Only a plain weave has one size of hole

A weave’s holes come in kinds. A plain weave has one; a 4/4 twill has four; a basket has three. The kind a hole belongs to is decided by where the four threads round it sit in the thickness of the cloth, which is decided by whether each of them is floating over, floating under, or on its way between — three states, read off the draft.

So there is an obvious question. Which weaves have only one kind of hole?

The obvious answer is the plain weave, and only the plain weave, with what looks like a one-line proof. A thread is at the middle of the cloth exactly where it transits, a plain weave transits everywhere, and nothing else does. Uniform holes therefore mean universal alternation, and universal alternation in both directions means plain weave.

The proof is wrong at its second step, and finding out where takes an enumeration rather than an argument.

The claim #

Having holes of one size is not a property of a weave. It is a property of a weave and a construction together, and only the plain weave has it at every construction.

At a muslin’s sett — 24 ends and 22 picks per centimetre — fourteen of the 22,874 drafts have uniform holes. Set the same yarn square and the count is 170. Turn the cloth over, so that the weft is the denser system, and it is fourteen again, but they are a different fourteen.

Two drafts are in all three lists. They are the plain weave and its complement, which are the same cloth read from its two faces.

The argument #

The waist of a hole is the smaller of two clear widths: one across the ends, one along the picks. Each is bounded by its own pair of threads, and each comes to the spacing less the diameter exactly where both of that pair are cut through their own axes — which happens when the two are level.

Now the step the one-line proof skips. For the waist to be the same at every hole, it is not necessary that every clear width be the same. It is only necessary that the smaller of the two be the same. And in a cloth whose warp is set closer than its weft, the gap across the ends is the smaller of the two before any weave is applied at all.

A muslin’s warp gap is 249.6 µm and its weft gap is 287.4 µm. So any draft whose ends all transit gives a warp clear width of exactly 249.6 µm at every hole, and whatever its picks are doing gives a weft clear width somewhere above 287.4 — which is never the smaller, and is therefore never consulted. Every such draft has uniform holes, and its picks are irrelevant to that fact.

How many drafts have all their ends transiting? An end transits at every gap exactly when its column alternates, and a column of four alternates in two ways. Four columns, two choices each: sixteen. Two of those sixteen have every row constant, which fails the interlacing condition on the picks and is not a cloth. Fourteen remain.

Fourteen. Which is the number the enumeration returned, and the coincidence is not one.

What the square-set control settles #

If the fourteen are an artefact of the warp being the binding direction, then making neither direction binding should change the count, and it does — dramatically, and in the direction that is at first surprising.

Set the same yarn square, so that the two gaps are equal to the last digit. Now a hole bound across its ends and a hole bound along its picks give the same answer, because the two gaps are the same number. Far more drafts come out uniform: 170 of them, twelve times as many.

That is the opposite of what a first guess suggests. Making a cloth more symmetric does not make uniform holes rarer; it makes them commoner, because it removes a distinction the arithmetic was using to tell holes apart.

Then turn the cloth over. Swap the two setts and the two counts, so that the weft is the dense system. Fourteen drafts come out uniform again — but now the fourteen are the drafts whose picks all transit, which is a different fourteen, related to the first by transposition.

Three constructions of one yarn: fourteen, one hundred and seventy, fourteen. The intersection is two.

What was counted, and how #

Running the direct computation over the census is not affordable: sampling four hundred levels in each of sixteen holes in each of 22,874 drafts is a hundred and fifty million evaluations of a chord.

It is also unnecessary, because the answer depends on very little. A hole’s waist depends on its four threads’ levels and on nothing else, and it depends on the two pairs rather than on the order within a pair — swapping the two ends bounding a gap reflects the picture and changes no width. So each hole reduces to a key: the sorted pair of end levels beside the sorted pair of pick levels. There are at most thirty-six such keys and this construction uses seven of them. Each key’s waist is computed once and looked up thereafter, and the census becomes 366,000 lookups.

The enumeration is the same one this collection has used since every cloth at four by four was counted, and which has since been asked about separable drafts, plane groups, shaft counts and colour-and-weave collapses: all 2¹⁶ matrices, filtered to those in which every end and every pick interlaces at least once, giving 22,874 — a count asserted rather than remembered, every time the census is run.

