Every cloth there is, at four by four

A four-by-four draft is sixteen yes-or-no decisions, so there are $2^{16} = 65{,}536$ of them. That is a number a computer disposes of in a fraction of a second, and the consequence is unusual for this subject: the four-by-four repeat can be surveyed completely. Not sampled, not characterised, not argued about — enumerated, every one, with every property of interest computed for each.

This essay is the tour. Most of the numbers in it appear elsewhere on this site as ingredients in some other argument; here they are the argument, because the shape of the whole census says something that no individual count does.

The filter, and what it removes #

Of the 65,536 matrices, most are not weaves in any sense worth arguing about.

An end that is on the face at every pick is not woven in at all: it lies on the surface and can be pulled straight out. A pick that is under the warp at every end is the same failure in the other direction. The honest minimum for calling a matrix a weave is that every end and every pick interlaces at least once, and applying it leaves 22,874.

That is about thirty-five per cent of the matrices, which is a higher survival rate than most people guess. The filter is weak on purpose: it removes only the drafts that are obviously not cloth, and everything it leaves is a candidate.

The 144 #

Of the 22,874 that survive, 144 describe more than one cloth.

They interlace everywhere. They pass every check a weaver would apply by eye. And the threads split into two sets such that at every crossing between the sets the same one is on top, so the upper set lifts off and the fabric is two fabrics lying on one another.

That is the site’s central check, and the census is where its rarity becomes visible: six drafts in a thousand. Rare enough that nobody would find one by accident, and common enough that a designer generating drafts by rule rather than by hand will meet one.

The distribution over float length is the interesting part. No draft with a longest float of one or two separates. All 144 have a longest float of three, out of the 22,784 drafts at that float length. A long float is not sufficient for separation — the overwhelming majority of long-float drafts are perfectly good cloth — but it is necessary, at this size, and the reason is not hard to see: separation needs a region where one system never comes to the face, and a region takes room.

The ninety #

Ninety of the 22,874 are balanced in the strict sense: two up and two down in every end and every pick, so every thread carries exactly the same share of the face. None of the ninety separates.

Nought out of ninety is not a proof and the site does not present it as one. Sampling at six by six found the same thing and did not find a counterexample either, which raises the evidence without changing its kind. The honest statement is that balance appears to force integrity at these sizes, that it is a measurement, and that it has no proof attached here.

What makes it worth recording is that the implication is not obvious in either direction. A balanced draft is one whose column and row sums are all equal, which is a statement about counts; integrity is a statement about connectivity. There is no evident reason why one should imply the other, and yet in ninety cases out of ninety it does.

Twelve of the seventeen groups #

Every draft is a periodic pattern in the plane, so it has a wallpaper group, and the census classifies all 22,874.

Twelve groups occur. The distribution is extremely lopsided: 14,848 drafts have no symmetry at all beyond translation, which is nearly two-thirds of the census, and the symmetric ones tail away quickly — 3,104 with a glide-reflected centred mirror, 2,624 with a plain mirror, and so on down to thirty-two drafts apiece in the smallest classes.

The five groups that never occur are exactly the five requiring a three-fold rotation, and their absence is a theorem rather than an artefact of the size. A symmetry of a draft has to permute the intersections of warp and weft, so its linear part maps the square grid to itself, and the symmetries of a square contain no element of order three. No woven draft of any repeat size can have three-fold symmetry, and the essay on plane groups works it through.

Twenty-two thousand drafts, two hundred and fifty-six surfaces #

The last count is about what an eye can recover, and it is the most deflationary of them.

Thread a cloth with a one-and-one colour order in both directions and the visible colour at an intersection is the warp’s where the warp is up and the weft’s where it is not. Where the two threads crossing are the same colour, the intersection looks identical whichever is on top — so the weave leaves no trace there at all.

On a one-and-one order, eight of the sixteen intersections are blind. The whole census collapses onto 256 distinct surfaces, which is eighty-nine drafts apiece on average, and no eye and no photograph can separate the members of a class.

