An irregular satin scatters where a regular one lines up

The six-end satin that does exist dropped one word from the definition. A satin is usually given as a construction with a move number — one mark per end, advancing by a fixed step — and the theorem that rules out six ends is a theorem about that step. Drop the step and keep the drawing’s own conditions — one mark per end and per pick, no two marks touching — and six ends has exactly one satin, unique up to where the repeat is started.

Most satins still have a diagonal then found why a regular satin usually fails at the thing it was chosen for. Its marks lie on a lattice; every lattice has a shortest step; its closest pairs of marks all run along that step; and a run of closest pairs all pointing one way is a row. Only a tie between two shortest steps avoids it, and the float limit leaves one order where that happens.

An irregular satin has no lattice, so its closest pairs need not point one way at all. Whether that freedom is ever available — whether, at the orders where every regular satin has a row, an irregular one exists at the same spread with its pairs scattered — is the question this account answers, and the answer is a census rather than an argument.

What a scatter is, measured #

The measurement is the one the closeness census already makes. For each satin, every pair of marks is taken on the torus the repeat lives on, the closest pairs are found, and their directions are counted. The largest share of the closest pairs pointing one way is the number: one means every closest pair runs the same direction, which is a row, and anything less is a scatter.

A regular satin’s marks lie on a lattice — which is the same lattice its plane group is read from — so its closest pairs run along the lattice’s shortest vector and its share is one — unless two vectors tie, in which case it is a half. Those are the only two values a regular satin can take, and that is the earlier account’s finding written in this account’s units.

An irregular satin has no constraint of the kind, so its share can be anything down to the reciprocal of the number of directions it manages.

It buys something at six, seven and nine, and nothing at eight or eleven #

The census is small enough to give in full.

Five ends. Two distinct satins, both regular, both at the best spread, both with two tied shortest directions — so a regular satin already scatters and there is nothing to buy. This is the order the float limit leaves standing.

Six ends. One satin, and it is irregular, because no regular one exists. Its closest pairs run four ways with a quarter in each. Irregularity is not an alternative here; it is the only thing there is.

Seven ends. Ten distinct satins, and all ten sit at the same best spread — four regular and six irregular. Every one of the regular ones has a share of one: a row. The best of the irregular ones scatters over four directions with a third in the largest. So at seven ends irregularity buys a row-free satin outright, at no cost in spread.

Eight ends. Forty-seven distinct satins, and only two reach the best spread — both regular, both with a share of one. No irregular satin is anywhere near it, so irregularity has nothing to offer at eight ends and the eight-end satin’s row is unavoidable.

Nine ends. Three hundred and fifty distinct satins, and all three hundred and fifty reach the best spread: four regular, all with a row, and three hundred and forty-six irregular, the best of them scattering over four directions with a quarter in the largest.

Ten ends. Two at the best spread, both regular, both tied — row-free already.

Eleven ends. Four at the best spread, all regular, all with a row, and no irregular satin among them.

So irregularity buys something at six, seven and nine and nothing at eight or eleven, and there is no pattern in that list a reader would have predicted.

The pattern is in how crowded the best spread is #

There is a pattern and it is not in the orders; it is in how many satins reach the best spread.

Where the best spread is crowded, irregularity buys something. At seven ends ten satins of ten reach it; at nine, three hundred and fifty of three hundred and fifty. With that many at the top, some of them are bound to be irregular, and an irregular satin at the top has no reason to line up.

Where the best spread is exclusive, it does not. At eight ends two of forty-seven reach it and at eleven four of twenty-eight thousand. A best spread that only two arrangements achieve is a best spread achieved by symmetry — and symmetry at that level of exclusivity is a lattice, which is to say a regular satin.

That is the mechanism and it is worth stating as a general shape. A maximum reached by very few objects is reached by the symmetrical ones, because symmetry is what lets an arrangement be extremal; a maximum reached by very many is reached by ordinary ones too, and the ordinary ones carry none of the symmetry’s consequences. The row is a consequence of the symmetry, so it survives exactly where the maximum is exclusive.

