Which satins are worth weaving

The coprimality condition says which satins exist. It does not say which to weave, and on eleven ends there are eight moves to choose between.

Manuals give the survivors as a list — on 11: 2, 3, 4, 5, 6, 7, 8, 9 — with no ranking, which reads as though they were interchangeable. Weavers do not treat them as interchangeable. There is a rule of thumb, usually called the counter: take the move nearest the square root of the order. On sixteen ends that is three or five, and those are the two anybody uses.

The rule is given without a reason, as rules of thumb are. It has one, and the reason is measurable.

What “scattered” means, exactly #

The satin condition has two halves. Every end must be tied once, which is the coprimality; and the interlacings must not line up, which is what excludes the moves of one and one less.

The second half is usually stated as an aesthetic requirement and enforced by removing two moves by hand. That is unsatisfying — it makes the definition look arbitrary — and it is unnecessary, because the property being reached for is a quantity.

The interlacings of an order-$n$ satin sit at the points $(k, km \bmod n)$ for $k$ from zero to $n-1$. They live on a torus, because the repeat tiles in both directions, so the right distance between two of them is the toroidal one: the smaller of the two ways round, in each coordinate.

The spread of a move is the distance from an interlacing to its nearest neighbour, minimised over all pairs. It is a single number, it is computed while the figure draws, and it says how close the eye’s best chance of finding a pattern actually is.

A twill’s move of one puts the interlacings on the main diagonal, adjacent points a distance $\sqrt2$ apart — as close as two distinct lattice points can be without sharing a row or column. Every satin move beats it, and the figures assert that rather than assuming it. That is the second half of the satin condition, recovered as a measurement instead of imposed by hand.

Why mirror moves tie #

The moves come in pairs, $m$ and $n - m$, and the two produce mirror-image drafts — the same cloth turned over, with its scatter running the other way.

Any honest measure of scatter has to give them the same number, and a measure that did not would be measuring the drawing rather than the weave. The figures check it: for every move the mirror is found and the two spreads are required to agree exactly.

That check is worth more than it looks. It is the sort of thing that passes trivially once the code is right and catches a whole class of error while it is not — an asymmetric distance, a loop that starts at the wrong index, a modulus taken before an absolute value. It found none of those here, which is the outcome an assertion is supposed to have most of the time.

The counter rule, and where it comes from #

Ranking the moves at every order a weaver actually uses gives the same answer the trade gives: the move nearest $\sqrt{n}$ is among the best.

It is easy to see why, once the spread is written down. The interlacing at $k$ and the one at $k+1$ are one apart in the first coordinate and $m$ apart in the second, so they sit at distance $\sqrt{1 + m^2}$ — small when $m$ is small. The interlacings at $k$ and $k+j$ are $j$ apart in the first coordinate and $jm \bmod n$ in the second, and the dangerous $j$ is the one for which $jm$ is close to a multiple of $n$.

A small move puts consecutive interlacings close together. A move close to $n$ has the same problem by symmetry. A move for which some small multiple lands near a multiple of $n$ is bad in a less obvious way, and that is the case the eye discovers as a faint diagonal at some angle nobody predicted.

The best move is therefore the one whose multiples stay furthest from multiples of the order for as long as possible — and the value that does that is near $\sqrt{n}$, because it balances the two coordinates against one another. The rule of thumb is a rough statement of a lattice-packing condition, arrived at empirically by people who could see the result.

That last point is worth pausing on. The orders in daily use — five, eight, ten, twelve — are largely the ones with exactly one mirror pair of moves, so there is nothing to choose and nothing to get wrong. The orders where the choice matters are the primes, and primes are rare in weaving for reasons of shaft counts and machine building rather than reasons of design.

The ranking is readable off a continued fraction #

The spread has to be computed over every pair, which is why the figures compute it. But there is a way of reading the same ranking off the move by hand, and it explains the two kinds of failure rather than merely separating them.

Expand the fraction m/n as a continued fraction. The interlacings approach one another exactly where m/n has a good rational approximation with a small denominator — because j steps landing near a multiple of n is the statement that j·m/n is near an integer — and a good approximation with a small denominator is what a large partial quotient in the expansion is. So the rule is: a good satin move is one whose continued fraction has no large partial quotient in it.

Read that against the eleven-end moves and it sorts them at sight.

