical for the two, and what separates them in the cloth is the yarn’s own handedness rather than anything in the draft.

Together the two generate a dihedral action on the even-length compositions, and the twills are its orbits. Twenty-one of them on eight ends.

The size of the redundancy is worth a moment. A sequence with $2k$ parts has $k$ even rotations, and each of those can be read backwards, so an orbit contains at most $2k$ strings — and it contains fewer when the sequence has a symmetry of its own. A 2/2/2/2 twill is fixed by every rotation, so its orbit is a single string; a 1/2/3/2 twill is fixed by the reversal but not by any rotation, so its orbit has half the maximum size. That is why the redundancy grows with the repeat rather than staying at some constant: longer sequences have more parts, more parts mean more rotations, and the symmetric sequences that resist them become a smaller share of the whole.

What is not quotiented out, and why #

Two things that might look like they should be identified are not, and the reasons are worth separating.

A twill and its inverse are different cloths. A 3/1 twill and a 1/3 twill have identical float lengths, identical interlacing counts and identical plane groups; the only difference is which system is on the face. That is not a symmetry of the sequence — reversing 3/1 gives 1/3 read as a string, but the operation that produces the inverse cloth is exchanging the over-runs with the under-runs, which is an odd rotation. It is excluded deliberately: a warp-faced denim and a weft-faced one are different fabrics with different wear, different balance and different costs, and a catalogue that merged them would be useless to anybody buying cloth.

Twills on different repeats are never identified. A 2/2 twill on four ends and a 2/2/2/2 twill on eight are the same fabric — the second is the first written twice — and the enumeration counts them separately because they sit in different rows. That is a real over-count across rows, and it is left in because each row answers the question how many twills does a loom with this repeat admit, which is the question a designer asks, and the answer to it does include the ones that happen to be periodic.

The second is a genuine limitation and it inflates the larger repeats slightly. On twelve ends the twills that are really twills on six, four, three or two ends are all counted again.

It is a limitation with a known remedy, which is worth naming rather than leaving as a caveat. A sequence that is a shorter sequence repeated is a periodic one, and the count of aperiodic sequences of a given length is recovered from the total by Möbius inversion over the divisors — the same manoeuvre that turns the count of all necklaces into the count of primitive ones. Applying it here would give a second column: twills that are genuinely new at this repeat, against twills the repeat merely admits. Both are meaningful and they answer different questions, and the one computed is the one a designer asks.

The eight twills on six ends, listed #

Six ends is small enough to write out and large enough to be interesting, and the list settles several of the questions above by inspection.

The sixteen sequences reduce to eight classes: 3/3, 2/4, 1/5, 1/1/2/2, 1/2/2/1, 1/1/1/3, 1/2/1/2 and 1/1/1/1/1/1. Two of them are worth remarks.

3/3 and 1/1/1/1/1/1 are each alone in their orbit. The first is fixed by every even rotation because all its rotations are itself; the second likewise. They are the two most symmetric sequences on six ends, and a symmetric sequence has a small orbit, which is Burnside’s observation doing the work it always does.

1/1/1/3 has the largest orbit, four strings. Its three even rotations are all different and reversal produces nothing new, because the sequence read backwards is one of its own rotations.

And the last entry, six runs of one, is the plain weave. It is a twill by every definition used here — a sequence of alternating runs, stepped one end per pick — and its step happens to reproduce the alternation exactly, so the diagonal has nowhere to go. A plain weave is the degenerate twill, and the enumeration finds it without being told, which is a mild but real check that the generation is not quietly excluding cases.

The second column, computed #

The over-count across rows is named above as a limitation with a known remedy, and the remedy is cheap enough to run here rather than to describe. A twill that is a shorter twill written out is periodic, and separating the periodic classes from the rest gives the column a designer would actually want: twills that are genuinely new at this repeat.

A sequence is periodic when it is a shorter sequence repeated. Repeating a composition of n/k some k times gives a composition of n, and the parts count multiplies by k — so a periodic even-length sequence on n ends comes from an even-length sequence on a divisor of n, or from an odd-length one repeated an even number of times.

Running that out at the three repeats this essay works in:

The classes being subtracted are exactly the ones a reader would name. On four ends it is the alternating sequence, which is the plain weave written twice. On six it is that and the 1/2/1/2, which is a 2/1 twill written twice — and both are in the list of eight this essay sets out, sitting at its end. On eight it is the plain weave again, the 2/2/2/2 which is a 2/2 twill written twice, and the pair 3/1/3/1 and 1/3/1/3, which are a 3/1 twill and its inverse written twice and form one class between them.

Three things are worth taking from the second column.

The over-count is small and it is not negligible. Three classes in twenty-one is fourteen per cent at eight ends, and it grows with the number of divisors rather than with the repeat — so a repeat of twelve, which has five proper divisors, loses proportionally more than a repeat of eleven, which has none at all. A prime repeat has no periodic twills whatever, which is a pleasing thing to be able to say and is the same divisor arithmetic that decides which satins exist.

The two columns answer two questions and both are asked. How many twills does a loom with this repeat admit includes the periodic ones, because a weaver setting up eight shafts really can weave a 2/2 twill on them. How many twills does this repeat contribute to the catalogue excludes them, because the 2/2 twill was already counted at four. The first is the designer’s question and the second is the enumerator’s, and quoting one for the other is the same conflation this essay is about at one level up.

