What a repeat repeats

A weaver hands over a draft on a piece of point paper eight squares by eight. The question this essay is about is the one nobody asks on receiving it: how much of that paper is the cloth, and how much is the paper saying the same thing again.

It is not a pedantic question. Almost every count on this site is taken off a draft written at some size, and several of them change when the size changes while the cloth does not.

The draft is a matrix and the paper is not #

A draft is a binary matrix: rows are picks, columns are ends, and a filled square means the warp is on the face. The cloth is that matrix tiled over the plane, which is what makes a weave a weave rather than a picture — it goes on for ever in both directions and the matrix says how.

Which means the matrix is not unique. Tile a two-by-two plain weave into a four-by-four and the second matrix describes exactly the same cloth as the first. Nothing in either drawing says which of the two is in hand. An eight-by-eight point paper filled with plain weave and an eight-by-eight point paper filled with an eight-end satin are the same object to every eye on this site and to every gate it runs.

So a draft carries two things and the drawing separates them nowhere: the cloth, and a decision about how much paper to use.

The smallest rectangle, which is the answer a weaver wants #

The first thing to compute is the obvious one. A horizontal period is a number of ends at which every column equals the column that far along; a vertical period is the same down the picks; and the two directions are independent, so there is a smallest rectangle that tiles the drawing and it is found by trying divisors.

That rectangle is what a weaver means by a repeat. It is what the point paper is ruled for, it is what a dobby chain stores, and it is what a mill quotes. For a plain weave written at eight by eight it is two by two, and asking for anything smaller is asking for a fabric with no interlacing in it.

The computation is cheap and it comes with a check that is not optional: tiling the cell back out has to reproduce the writing exactly. A pair of periods that fails that test is not a pair of periods, and the failure mode of a search like this is a plausible wrong answer rather than a crash.

The rectangle is not the answer #

Count the marks in the right-hand panel of the figure above. A plain weave written at eight by eight has thirty-two translations that leave it exactly as it was — not the sixteen that the two-by-two rectangle accounts for, but twice that many.

The extra ones are diagonal. Move a plain weave one end along and one pick down and warp-up lands on warp-up: the parity of i plus j is what decides a plain weave and adding one to each leaves the parity alone. That translation is a period, and it is not a period of the rectangle in either direction on its own.

Translations that fix a matrix are closed under addition — two of them done in succession are a third — so they form a subgroup of the repeat’s own translations. The area of a fundamental domain, the smallest patch that tiles the cloth when the whole subgroup is allowed rather than the axis-aligned part of it, is the repeat divided by how many periods there are.

For plain weave that comes out at two intersections. One warp-up and one weft-up, and everything else is those two repeated. Point paper cannot draw it, because a fundamental domain of area two under a lattice generated by (1, 1) and (2, 0) is not a rectangle of squares, and the ruling of the paper is the whole problem.

Every twill and every satin is n, not n² #

The plain weave is not a special case; it is the smallest member of a family that includes almost everything ever woven.

A twill or a satin is built by one rule: pick a column pattern and shift it by a fixed number of ends for each successive pick. Written out, that is a matrix where the entry at pick i and end j depends only on j − mi taken modulo the repeat. Move one pick down and m ends along and the two changes cancel exactly. So (1, m) is a period of every twill and every satin, and the subgroup it generates has as many members as the repeat has ends.

The fundamental domain of an n-end twill or satin therefore holds n intersections, not n². An eight-end satin drawn on sixty-four squares is a cloth whose unit is eight. A 2/2 twill drawn on sixteen has a unit of four. A 1/2 twill on nine has a unit of three.

The exception is instructive. A basket weave is a plain weave with its threads doubled, and doubling breaks the diagonal: 2/2 basket written at four by four has only two periods, so its unit is eight intersections — twice the twill’s on the same paper, from a construction that looks simpler. Nothing about the drawings says which of the two is the smaller cloth.

What was counted, and how #

The site has run an exhaustive sweep since its earliest essays: every four-by-four binary matrix in which each end and each pick interlaces at least once, which is 22,874 of the 65,536 there are. Every census on the site — how many fall apart, which plane groups occur, how many twill classes a repeat admits — is a summary of that sweep.

Running the lattice computation over all 22,874 gives two numbers and they are far apart.

22,688 of them have no period but the trivial one. Those are the drafts whose unit really is sixteen intersections, and they are the overwhelming majority: a matrix chosen at random almost never has a symmetry.

And there are 1,446 cloths. Not 22,874. The difference is not mostly the drafts that repeat inside themselves — those are only 186 of the total — it is that a cloth can be started anywhere. Sixteen translations of a draft with no period give sixteen distinct matrices, all of them in the sweep, all of them the same fabric with the pencil put down in a different place.

The counting is orbit–stabiliser and nothing more: the translations of a four-by-four form a group of order sixteen, the periods of a given draft are its stabiliser, and the number of distinct writings is sixteen divided by the size of that stabiliser. Writing the two computations separately and requiring them to agree is what catches the mistake this kind of arithmetic invites — a canonical form that misses a symmetry and an orbit count that assumes one both produce numbers in the right range.

Which counts move and which do not #

Not every quantity on this site is affected, and sorting them is the practical half of the whole business.

Invariant under rewriting. The float lengths, the interlacings per intersection, the layer count, the cover factor at a given sett, the crimp, and the number of shafts. Every one of those is a property of the cloth: tile a plain weave into a four-by-four and it still has floats of one, still interlaces at every crossing, is still one cloth, and still needs two shafts, because the shaft count is the number of distinct columns and retiling adds no new ones.

Not invariant. Every census that counts drafts. The plane group census, the colour-collision count, the twill classes, the count of separable drafts — all of them are sums over the sweep, and the sweep counts writings. A statement of the form “x per cent of four-by-four drafts do such and such” is a statement about a set that contains a plain weave once and an arbitrary asymmetric draft sixteen times.