Half the information in the draft is destroyed, and it is destroyed by the colouring rather than by anything about the cloth. The blind intersections are exactly the ones where the two threads crossing are the same colour, which is a property of the colour order alone — so the eight is not a fact about weaving and would be a different number under a different order. What is a fact about weaving is that the surviving eight intersections are not enough to reconstruct sixteen, and no amount of looking will make them so.

What the shape of the census says #

Four counts, and read together they say something none of them says alone.

The named weaves are a vanishing fraction. This site draws perhaps a dozen four-by-four drafts with names — plain, the basket, three twills, a handful of others — out of 22,874. Weaving’s vocabulary is not a survey of the possibilities. It is a very small, very heavily used corner of them, selected over centuries for reasons that have nothing to do with completeness.

And the selection is not arbitrary. Every named weave is highly symmetric, has short floats, and is one cloth. Those are exactly the properties that make a draft easy to tie up on few shafts, predictable to weave, and safe. The census’s fourteen thousand asymmetric drafts are all perfectly good fabric; they are simply harder to specify, harder to remember and no better at anything.

Most of the interesting properties are rare. Symmetry is rare, separation is rare, balance is rare — ninety drafts out of twenty-two thousand. A property that a designer wants is generally a property that constrains hard, and the census is the way to see how hard.

And the whole thing is a small world. Four by four is where exhaustive enumeration stops being possible. Six by six is $2^{36}$, about seventy billion matrices — enumerable in principle on a large machine and not in a build, which is why every claim on this site about six-by-six behaviour is a sampled claim and says so.

Twenty-two thousand eight hundred and seventy-four, derived #

The survivor count is reported above as a measurement, and it does not have to be. It is a straightforward count of the matrices with no constant row and no constant column, and doing it by inclusion and exclusion both checks the sweep and explains the thing the section above found surprising — that the survival rate is so high.

Write it as removing the bad cases. Let a bad row be one that is constant, and a bad column likewise. There are four of each kind, each realisable two ways.

Remove the bad rows alone. Fixing s rows constant costs 2 per row and leaves the other 4 − s rows entirely free, which is 2^s × 2^(4(4−s)) matrices. Alternating over s = 1 to 4 gives −32,768 + 6,144 − 512 + 16, or −27,120. The columns give the same figure.

Then put back the overlap, and this is where the arithmetic gets its shape. A constant row and a constant column meet, at the cell they share, and the row’s value has to equal the column’s there. So once at least one row and at least one column are constant, every one of them carries the same value — two choices for the whole configuration, not two per row — and the free cells are the (4 − s)(4 − t) that lie in neither. Summing 2 × 2^((4−s)(4−t)) over all sixteen combinations with signs gives +11,578.

So the count is 65,536 − 27,120 − 27,120 + 11,578 = 22,874, which is what the sweep reports.

Two things are worth taking from that beyond the agreement.

The high survival rate is the overlap term. The first-order removal is 54,240, which would leave 11,296 — under a fifth of the matrices, and close to what most people guess. The overlap adds back 11,578, more than doubling the survivors, and it does so because a constant row and a constant column are strongly correlated failures rather than independent ones: a matrix with a constant row is very much more likely than average to have a constant column too, since the pinning of the shared cell propagates.

And the derivation says how the count scales. The same argument at n by n gives a survival fraction rising towards one, because the leading removal is 2n·2^(−n) of the matrices and the repeat’s area grows as n². At four by four the filter removes two thirds; at six by six it removes about six per cent; at ten by ten, four parts in a thousand. The filter is severe only at this size, which is worth knowing before reading four-by-four proportions as though they were properties of weaving.

That last point is a caution about the whole census. A fraction measured here is a fraction measured where the boundary effects are largest, and the ones that involve counting arrangements within a repeat — the separations, the symmetries, the collision classes — will not scale like the ones that involve counting the repeat’s edges.

What is not in the census, and could be #

Three columns the sweep does not compute, listed because each is cheap and each would answer a question the current numbers raise.