The two bad moves are bad in the two ways the essay above separates, and the expansion shows that they are the same fault written at different depths. A move of two has a large first partial quotient, which is the statement that the move itself is small — a near-approximation to zero. A move of five has a large second one, which is the statement that two steps of it nearly close the circuit. Neither is visible in the number; both are visible in three terms of an expansion.

It also puts the counter rule on a footing. A move near √n makes the first partial quotient about √n as well, which is as small as the first quotient can be made — and the expansion of such a fraction tends to stay short and flat rather than developing a large term later. The counter is a rule of thumb for keeping the whole expansion small, stated by naming the value that keeps the first term small. That is why it usually works and why it fails exactly when the value it names is unavailable.

Twenty-five is that case and reads correctly here too: 4/25 expands as [0; 6, 4] and 7/25 as [0; 3, 1, 1, 3], so the counter’s substitute carries a six where the better move’s worst term is a three. The rule picks the move with the larger quotient, which is the failure, and the expansion says so in one line where the pairwise measurement needs eighteen moves and three hundred pairs.

The connection is not a coincidence of notation. It is the same fact that makes continued fractions the tool for choosing multipliers in a lattice rule, and it is why the satin question and the numerical-integration question have the same answer.

It is a reading, though, and not the measurement. At sixteen ends it prefers seven, whose expansion [0; 2, 3, 2] has no term above three, over three and five, whose expansions carry a five — and the spread prefers three and five. Between sixteen and thirty the two disagree at eleven orders. The expansion flags a move whose few steps nearly close the circuit; it does not see a move whose two steps land a short diagonal away, which is what seven does on sixteen ends, and the lattice the marks lie on is what says which of the two a given move suffers from.

Where the rule breaks #

An enumeration is worth running because it can disagree, and this one does. The counter rule holds at every order from five to twelve that has a satin, and it fails first at thirteen.

The square root of thirteen is a little over three and a half, so the counter names four. Four scores $\sqrt{10}$. The moves of five and eight score $\sqrt{13}$, which is better — and their marks sit on a square lattice, with no diagonal at all, because five squared is minus one modulo thirteen. So the rule picks the wrong move at an order where the best move is not merely better but a different kind of satin.

It fails again at twenty-one, twenty-five and twenty-nine. Twenty-five is the cleanest case to see: the square root is five, five is not coprime with twenty-five, so the nearest available move is four, which scores $\sqrt{17}$, while seven scores $5$.

The failures are the ones the rule’s derivation predicts. The counter is a proxy for balancing the two coordinates, and it does that by naming a number rather than by stating the condition, so it goes wrong whenever the move nearest the number is not the move whose multiples keep furthest from closing the circuit — which a perfect square with an excluded root forces, and which thirteen does without any exclusion at all.

Thirteen-end satins are uncommon and nobody weaves a twenty-five-end one, so the failures have little practical consequence. They are recorded here because a rule that has been tested only where it works has not been tested, and because the shape of the failure explains what the rule was standing in for.

There is a second, gentler way the rule can go wrong that the enumeration also shows. At sixteen ends the counter is three and three is among the best — but so is five, and so are eleven and thirteen, their mirrors. The rule names one move where four are equally good, which costs nothing but does conceal that a choice existed. A weaver following the rule at sixteen ends will produce a perfectly scattered satin and will not know that the move of five would have produced an equally good one running the other way, which is occasionally what is wanted when a cloth has to be matched to another piece.

Eleven ends, which is where the choice is real #

Eleven is the smallest order with a genuine decision in it, and it is worth working through because it shows the ranking doing something a list cannot.

Being prime, eleven is coprime with everything below it, so the candidate moves are two through nine — eight of them, four mirror pairs. The enumeration splits them cleanly. Moves three, four, seven and eight score $\sqrt{10}$; moves two, five, six and nine score $\sqrt5$. Half the legal satins on eleven ends are noticeably worse than the other half, and a manual’s list gives no indication of which half is which.

Four legal satins, four inferior ones, and no manual distinguishes them. The bad half is bad for two different reasons and that is instructive. The move of two is bad in the obvious way: consecutive interlacings are one apart across and two apart down, so they are nearly adjacent and the eye finds them at once. The move of five is bad in the non-obvious way: five is close to half of eleven, so two steps of five land one short of a full circuit and the interlacings two apart in the first coordinate are one apart in the second. Nothing about the number five looks dangerous, and the arrangement it produces has a fault in it that only appears when the second-nearest relation is examined.

