The six-end satin that does exist

A satin steps by a fixed number of picks from one end to the next, and must visit every pick exactly once before it returns. So the step has to be coprime with the order, and it must not be one or one less than the order, since that would put two marks side by side and draw a line on a surface whose whole purpose is to have none.

At six ends the only numbers coprime with six are one and five, and both are excluded. There is no six-end satin, and this collection proved it among its founding essays and has quoted it since.

The proof is correct. It is also a proof about satins that have a move number, and that condition was never in the definition of what a satin is for.

What a satin is, if it is not a move #

Say what the cloth has to be rather than how it is made.

One warp mark in every end and in every pick. That is what makes it a satin rather than a twill: exactly one interlacing per thread per repeat, which is the fewest a thread can have and still be woven in, and is why a satin has the longest floats and the highest lustre of any weave of its order.

And no two marks in adjacent ends in adjacent picks. Two marks touching would bind their two ends together at neighbouring picks and put a visible short diagonal on the face. The condition is cyclic: the last end is adjacent to the first.

That is the whole of it. The usual construction adds a third condition nobody states — that the mark advances by the same number of picks at every end — and it is exactly that condition which fails at six.

Enumerate arrangements satisfying the first two and nothing else, and the counts are these.

Four ends has none of any kind #

At four ends the search comes back empty. Not “no regular one”: none.

The reason is short. One mark per end and no two adjacent means each mark must be at least two picks from its neighbours in both directions, and with four picks in a cyclic arrangement of four marks there is nowhere for the fourth to go. Three ends is the same and worse.

So the four-end sateen a weaver draws — and weavers do draw one, and cloths are woven from it — is a construction with two marks touching in it. It is a twill, or something twill-adjacent, and its surface has a short diagonal in it that a five-end satin does not. That is not a defect of anybody’s drawing; it is the only thing available at that order, and calling it a satin is a courtesy the arithmetic does not extend.

Six ends has exactly one #

Thirty-six arrangements satisfy the two conditions at six ends. They form one translation class, of orbit thirty-six — which is the full group of translations of the repeat.

So there is exactly one six-end satin, unique up to where the repeat is started on the paper, and by the characterisation below it has no move number and cannot have one.

It has a warp float of one and a weft float of five, which is what one mark per end gives: the cloth is weft-faced, with each end appearing on the surface once in six. Its complement is the six-end sateen, warp-faced, and the two are the same picture inverted.

And it is one cloth. The integrity criterion says so, and it is not a foregone conclusion for a draft that is five-sixths one thing: a cloth that is nearly all weft on the face is exactly where a separable fabric would be expected. Every satin arrangement at every order enumerated here is one cloth, and the reason is that one mark per end and per pick means every thread of both systems is bound at least once.

What the one six-end satin looks like #

It is worth writing the arrangement out, because it is a specific cloth and not a family.

Reading the ends from left to right, the mark sits at picks 0, 2, 4, 1, 5, 3. The steps between consecutive ends are therefore +2, +2, +3, +4, +4 and then +3 to close the cycle — which is what “no move number” means in practice: two steps of two, two of four, two of three, in that order.

Every other one of the thirty-six arrangements is this one shifted. Shift the whole thing one end along and one pick up and the step sequence rotates and shifts; there are thirty-six places to put it and thirty-six arrangements, so nothing is left over.

That is worth dwelling on because it is the strongest form the result takes. A weaver drawing a six-end irregular satin has no design decisions to make beyond where to start: any six-end satin anybody has ever drawn is this cloth, whatever route they took to it, and two manuals that print apparently different six-end satins are printing the same one at different origins.

At seven ends that is no longer true — there are ten distinct classes, four regular and six not — and a designer has a real choice among them.

Regularity is a translational symmetry #

This is the part that makes the theorem a theorem about drawings rather than about recipes.

A satin with move m is carried to itself by shifting one end along and m picks up. So its orbit under the n² translations of the repeat has n members rather than n², and a satin is regular exactly when a one-end step fixes it.

