How many cloths are there

Every enumeration on this site has taken the same denominator: 22,874, the number of four-by-four drafts in which every end and every pick interlaces at least once. How many fall apart, which plane groups occur, how many a two-colour surface fails to distinguish — all of them are fractions of that number.

It is a count of matrices. It is not a count of fabrics, and the difference is a factor of fifty-four.

Why a draft is not a cloth #

Take a plain weave and write down its four-by-four repeat. Now start the repeat one end further along. A different matrix, the same cloth — the repeat has no origin, and where a notation begins is a decision of the person writing it.

That alone is sixteen matrices per fabric before anything interesting has happened, and there are three more identifications that all have the same character.

Turn the cloth end for end. Rotate it by a half turn in its own plane. Warp is still warp, weft is still weft, the same side is still up, and the matrix has had its rows and its columns both reversed.

Turn the cloth over. Now the back is facing, so where the warp was on the surface the weft is — the matrix is complemented — and the ends run in reverse order because looking at the other side reverses left and right.

Turn it through a quarter turn. Warp becomes weft. This one is arguable and is treated separately below.

Each is an operation a person can perform on a piece of cloth without changing the cloth, and each produces a different sixteen-bit number.

The group, and why the complement matters #

Composing those operations gives a group, and it is built here by closure from generators rather than written out, because writing out sixty-four elements by hand is a way to omit one.

Three groups are built rather than one, because the honest answer depends on which identifications are being claimed and a single number would hide the choice.

Translation, sixteen elements. The repeat has no origin; nothing else is claimed. The most conservative reading.

Cloth, sixty-four. Adds the half turn and the two ways of turning over. These are exactly the operations that carry a piece of cloth to itself with warp still warp — the physical symmetries of a rectangle of fabric.

Construction, a hundred and twenty-eight. Adds the quarter turn, which exchanges warp and weft. The most generous reading and the most arguable: warp and weft come off different beams under different tensions in different yarns, so a quarter turn gives the same structure and not the same cloth.

The subtlety worth flagging is the complement. Turning a fabric over is not a reflection; it is a reflection composed with a complement, because the thread that was on the face is now underneath. A group built with the reflections and without the complements would be a group of symmetries of the drawing, and it would give a different and meaningless answer. That is the one place this construction could go quietly wrong, and the check for it is described below.

The counts #

Counted over the 22,874 drafts:

So the honest answer to “how many four-by-four weaves are there” is 426, and the catalogue as usually counted is inflated fifty-fourfold.

The inflation is not quite the group order, and the shortfall is informative. If every draft had a trivial symmetry group its orbit would be the full sixty-four elements long and the count would be 22,874 ÷ 64 = 357. It is 426, which is larger — because symmetric drafts are fixed by part of the group and therefore have shorter orbits, and a shorter orbit means more orbits for the same number of drafts. Orbits here run from two members to sixty-four.

The finding, which was not expected #

An earlier essay here of this site recorded this enumeration as not done, with a reason for thinking it safe to defer:

The census counts matrices, not cloths. Two drafts related by a translation, a reflection or an exchange of face and back are one fabric, and 22,874 is therefore a count of notations in exactly the sense that inflates a twill catalogue. Every downstream use takes the same denominator, so no fraction on the site is affected; the orbit count is one Burnside sum away and is not done.

The emphasised clause is wrong, and the way it is wrong is worth dwelling on because the reasoning behind it is very natural. It is an argument about the denominator. Change 22,874 to 426 and every fraction on the site would indeed be unaffected — if the numerator changed by the same factor.

It does not. Fractions have two ends.

The 144 separable drafts are 6 cloths. Not 144 ÷ 53.7 ≈ 2.7, which is what a uniform shrinkage would give, but six — because the separable drafts sit in unusually short orbits. They are more symmetric than average, so the group moves them to fewer distinct places, so they survive the identification in greater numbers than their share would suggest.

The rate therefore rises:

by draft — 144 of 22,874, which is 0.63 per cent. By cloth — 6 of 426, which is 1.41 per cent.

More than double. And under the widest identification, 4 of 219, which is 1.83% — nearly triple.

Six fabrics, written a hundred and forty-four ways #

The reframing is worth stating in words, because “six” is a much more comprehensible object than “one hundred and forty-four”.

There are six four-by-four fabrics that interlace everywhere and nonetheless fall into more than one piece. Every one of the 144 drafts this site has been quoting is one of those six, written down from a different starting end, or from the back, or upside down.