And whether the maximum is exclusive is an arithmetic accident of the order, which is why the list of orders looks random. At eight ends the best spread is 8 and only a lattice achieves it; at nine it is 5 and everything does.

The nine-end case, which is the extreme one #

Nine ends is worth drawing out because it is where the two populations are furthest apart and the arithmetic is most nearly absurd.

Every one of the 350 distinct nine-end satins reaches the best spread. The best spread at nine is 5 — the same as at five and seven ends — which means the condition is so easily met that no arrangement fails it, and the ranking the account has been using to choose between moves has nothing to say at all.

Of those 350, four are regular and every one of the four has a row. The other 346 are irregular and the best of them scatters its closest pairs over four directions with a quarter in each.

So at nine ends the move number selects, out of 350 equally good arrangements, the four that have a row. That is a strong statement and it is what the definition does: the move is exactly the condition that makes the marks a lattice, and a lattice at this spread has one shortest vector. A designer who draws a nine-end satin by the book gets one of four; a designer who draws one freehand, avoiding touching marks, gets one of 350 and almost certainly one without a row.

It is also the clearest case of the crowding argument. At nine ends the best spread is universal, so nothing about reaching it requires symmetry, and the symmetric arrangements are simply four members of a large population — carrying, as a side effect of their symmetry, the one property the construction was supposed to avoid.

Which is the same finding the account keeps producing #

This is the third time these satin accounts have found that dropping the move number changes the answer, and the three are one fact seen three ways.

At six ends it changes existence. No regular satin exists; one irregular one does.

At seven and nine it changes a property at unchanged quality. Regular and irregular satins reach the same best spread, and only the irregular ones avoid the row.

At eight and eleven it changes nothing, because the quality itself is a symmetry the irregular satins cannot reach.

So the move number is a restriction whose cost depends entirely on the order, and it is not the orders with fewer moves that pay. Eleven has eight moves and pays nothing because its best spread is exclusive; nine has four and pays a great deal because its best spread is universal.

What it costs to weave one #

An irregular satin is a draft like any other and the loom does not know the difference, so the cost is not in the weaving. It is in three other places and they are worth naming because they are the reason the trade uses regular satins.

It cannot be written as a move. Four ways to write a weave down prices notations against each other, and this is the same trade at one construction: A regular satin is specified by two numbers — its order and its move — and an irregular one needs its whole arrangement: seven numbers at seven ends, nine at nine. That is a real cost in a vocabulary and a card-cutting instruction, and it is the cost the move number was invented to avoid.

It has no translational symmetry. A regular satin repeats under a shift of one end and m picks, so its point paper can be checked by eye against itself; an irregular one cannot. The orbit of an irregular satin under the repeat’s translations is the full $n^2$, where a regular one’s is $n$ — which is the property the census uses to tell them apart and is also what a weaver would notice.

And nothing else is different. The float is n − 1 either way, so the float limit falls identically; the shaft count is n either way; the interlacing count is n either way; and the soundness criterion finds every satin at every order to be one cloth. The irregular satins are satins by every measure the site takes except the one the definition mentions.

So the trade’s preference is a preference about notation, and the arithmetic says what it costs: at seven and nine ends it costs the row-free property, at unchanged spread, unchanged floats and unchanged shafts.

Which satins the float limit actually leaves, counting both kinds #

The previous account crossed the row-free orders with a float limit and found one survivor. That census was over regular satins, because row-freeness there was a lattice property and only regular satins have a lattice. Adding the irregular ones changes the answer.

At a float limit of eight the four admissible orders are five, seven, eight and nine. Five is row-free regularly. Seven and nine are row-free irregularly. Eight is not row-free at all.