That is a property of the picture, not of the construction that produced it. It lets the census sort regular from irregular without being told which move each was built from, and it is asserted as an equivalence: every class the census calls regular must have a move number and every class it does not must not.

The orbit sizes at each order say the rest. Five ends: two classes, both of orbit five, both regular — the moves two and three, which is exactly what the coprimality argument gives. Six ends: one class of orbit thirty-six, so no symmetry at all. Seven: ten classes, four of orbit seven and six of orbit forty-nine, so four regular and six irregular.

At five ends the two definitions agree completely and the extra freedom buys nothing. That is what makes six the first interesting order rather than the only one anybody looked at.

The middle category, which appears at eight #

At eight ends something arrives that has no name in the trade.

Forty-seven distinct classes, of which two are regular — the moves three and five. Of the remaining forty-five, two have a translational symmetry that is not a move: their orbits are thirty-two rather than sixty-four, so something carries them to themselves, and it is not a one-end step. A half-repeat shift does it.

So the classification is not two-way. There are satins with a move, satins with a symmetry and no move, and satins with no symmetry at all, and the third is much the largest group at every order past six.

Nine ends does the same thing: three hundred and fifty classes, four regular at moves two, four, five and seven, two part-symmetric at orbits of twenty-seven, and three hundred and forty-four with nothing. Ten ends has three thousand and five classes, two regular, and twenty part-symmetric at orbits of twenty and fifty.

This was the mistake this rung made first, and it is recorded because the correction is the interesting part. Regularity was initially taken to be “has a short orbit”, which is true of a regular satin and is also true of the part-symmetric ones — so the eight-end regular column came out at four with two blank move numbers beside it. A blank in a column that should not have one is what caught it. The characterisation had to be tightened from some translation fixes it to a one-end step fixes it, which is the definition rather than a consequence of it.

How many satins there are, in the limit #

The census counts to ten ends and stops. It is worth asking what it is counting towards, because the answer says how unusual a satin is, and the answer is: not at all.

An arrangement with one mark per end and one per pick is a permutation. The second condition forbids, at each of the n adjacent pairs of ends, two of the n available picks — so a fraction two over n of arrangements fail at each pair, and if the pairs were independent the survivors would be n factorial times (1 − 2/n) raised to the n, which tends to n! ÷ e².

The pairs are not independent, so that is a limit rather than a formula, and the census is exactly the evidence for it. Take each order’s arrangements — the class counts multiplied by their orbits — against n factorial over e squared:

The ratio climbs steadily, which is what a slowly-approached constant looks like.

Two things follow. A satin is a generic object, not a rare one — the two conditions that define it cost a constant factor of about seven, not a factor that grows, so the number of satins of order n runs away as fast as n factorial does.

And the regular ones do not. There are at most φ(n) − 2 of them at any order, which is fewer than n, so the fraction of satins reachable by a move number falls to zero like n over n factorial. At five ends the construction finds both satins that exist. At nine it finds four of three hundred and fifty. The recipe was never a description of the objects; it was one way of writing down a few of them, and the fact that it fails at six is a small instance of a much larger blindness.

What the trade did instead #

Weavers have wanted a six-end satin for as long as they have wanted six-end cloths, and the trade’s answer is the irregular satin or satinette: a six-end weave with one mark per end, marks not touching, and no constant step. It appears in weaving manuals as a construction to be drawn out by hand rather than derived, and it is often presented as a compromise or a special case.

It is neither. It is the unique cloth satisfying what a satin is for, and it is a compromise only against a condition that was never the point.

The same is true of the eight-end and ten-end irregular satins, which the trade also uses and which the census counts in their forty-fives and three thousands.

A rule of thumb that names a construction has quietly changed the definition of the object. That is the shape of this rung, and it is the shape of several results in this collection.

What was counted, and how #

The search is over permutations with a pruning test at each end, which is exhaustive up to about eleven and is the whole cost of every satin result here.

Four ends has none, asserted as a count of zero rather than as an absence.

Six has exactly one distinct class, of orbit thirty-six, with no regular class in it, and it is one cloth. Four separate assertions, because there are four ways for that claim to be wrong.