And the subtraction is a check. The periodic classes are identifiable two ways — as sequences that repeat, and as classes whose canonical form is already in a shorter repeat’s list — and the two must agree. They do at all three repeats, which is one more independent route to the same objects and is the reason the count is worth quoting at all.

What was counted, and how #

The compositions are generated exhaustively by recursion: at each step take a run of every permissible size and recurse on what is left, keeping the sequences of even length. Nothing is sampled and nothing is looked up.

The canonical form of a sequence is the alphabetically smallest string among all its even rotations and their reversals, and two sequences are the same twill exactly when their canonical forms agree. Counting distinct canonical forms counts orbits.

Then every class is built as an actual draft and put through the integrity check. A twill is one cloth by construction — the diagonal ties every end to every pick eventually — so any class whose layer count came out other than one would mean the enumeration was generating something that is not a twill, and the figure refuses to draw. That assertion has never fired, which is what an assertion of a construction’s own correctness should do.

The counts are asserted to grow with the repeat as well, so a change that quietly dropped a family of sequences would be caught rather than reported as a smaller catalogue.

A twill is one cloth, and that is not obvious #

The integrity assertion above deserves more than a sentence, because the corresponding statement for reversed twills is false.

A draft describes one cloth when its above-and-below relation is strongly connected. For a twill the argument is short: end $j$ is over pick $i$ exactly when $j - i$ falls in the over-part of the sequence, so every end has the same relation to the picks up to a shift, and the shift is by one each pick. Following the diagonal walks every end and every pick in turn, which connects everything to everything.

Reverse the twill at intervals and the shift is no longer uniform. At the tightest reversal — turning every two ends — the cloth falls into three components, which is a herringbone that is not a fabric. The reason the plain twill is safe is the regularity of the step, and the moment the step stops being regular the guarantee goes with it.

What the count is good for #

Three uses, and the third is the one this site cares about.

It is a denominator. Any constraint on twills — a float limit, a balance requirement, a minimum interlacing count — removes some fraction of the catalogue, and a fraction needs something to be a fraction of. Without the orbit count the denominator would be the sixty-four, and every such fraction would be wrong by a factor of three on eight ends and by more than five on twelve — an error that grows exactly where the catalogue is largest and the fraction is most worth knowing.

It bounds the design space honestly. A designer told there are sixty-four twills on eight ends and then finding that many of them are the same cloth loses confidence in the whole catalogue. Twenty-one is a smaller and truer number.

And it is a check on the machinery. The compositions are generated one way and the orbits are counted another, and every class is then built and tested. Three independent routes to the same objects is how an enumeration earns being quoted, and it is the reason this site prefers a count it can re-run to a count it can cite. A table copied from a manual is right until a definition shifts underneath it; an enumeration that rebuilds every object it counts fails loudly when that happens.

Where the model stops #

The enumeration counts twills in the narrow sense — a single sequence, stepped by one end per pick, repeating. That excludes a great deal of what a weaver would call a twill.

Steps other than one. A twill can advance two or three ends per pick, which changes the angle and, where the step shares a factor with the repeat, destroys the cloth. Those are not in the count.

Reversals, breaks and displacements. Herringbones, broken twills, pointed and waved twills are all built from a twill and none of them is one.

Combination and shaded twills. A cloth that runs one twill for some ends and another for the rest is a compound draft, not a sequence.

So the number is a floor on what a repeat admits, and a low one. What makes it worth having anyway is that it is exact, that it counts objects rather than notations, and that everything built on top of it is built on twills in exactly this sense.

The comparison with the full census makes the size of the omission plain. Every twill on four ends is a four-by-four draft, and there are 22,874 of those in which every thread interlaces. Three of them are twills. A twill is an extremely special kind of weave — one sequence, one step, repeated — and the reason the subject is built on them anyway is that the regularity is what makes them predictable, one cloth, and easy to tie up on a loom with few shafts.

Who worked it out #

Weaving manuals give tables of twills by repeat and they are tables of sequences, produced by writing out the compositions in order. They are correct as tables and they do not claim to be counts of cloths; the reduction is left to the reader’s judgement, which is fine in a workshop and not fine in an arithmetic.

The mathematical treatment came with the interest in fabrics as periodic structures. Grünbaum and Shephard’s papers from 1980 onward set out weaves as two-coloured periodic patterns and classified them by symmetry, and the equivalences used here — rotation of the sequence, reversal of the cloth — are the ones that fall out of asking which symmetry operations of the plane carry a draft to itself.

Enumerating the orbits is elementary once the group is identified, and the answer for small repeats has been rediscovered several times in different notations, which is the usual fate of a count nobody thought worth publishing. The objects being counted are necklaces with a parity condition, and the counting of necklaces is one of the oldest exercises in combinatorics; that the beads happen to be runs of thread over and under changes nothing about the arithmetic and everything about who needs the answer. What has not been common is the third step: building every class and putting it through an integrity test. That step is cheap, it catches an enumeration that has drifted off its objects, and it is the difference between a list and a measurement.

Where the ladder goes next #

The rung below is twill direction, which is one of the two operations quotiented out here, and below that the three foundation weaves.

Sideways, the float limit uses this count as its denominator, and the plane groups are the same equivalence question asked of all drafts rather than only of twills.

What the pictures here cannot show. A count is not a picture and the bars on this page are the closest thing available. No drawing can demonstrate that twenty-one is the right number, because the claim is about a completed search; what a drawing can do is show two sequences that turn out to be the same cloth, and the reader is asked to accept that the machine did the rest.