So the practical answer changes from one order to three, and the two extra ones cost nothing but a notation. That is the finding at its most useful, and it inverts the earlier account’s conclusion rather than qualifying it: the row question was said to be unavailable at a staple yarn’s float limit because there was only one row-free order to choose. There are three, and two of them were excluded by a definition rather than by a yarn.

Eight ends remains the exception and it is the awkward one, because the eight-end satin is the second commonest in the trade after the five and is what a damask ground usually is. At eight ends the best spread is exclusive, only regular satins reach it, and every one of them has a row — so a damask on an eight-end ground carries a row that nothing available at that order removes.

That is a concrete prediction about a cloth anybody can examine. An eight-end satin under raking light should show a row and a nine-end drawn freehand should not, at the same sett in the same yarn, and the difference is not a matter of degree: one has every closest pair on a line and the other has them in four directions.

What was counted, and how #

Every satin at each order is enumerated as an arrangement — one mark per end and per pick, no two marks adjacent cyclically in either direction — and reduced to distinct classes by the translations of the repeat, which is the property that also separates the regular from the irregular: a regular satin’s translation orbit has $n$ members and an irregular one’s has $n^2$. For each class, every pair of marks is measured on the torus, the closest pairs are collected, their directions are counted up to sign, and the largest share pointing one way is the scatter. The order’s best spread is the largest closest distance any of its satins reaches, and the comparison is made only among the satins that reach it.

Four things are checked. Irregularity buys a row-free satin at some orders and not at others, which is the finding stated so that a uniform answer either way would fail it. Where it buys, no regular satin at the best spread scatters — either every one lies on a line or none exists — and the irregular one scatters over several directions with under half its pairs in the largest. And it does it at the same best spread, so the scatter is free by the measure the account ranks with. At eight ends the best spread is reached by regular satins alone, which is the negative half checked rather than inferred.

The orders run from five to eleven because the arrangements are enumerated exhaustively and eleven already holds twenty-eight thousand of them.

Where the census stops #

It stops at eleven ends. The arrangement count grows very fast — twenty-eight thousand distinct classes at eleven — and the orders beyond are the ones a float limit rules out anyway, so the range is the practical one rather than an arbitrary cut. Whether the alternation between orders that buy and orders that do not continues past eleven is not known here.

Distance is on the torus and is Euclidean in draft coordinates. A draft’s cells are not square in cloth — the setts differ — so the closest pairs in the cloth are not necessarily the closest pairs in the draft, and a row’s visible direction is the draft’s direction scaled by the sett ratio. That correction is the same one a motif drawn at the wrong shape is about and it has not been applied here.

And a scatter is not an appearance. Four directions with a quarter in each is a statement about where the closest interlacings are, not about whether a reader sees anything. The previous account said at length that the arithmetic ranks arrangements by a distance and a distance is not a judgement, and every word of that applies here.

Still open: whether the scatter survives an unequal sett #

The one correction the census most wants is the sett ratio, and it could change the answer rather than merely the numbers.

A row is a line of closest pairs, and which pairs are closest depends on the metric. In draft coordinates the cells are square; in cloth they are a rectangle whose sides are the two thread spacings, and a cloth set at twenty-eight ends and twenty-six picks has a metric about eight per cent from square. A satin whose closest pairs tie in the draft’s metric need not tie in the cloth’s, so a row-free order could acquire a row when it is woven — and an order with a row could lose it.

That is a computation this collection is fully equipped for: the lattice reduction takes a metric, the closeness census takes a distance, and both would be run at the cloth’s own spacings rather than at unity. It would turn a property of the draft into a property of a construction, which is the direction nearly every result here eventually travels, and it would say for the first time whether a satin’s row is something a weaver can dent away.

Who found it, and when #

Irregular satins are old practice — a designer who wants a satin at six ends draws one — and their classification without a move number is new here. The closeness measurement is from two accounts below. Asking whether irregularity buys a row-free satin, and finding that it does at six, seven and nine and not at eight or eleven, was done here.