How many drafts are the same cloth. The census counts matrices, and two matrices related by a translation, a reflection, a rotation or an exchange of face and back describe one fabric. The plane-group classifier already computes every symmetry each draft has, so counting orbits would be one Burnside sum away — and the answer would be several times smaller than 22,874. The count as it stands is a count of notations in exactly the sense that inflates a twill catalogue, and it is left uncorrected here because every downstream use of it — fractions of the census that separate, distribute by group, collapse under colour — takes the same denominator and is unaffected by a uniform division.

How many shafts each draft needs. Two ends that lift identically at every pick can share a shaft, so the number of shafts a draft requires is the number of distinct columns in its matrix, which is one pass. That single integer decides whether a draft is weavable on ordinary machinery, and it is the strongest predictor of which of these twenty-two thousand anybody ever wove.

How the census looks under other colour orders. The collapse onto 256 surfaces is computed for a one-and-one order in both directions. A two-and-two order blinds a different set of intersections and would collapse the census differently, and the question of which order blinds fewest is a small optimisation nobody appears to have run.

None of the three is difficult. They are listed rather than done because an essay that adds three columns to a census has added three columns and no argument, and the arguments are what the essays are for.

What was counted, and how #

The sweep runs once per build and every figure reads its summaries off the same run, so two figures cannot disagree about how many drafts there are.

Each matrix is generated from a sixteen-bit integer, filtered for interlacing, and then measured: longest float cyclically, separable layers by strong connectivity, plane group by its translation lattice and point group, and balance from the row and column sums. The results are cached in memory for the build and not to disk, deliberately — the whole sweep takes about a fifth of a second, which is worth paying once and not worth the risk of a stale file disagreeing with the code that would have produced it.

Several counts are then asserted rather than reported. The survivor count must be 22,874; the separable count must be 144; the balanced separable count must be zero; the five hexagonal groups must be absent; and the group census must account for every draft, so a classifier that silently failed to classify something would be caught rather than producing a slightly short table.

The two ends of the float distribution are asserted too — the shortest floats never separate, the longest sometimes do — because a figure whose whole content is a gradient should fail if the gradient disappears.

Where the model stops #

The census is exact about the matrix and silent about everything else, and the silence is total.

It knows nothing about yarn. Two drafts with identical entries in every column of the census can be different setts, different counts, different fibres and utterly different cloth.

It knows nothing about the loom. A draft’s difficulty — how many shafts it needs, whether it can be woven at all on ordinary machinery — is not a property the census computes, and it is the property that decided which of these twenty-two thousand drafts anybody ever used.

And four by four is small. A repeat of four cannot hold a satin above five ends, cannot hold a herringbone with a reversal worth the name, and cannot hold a jacquard figure at all. The census is a complete map of a small region, and its completeness is exactly as valuable as the region is representative — which for the foundation weaves is very, and for figured cloth is not at all.

Who worked it out #

Nobody, before computers, and it is worth saying why the enumeration is recent rather than old.

The mathematical interest in weaves as periodic structures dates from Grünbaum and Shephard’s work from 1980 onward, and their concern was classification and existence rather than census: which weaves are possible, which symmetry types occur, what makes a fabric hang together. The connectivity criterion used here is theirs in substance.

Exhaustive enumeration of a design space is a different activity and it requires a machine. Sixty-five thousand matrices is nothing now and was a serious undertaking in 1980, so the census as an object is a recent thing to be able to have — and being able to have it changes what questions are natural. Which weaves are possible becomes how many, and how are they distributed, and the second question is the one the numbers on this page answer.

The one result on this page that seems to be genuinely new here is the ninety. That balance forces integrity at four by four is not a statement this site has found in the literature, and it is offered as a measurement with a sampled corroboration at six by six and no proof — which is the weakest of the four claims and the one most worth someone else’s attention.

Where the ladder goes next #

Below this rung are what cloth is and the draft as a matrix, which is the move that makes a census possible at all.

The counts here are the raw material for the integrity check, the plane groups, colour and weave as a two-colour problem and balance, each of which takes one column of it and argues about that. The full catalogue of the drafts this site draws, with every column computed, is the weave index.

What the pictures here cannot show. No figure on this page shows the census. Twenty-two thousand drafts cannot be drawn, and every picture here is either a summary of them or one example pulled out. The claims are about a completed search, and a picture can only ever illustrate what was searched for.