That second case is the argument for computing the spread rather than eyeballing the draft. The first failure is visible immediately in the drawing; the second is not visible until the cloth is woven and the light is right, which is the most expensive moment to discover it.

What was counted, and how #

For each order the coprime moves are generated from the condition rather than looked up. For each move the interlacings are placed and every pair is measured on the torus, taking the smaller of the two distances in each coordinate before combining them. The minimum over all pairs is the spread.

Three things are asserted while the figure draws.

Every satin move must beat a twill’s move. This is the second half of the satin definition, and if it ever failed the definition would be arbitrary rather than derived.

Every move must tie with its mirror, exactly.

And the figure reports, rather than asserts, whether the counter is among the best — because it is not always, and an assertion that must be true would have hidden the finding instead of producing it.

What the measure does not capture #

The spread is the nearest pair, and a single minimum is a crude summary of a point set.

Two arrangements with the same nearest-neighbour distance can differ in how the rest of the points are distributed, and the eye is sensitive to more than the closest pair — a faint line made of six points at moderate spacing is more visible than a single close pair, because it is a line. A fuller measure would look at the whole distribution of pair distances, or at how much energy the arrangement puts into each spatial frequency, which is closer to what a visual system is doing.

There is also nothing here about the cloth. The spread is computed on the lattice of interlacings and says nothing about yarn diameter, sett or crimp, and two satins with the same spread in very different yarns will not look alike. The claim is narrow: among the moves one order admits, in one cloth, this is the ranking.

And it says nothing about how densely the cloth can be set, which every move of one order shares — the interlacing count is the same whichever move is chosen, so firmness, maximum sett and freed length are identical across the whole row. The move changes where the interlacings are and nothing else, which is exactly why a measure of where is the right measure and why every other quantity on this site is silent on the question.

Who worked it out #

The counter is old and anonymous. Nineteenth-century weaving manuals give tables of satins by order with the usable moves marked, and the language they use — a counter of five on an eight-end satin, meaning the move — is the language of the workshop rather than of a theory. The tables are correct and the reason for the choice is not given.

The mathematical treatment of satins came much later and approached the question differently. Grünbaum and Shephard’s 1980 paper Satins and twills: an introduction to the geometry of fabrics classified periodic weaves by their symmetry and derived the coprimality condition as a corollary, and their concern was which arrangements are possible rather than which are good. That is the usual division: the mathematics settles existence and the workshop settles preference, and the two literatures rarely meet.

There is a third literature that has the answer and does not know it is about weaving. The question which multiplier scatters the points of an arithmetic progression most evenly on a lattice is the question behind low-discrepancy sequences and behind the choice of multiplier in a lattice rule for numerical integration, and it has been studied properly since the 1950s under names like the Korobov construction. The good multipliers there are chosen by very nearly the criterion used above, and the continued-fraction expansion of $m/n$ is what decides the answer. A satin move is a rank-one lattice rule on a repeat, which is a sentence no weaving manual has ever needed and which happens to be exactly true.

What is unusual about the counter is that it is a preference with an existence-flavoured reason behind it. The best move is a lattice question and lattice questions have answers. That the answer was found by weavers looking at cloth, several centuries before anybody wrote down the distance function, is the same pattern as the satin theorem itself — the practice was right and the explanation arrived from elsewhere.

Where the ladder goes next #

Upward from here the satin anchor runs into the float limit, which caps the order and therefore caps the whole question, and into lustre, which is what the scatter is in service of.

Sideways, the plane groups a draft can have is the same lattice read for symmetry rather than for spacing, and a satin’s quarter turn — whether its move squared is congruent to minus one — is a second congruence on the same number, with its mirror, at plus one, a third.

What the pictures here cannot show. The bars rank arrangements by a distance, and a distance is not a judgement. Whether a reader finds move two on eleven ends objectionable is a question about a visual system, and nothing on this page models one. What the figures establish is that the moves differ measurably, and that the measurement agrees with the choice weavers have been making for centuries — which is a weaker claim than “this one looks better” and the only one the arithmetic supports.