Three things are asserted as the census runs, and each would catch a different mistake:

• How many drafts have every thread transiting is the same at all three constructions. It has to be: it is a statement about the matrix and no sett appears in it. If that number moved, the level rule would be reading the construction when it should be reading the draft.
• The warp-dense uniform drafts are a subset of the square-set ones. This is why a single control was not enough. Intersecting fourteen with a hundred and seventy that contain them returns fourteen and proves nothing, which is what the first version of this census did.
• The count at one construction is not the count at another. Asserted as an inequality, because the whole claim is that this number is not a property of the weave.

The rest of the distribution, which has a ceiling in it #

Fourteen drafts sit at the bottom of a distribution, and the distribution is worth having whole. Counting how many sizes of hole each of the 22,874 drafts has, at a muslin’s construction:

Two things fall out of it.

Nothing has more than five. The four threads round a hole have three states each, which is eighty-one combinations before the two symmetries within a pair are removed — and the number of combinations that actually occur across every draft at this repeat is seven. The levels are not independent: an end transits at a gap exactly where the two picks it lies between disagree about it, so the quadruple is constrained by the same entries that produced it. Seven keys, and only five distinct waists among them, because two pairs of keys turn out to give the same minimum.

The widest hole is bounded, and the bound is reached. No draft at this repeat can produce a passage wider than one bounded by threads at opposite extremes in both directions, which is 287.4 µm against a shown opening of 249.6 — 15.2 per cent over, and that is the ceiling on how wrong a specification can be from this cause. The weave that reaches it is the basket, which is not an exotic draft found by search but the third weave in every beginner’s book.

The ceiling is checked from the other direction as it is computed: the basket is built by hand, its own census run, and the two numbers are required to agree. A ceiling found by taking a maximum over a loop is the largest thing that loop happened to see; a ceiling that a separately constructed weave lands on exactly is a bound.

Which twelve, and what they are #

The twelve coincidental drafts are the alternating-column drafts that are not plain weaves, and they are recognisable: they are plain weave with its ends taken in blocks. 0011 / 1100 / 0011 / 1100 is a warp rib — two ends up, two down, alternating pick by pick — and 0111 / 1000 is the same idea with an uneven partition.

Two of them carry a longest float of three. So it is not the case that a uniform-holed weave must be a short-float weave, even by accident: a draft can have a thread lying across three quarters of its repeat and still measure the same at every hole, provided the direction that measures is the one whose threads all turn.

They are also, as a set, unusually symmetric — every one of them has a plane group with a mirror in it, which is what one would expect of a set defined by a condition on whole columns, and which is the same phenomenon that made the separable drafts sit in unusually short orbits when the census was quotiented by the symmetries of the cloth.

They are also, none of them, weaves anybody names. A warp rib is a construction rather than a weave in most books, and 0111 / 1000 is a rib with an uneven partition — the sort of thing a stripe produces as a by-product rather than the sort of thing a table of basic weaves contains. That is exactly why the seven-weave catalogue missed them.

Where the model stops #

The level rule is a plain weave’s amplitude applied to every weave. A satin’s threads have less crimp than a plain weave’s at the same sett, so they do not swing as far from the middle, and the excess of the widest hole over the projection should be smaller than the arithmetic here gives. That does not touch the census, which asks whether the sizes are equal rather than what they are — but it is the reason the individual micrometres in these tables are softer than the counts.

The enumeration is at four by four only. That is the repeat this collection can exhaust, and the same caution applies here as everywhere it has been used: the pattern that fourteen equals the number of alternating-column matrices is a derivation rather than a sample, so it should hold at any even repeat, and nothing here has checked it at six.

The construction is Peirce’s throughout, so everything the circular section assumes is assumed here: round threads, uniform along their length, at a stated packing factor. A flattened thread intrudes further at the mid-plane and less at the extremes, which would widen the gap between the uniform drafts and the rest rather than close it.

And uniformity is being judged at a tolerance. Two waists are called the same if they agree to a micrometre, and the sampling that finds them has its own tolerance, which travels with the answer. A pair of drafts differing by a nanometre would be called uniform here and would be a different cloth to nobody.

Fourteen is 2⁴ − 2, and that closes the six-by-six question #

The census’s fourteen was matched against a hand count of alternating-column matrices and the two agreed. That hand count generalises, and generalising it settles the open question the limits section records — what happens at a repeat of six — without running an enumeration that would not fit.

A column of length n alternates in exactly two ways: starting up or starting down. Call them A and B. Any assignment of A or B to the n columns gives a draft in which every end transits at every gap, so in a warp-dense cloth every such draft has uniform holes. That is 2ⁿ assignments.