That is a considerably more useful thing to know than the draft count. Six is small enough to look at. It also sharpens the site’s central claim rather than weakening it: the failure is rarer among fabrics than the draft count suggested in absolute terms, and more common as a proportion, and both of those are the sort of thing worth having stated correctly.

The orbits are not all the same length #

The orbit lengths carry information beyond the count and are worth a paragraph, because they are what makes the finding of this essay possible rather than an accident.

An orbit is as long as the group unless something fixes part of it. A draft with no symmetry whatever sits in an orbit of sixty-four; a draft fixed by half the group sits in one of thirty-two, and so on. So orbit length is an inverse measure of a draft’s own symmetry, and it is measured here rather than assumed: the orbits in this census run from two members to sixty-four.

A plain weave is at the short end. It is fixed by a great deal of the group — a half turn leaves it alone, and so does a shift of two ends — so it has very few distinct notations. A draft chosen at random is at the long end, because a random matrix has no symmetry to speak of.

That is the mechanism behind the rate shift. The 144 separable drafts average 24 members per orbit against the census-wide mean of 53.7, which is to say they are markedly more symmetric than a typical draft. Averaging out to fewer than half the usual orbit length is not a small effect, and it is what turns 144 drafts into six cloths rather than the two or three a uniform shrinkage would predict.

Why the broken drafts should be the symmetric ones is a separate question and this essay does not settle it. The plausible account is that a draft with a thread that never interlaces, or with a clean division into two systems, is a very regular object — and regularity is what shortens an orbit. That is an explanation offered after the measurement, and it is recorded as such.

Counted twice, on purpose #

Two methods with no shared code path, and the assertion that they agree is the only reason to run both.

Canonical form. For each draft, apply every group element and keep the smallest sixteen-bit code produced. Two drafts are in the same orbit exactly when their canonical forms match, so the number of distinct canonical forms is the number of orbits. Direct, obvious, and vulnerable to any error in the group — a missing element merges nothing and splits orbits that should be one.

Burnside’s lemma. The number of orbits equals the average, over the group, of the number of elements each group member fixes. This never forms an orbit at all; it counts fixed points. It is vulnerable to entirely different errors, and it has a free check of its own: the average must come out a whole number, which it has no reason to do if the group is wrong.

They agree, at every one of the three levels, and the equality is asserted while the census runs.

The check that would catch a missing complement #

Two more assertions guard the group itself, and the second is the one that would catch the specific error this construction invites.

The set is closed under the group. Every group element applied to every draft in the census must land back in the census. That holds because complementing swaps a column sum of zero for a sum of four, and both are excluded; and transposing exchanges the row and column conditions, both of which are required. If it failed, neither count would mean anything.

Every draft in an orbit measures the same. All the members of an orbit are supposed to be one fabric, so they must agree about the properties this site measures — layer count and longest float. This is a check on the group rather than on the fabrics, and it is where a reflection without its complement would show up immediately: complementing a draft preserves the layer count and swaps warp floats for weft floats, so a group element that reflected without complementing would put drafts with different measurements in one orbit and the assertion would fail.

What else the identification would move #

The separation rate has been recomputed because it is this site’s central claim. It is worth being explicit about which other numbers on the site are quoted per draft, because each of them would move by its own factor and none of them has been redone.

The plane-group census. Twelve of the seventeen groups occur among four-by-four drafts, and the counts per group are draft counts. Under the identification, drafts related by symmetry are one cloth — and symmetry is precisely what the plane group measures, so this census would move more than any other, and in a direction that is hard to guess without running it.

The colour-collision count. 22,874 drafts collapse onto 256 surfaces under a one-and-one colour order, eighty-nine apiece. The numerator and denominator are both draft counts and there is no reason for them to shrink by the same factor.

The twill classes. Already counted up to the symmetries that make two twills the same twill, and by a different group from this one — rotations and reversals of the run sequence rather than operations on the matrix. Whether the two identifications agree on twills is a question this essay has not asked.

The honest summary is that one fraction has been checked, it moved, and the rest are outstanding. Recording that as outstanding is the point; the failure this whole essay is about was a claim that a computation could be skipped because its result was known in advance.

What the two counts say about symmetry, without a third census #

The orbit lengths are reported as running from two to sixty-four, and the two published numbers — 22,874 drafts in 426 orbits — pin down considerably more than that without any further enumeration.

An orbit is sixty-four long unless something fixes part of it, so 426 orbits would account for 27,264 drafts if none of them were symmetric. They account for 22,874, so there is a deficit of 4,390 drafts, and every one of them is a place where the group failed to move a draft somewhere new.