Five has two distinct classes and no irregular one, which is the assertion that keeps the whole rung from being vacuous: if the extra freedom bought something at every order, the six-end result would be unremarkable.

Past six the irregular satins outnumber the regular ones, asserted at seven, where it is six to four.

Every class is accounted for by its own orbit. The sum of the orbit sizes must equal the arrangement count exactly, at every order, which is the check that the classification partitions what it claims to partition.

And regularity is equivalent to having a move number, checked class by class at eight ends together with the requirement that a regular class’s orbit is exactly its own order.

What the irregularity costs on the cloth #

A regular satin has a translational symmetry and an irregular one does not, and the practical question is whether that shows.

In the floats, it does not. Every satin arrangement of order n has exactly one warp float of one and one weft float of n − 1 in every thread, because that is what one mark per end means. So the six-end satin’s lustre, its abrasion behaviour, its firmness and its setting limit are all identical to what a regular six-end satin would have had.

In the surface, it might. A regular satin’s marks lie on a lattice, so any residual visual texture repeats on a lattice finer than the repeat; an irregular one’s do not, so whatever texture there is repeats only at the full six by six. Whether that is better or worse is a question about what the eye does with a near-random arrangement against a regular one at a scale of a few threads, and this collection has no model of that.

And in the threading, it does. A regular satin can be woven on a straight draw with a stepped treadling, which is why it is easy. An irregular one needs its lifting plan written out, which is trivial on a dobby and a nuisance on a treadle loom — and is very likely the reason six-end satins are rare in hand weaving and unremarkable in power weaving.

So the cost of irregularity is entirely in the making. The cloth is a satin in every respect the cloth has, and it is only the loom that can tell.

Where the model stops #

The enumeration stops at eleven ends because the search is over permutations and the counts grow accordingly — three hundred thousand arrangements at ten. The interesting orders are all below that, but a general statement about every order is not available from this and would need an argument rather than a count.

Mirroring and warp-weft exchange are not quotiented out, which is the same choice the four-by-four census had to make and settled differently. A satin and its mirror image are counted as separate classes here, which is right for a weaver — they are different to thread and produce different-handed surfaces — and would be wrong for somebody counting abstract patterns. The census reports both figures and this essay quotes the translation-only one.

The adjacency condition is the plain one. A more demanding designer might want no two marks within two picks rather than one, which is a different enumeration with different and smaller answers, and this collection has not run it.

And nothing here is about which of them is worth weaving. At eight ends there are forty-five irregular satins and a weaver wants one; choosing needs a criterion — evenness of the surface, freedom from any visible line — that this rung does not supply.

The generalisation #

When a theorem says an object does not exist, check whether the definition it used was the object’s or the construction’s.

The coprimality argument is airtight and its conclusion is exactly true. What it is true of is a satin defined by its move number, and a move number is a description of how satins are usually laid out on paper. It got into the definition because it is how everybody draws them and because for the orders that occur most — five, seven, eight — it costs nothing.

This is not an unusual accident. A construction that always works becomes the definition, and then the definition rules out the cases the construction cannot reach. The cure is to write down what the object is for — here, one interlacing per thread with no two touching — and enumerate against that.

A count against the right definition is cheap. Deciding what the right definition is is the work.

Who found it, and when #

The coprimality condition on satin moves is old and standard, in every weaving manual, and so is the observation that four and six have no regular satin.

The irregular six-end satin is equally old as a practical construction and is drawn in the manuals. Whether anybody has enumerated the arrangements and found that there is exactly one, this collection does not know; the count is easy enough that somebody may well have, and it does not appear in the sources consulted here.

The combinatorial object — permutations with no adjacent values in adjacent positions, cyclically — is a known one and its counts are in the mathematical literature under other names. It was not enumerated here from that literature but by direct search, which is this collection’s standing habit: a quoted count is a count somebody else made.

Where the ladder goes next #

Sideways into the texture weaves, where a relief made at the loom turns out to be an order of magnitude deeper than one made in the finishing — and a square root says why it cannot be deeper still.