Two of them are excluded, and only two. If every column is the same type, then every row is constant — all ends up, or all down, on each pick — so no pick interlaces and it is not a cloth. Any other assignment has two columns of different types, so every row has both symbols in it and every pick interlaces.

So the count is 2ⁿ − 2.

At n = 4 that is fourteen, which is the number the census returned. At six it is sixty-two; at eight, two hundred and fifty-four.

And for odd n it is zero, because a column of odd length cannot alternate cyclically at all. A repeat of five has no uniform-holed draft whatever — not even a plain weave, since a plain weave does not exist on an odd repeat.

Which says how fast the property vanishes #

The count grows as 2ⁿ and the census it is drawn from grows as 2^(n²), so the fraction of drafts with uniform holes collapses.

At four by four it is fourteen in 22,874 — six in ten thousand. At six by six it is sixty-two in something over sixty-eight billion, which is one in a billion. At eight it is unmeasurable.

Uniform holes are not rare; they are asymptotically absent. Only the plain weave and a handful of rib-like relatives ever have them, and the handful does not grow at anything like the rate the space of weaves does. So the essay’s headline is if anything understated: the further a designer goes from a plain weave, the more certain it is that the cloth’s holes come in several sizes, and by a repeat of eight the certainty is total for all practical purposes.

And the intersection is exactly two, at every repeat #

The three-construction comparison found fourteen, a hundred and seventy, and a different fourteen, with two drafts in all three lists. That two is also a derivation rather than an observation, and it holds at every repeat.

A draft is uniform in a warp-dense cloth when every column alternates, and uniform in a weft-dense one when every row does. Both at once means the column types must alternate along the row as well — A, B, A, B — which is two assignments and no more, whatever n is.

Those two are the plain weave and its complement, which is the plain weave seen from its other face.

So the plain weave is not merely the commonest member of a list that happens to be short. It is the unique cloth, at every repeat, whose holes are one size regardless of how the cloth is set — and the twelve others at four by four, the sixty at six by six, and the two hundred and fifty-two at eight are all cloths that get the property from their construction rather than from their weave, and lose it the moment the cloth is turned over.

That is the essay’s claim in its sharpest form, and it now rests on an argument about alternating columns rather than on three counts from one enumeration.

The generalisation #

The shape of this is a familiar trap wearing an unfamiliar coat.

A minimum over two quantities hides everything about the larger one. Whenever a system’s behaviour is set by whichever of two constraints binds, every property of the slack constraint becomes invisible — and any classification built on the binding one will look like a classification of the system when it is a classification of which constraint happens to be tight. Change the balance and the classification changes, without anything in the system having changed at all.

The test is the one used here, and it is cheap: run the same classification with the balance reversed. If the answer moves, the classification was about the balance. The square-set control was not enough on its own precisely because it made neither constraint bind, which loosens rather than reverses; turning the cloth over is what reverses it.

And the second lesson is about catalogues. Seven weaves chosen because weavers weave them is a sample from a distribution nobody drew at random, and the twelve drafts that break the obvious claim are exactly the sort of thing such a sample omits — they are ribs and blocks rather than named weaves, so they appear in no table of the basic constructions.

Who found it, and when #

The 22,874 count and the enumeration it comes from are this collection’s, first run when it asked how many cloths there are at four by four, and the machinery has been reused for the separable drafts, the plane groups, the shaft counts and the colour-and-weave collapses.

The classification of holes by level appears to be new here, so there is no prior count to compare against. What is not new is the observation that a warp-dense cloth is governed by its warp: it is why a filter cloth’s opening size is quoted as the smaller side of its hole and why the setts a loom can reach are quoted per direction. The step taken here is to notice that a quantity defined as a minimum over the two directions inherits that asymmetry, and that a census of such a quantity therefore measures the asymmetry unless it is deliberately removed.

Where the ladder goes next #

Having a distribution of holes rather than one hole raises the question of which statistic of it matters, and the answer is that it depends what the cloth is for: a filter is rated by the hole it does not show, while what a fabric passes in bulk depends on the mean, so a weave chosen to improve one worsens the other.

Sideways, the shape of the hole matters as much as its size once something has to flow through it, and a weave with floats makes its holes long rather than square: a satin’s hole is a slot, and a slot of a given area passes less than a square of the same area, exactly.