Whether that matters depends on the claim. A ratio between two subsets of the sweep is often close to the ratio between the corresponding sets of cloths, because both are dominated by the same overwhelming majority of asymmetric drafts. A statement about the symmetric drafts is not, because those are exactly the ones counted fewer times.

The compound-cloths field recorded this as an open shortfall and left it there. This does not close it — each census needs its own quotient, and the quotient for the plane-group census is a different one from the quotient for the twill classes — but it is the first measurement of the size of the gap.

What the overcount does to a ratio #

The essay divides the site’s quantities into those that survive a rewriting and those that do not, and the second list is long enough to be alarming. It is worth saying how alarming, because for most of the statements actually made the answer is: hardly at all.

Every census on the sweep counts writings, and almost every draft has sixteen of them. So a subset of the sweep and its complement are both inflated by very nearly the same factor, and a ratio between them is very nearly the ratio between the corresponding sets of cloths. The 144-in-22,874 figure for separable drafts is a statement about drafts; the corresponding statement about cloths differs by however much the separable drafts’ own symmetries differ from the average, and the average is sixteen.

The correction is therefore bounded and computable rather than unknown. A draft’s orbit is sixteen divided by its stabiliser, and stabilisers are one, two, four, eight or sixteen — so no draft is over-counted by more than sixteen and none by less than one. A ratio can be wrong by at most a factor of sixteen and is in practice wrong by a per cent, because only 186 of 22,874 drafts have any symmetry at all.

Where the correction bites is exactly where the symmetric drafts are the subject. A census of which plane groups occur is a census about symmetry, so its rows are populated by the drafts that are counted fewest times, and the inflation is wildly uneven across them: the asymmetric row is inflated sixteenfold and the most symmetric row not at all. A statement about symmetry taken off a sweep that counts writings is the one that has to be re-derived, and a statement about anything else is very likely safe.

That is a more useful conclusion than the shortfall as recorded. The quotient is owed, and the reason it has not been urgent is arithmetic rather than neglect.

There is one more class of statement that needs care and it is easy to miss: any count expressed as an absolute number rather than as a ratio. “One hundred and forty-four drafts separate” is a count of drawings, and the corresponding count of cloths is smaller by whatever those drafts’ orbits are. A ratio survives; a headline does not, and several headlines on this site are absolute counts.

The loom already knew #

There is one place in weaving where the diagonal period is exploited rather than ignored, and it has been there since before anybody wrote a matrix down.

A harness factors a draft into a threading and a lifting plan: end j is up on pick i exactly when shaft threading[j] is up on pick i. For a twill the threading is a straight draw — shaft one, shaft two, shaft three, shaft four, over and over — and the lifting plan is one row shifted along for each pick. That is not a coincidence of notation. The straight draw is the diagonal period, built into the hardware.

Which is why a four-shaft loom weaves a twill of any repeat width without storing sixteen cells: it stores four heddle assignments and four lifts, and the shift does the rest. A basket weave on the same loom needs two shafts and four lifts, because it has no diagonal period to exploit and its lifting plan cannot be one row slid along.

The general statement is worth having. A cloth with a period (1, m) can be woven from a straight draw and a lifting plan that shifts by m each pick, so its machinery is linear in the repeat where the drawing is quadratic. The economics of a dobby rest on exactly that, and the harness census the compound-cloths field ran is the same fact counted a different way.

Where the model stops #

This is translations only. A draft also has reflections, rotations and the warp–weft exchange, and the site counts those separately because they are not the same kind of thing: a translation moves a cloth without changing which face is up, and a reflection produces a cloth that is a mirror image and may or may not be the one somebody wanted. Quotienting by the full symmetry group would give a smaller number again, and it would be answering a different question.

“The same cloth” is a choice, not a fact. Two drafts that differ by a translation are the same fabric with a different starting point, and calling them one thing is safe. Two drafts that differ by a reflection are the same fabric turned over, which is safe for some purposes and not for others — a twill’s direction is exactly the property a reflection destroys, and a weaver who has been sold a Z twill does not want an S one.

And a fundamental domain is not a thing anybody can weave. It is the smallest patch that tiles the cloth, and for a plain weave it is a two-square shape that no loom is set up around. The rectangle remains the useful unit for weaving; the lattice is the useful unit for counting. Confusing them in either direction is what this essay exists to prevent.

Who found it, and when #

None of this is textile mathematics. It is crystallography, and it is a hundred and seventy years old.

The idea that a periodic pattern is described by a lattice of translations rather than by whichever cell somebody drew is Bravais’s, from 1850, and the classification of the possible lattices is his. The observation that the choice of unit cell is arbitrary while the lattice is not is the first thing any crystallography course says, and it is said because the same confusion arises every time: a diffraction pattern does not know which cell was drawn.

The counting argument is Burnside’s lemma, or rather the orbit–stabiliser theorem behind it, which the site has already used to count weaves up to symmetry. What is added here is the observation that it applies to the sweep itself — that the site’s own enumerations have been counting orbits’ members rather than orbits since its earliest essays, exactly and by a factor of about sixteen.

The textile literature’s version of this is the phrase “the repeat”, used without qualification, in every book. It means the rectangle, and for the purposes those books are written for that is right. The gap between the rectangle and the lattice has no name in the trade because the trade has never needed one.

Where the ladder goes next #

The next rung on this anchor asks the question this one raises and does not answer: what a census should count, and what a plane group census over cloths rather than drafts would say. That needs the quotient by the full symmetry rather than by translation alone, and it needs a decision about whether a cloth and its mirror image are one thing — which is a decision about weaving and not about mathematics.