By Burnside’s lemma that deficit has a second reading. The lemma’s sum is the identity’s 22,874 plus whatever the other sixty-three elements fix, and it must come to 426 × 64. So

the sixty-three non-identity operations fix 4,390 drafts between them — an average of seventy apiece.

The deficit and the fixed-point count are the same number seen from two sides, which is the lemma restated, and it means the essay’s two counts already contain the census of symmetry it does not run.

How many cloths carry a symmetry #

The deficit also brackets how many of the 426 are symmetric, because orbit lengths must divide sixty-four.

If every short orbit were as long as possible — thirty-two, the longest a symmetric draft can have — each would contribute a deficit of thirty-two, and 4,390 ÷ 32 gives 137 orbits. If every short orbit were as short as possible — two, contributing sixty-two each — it would take 71.

So between seventy-one and a hundred and thirty-seven of the 426 four-by-four cloths are fixed by some operation other than the identity: between a sixth and a third of them.

That is a large fraction and it is the arithmetic behind the essay’s observation that the inflation factor is 53.7 rather than 64. A third of the catalogue at most, and a sixth at least, is symmetric enough to have fewer notations than the group has elements, and the shortfall from sixty-four is entirely those cloths.

And how much of that symmetry belongs to the broken ones #

The same accounting prices the essay’s central observation, which is otherwise stated as an average.

The 144 separable drafts sit in 6 orbits, so they would account for 384 drafts if none were symmetric and they account for 144 — a deficit of 240.

Against a total deficit of 4,390, that is 5.5 per cent of all the symmetry in the census, carried by drafts that are 0.63 per cent of it.

The separable drafts carry nearly nine times their share of the census’s total symmetry. That is a much sharper statement than a mean orbit length of twenty-four against 53.7, and it is the quantity the finding actually turns on: the rate more than doubles under the identification because the broken drafts are, by this measure, an order of magnitude more symmetric than the census average.

It also gives the essay’s own open question — why the broken drafts should be the symmetric ones — a size to be explained rather than a direction. Any account of it has to produce a factor of nine, not merely a tendency, and an argument that a separable draft is “a very regular object” would want to say why that regularity is worth nearly an order of magnitude rather than a few per cent.

None of that needed a new enumeration. It is two numbers and the lemma, and it is the kind of thing worth extracting before running a third census — because a census run without a prediction to check against can only report.

Where the model stops #

Four by four is a small repeat and the identifications are size-dependent. The quarter turn requires a square repeat; at four-by-six there is no such operation and the largest group is thirty-two rather than a hundred and twenty-eight. The inflation factor is not a constant of nature and should not be quoted as one.

The group is a choice and the essay makes it visible rather than resolving it. Whether a quarter turn gives “the same cloth” is a real question with a real answer that depends on what is being asked — for a designer choosing a construction, probably yes; for a weaver setting up two beams, certainly not. Three numbers are given because there are three questions.

Nothing here re-derives the downstream fractions. The separation rate has been recomputed because it is this site’s central claim. The plane-group census, the colour-collision count and the twill classes are all still quoted per draft, and every one of them would move by its own factor under this identification. That work is not done and is recorded as not done.

And a cloth is still not a fabric. Two drafts identified here are the same structure. In real yarn at a real sett they may behave differently, and the whole of this site’s setting and mechanics fields is about the ways they can.

Who found it, and when #

Burnside’s lemma is not Burnside’s — it is due to Cauchy and Frobenius, and Burnside stated it in his 1897 textbook without attribution, which is how it acquired the name and why careful sources now call it the orbit-counting lemma or the not-Burnside lemma.

Counting weaves up to symmetry is a natural thing for anyone with the lemma to try, and versions of it appear in the combinatorial literature on binary matrices under group actions. What is specific here is the group: the operations chosen are the ones that carry a physical piece of cloth to itself, which is why turning over is a complement, and that requirement is a fact about fabric rather than about matrices.

The finding — that the site’s own failure rate more than doubles under the identification — was not sought. The enumeration was run to close a shortfall recorded much earlier, and the shortfall’s own note said the result would change nothing. Checking a claim that says it is safe to skip a computation, by doing the computation, is not a sophisticated technique and it is the one that worked here.

Where the ladder goes next #

This ladder has spent six rungs on the draft as a plane pattern and this is the last of them for now. The remaining essay of this field steps back from the counting entirely: what the matrix cannot say collects the boundary of the encoding across every construction this collection has met, of which the gap between a notation and a fabric is one instance among several.