Most satins still have a diagonal

A twill is a line, and a satin is the weave designed not to be one. Its move is chosen so that no interlacing sits beside the next, and among the moves an order admits, the best is the one whose interlacings sit furthest apart. That is the whole of the usual account, and it leaves standing something nobody who has looked closely at an eight-end satin needs to be told: the cloth still has a faint diagonal in it.

The account never said it could not. Scattering the interlacings is a statement about the closest pair. A line is a statement about where the closest pairs point, and the two are different questions with different answers.

The marks of a regular satin lie on a lattice, and every lattice has a shortest step. Every mark has a neighbour along that step and another the same distance the opposite way, so the closest marks form rows, and the rows are the diagonal. The only way out is a tie — two shortest steps of equal length pointing different ways, with nothing to choose between them. Which satins get a tie is decided by the same number that decides whether a satin exists at all: the move, squared.

Every regular satin has a shortest step #

On n ends with move m, end k is tied at pick km, counted round the repeat. Shift the whole draft one end along and m picks up and every mark lands on a mark; shift it n picks up and they land on marks again. So the marks are exactly the points reachable from any one of them by whole numbers of those two steps. That is a lattice, with one mark in every n cells, and it continues unbroken across every copy of the repeat.

Two steps generate it, but many other pairs generate the same lattice, and the pair the construction hands over is usually long and nearly parallel. The shortest pair is found the way a greatest common divisor is. Subtract from the longer step the whole number of copies of the shorter one that brings it closest to nothing; if what remains is now the shorter, swap; repeat until neither shrinks. For a satin the procedure ends in two or three rounds and returns two steps, u no longer than v, with no shorter step anywhere in the lattice.

For the eight-end satin on three, u runs two ends across and two picks down — a length of √8, at exactly forty-five degrees — and v runs one end across and three picks up, at √10. Every mark has two neighbours at √8, one each way along u, and four at √10. So rows of marks at the closest spacing run at forty-five degrees through the whole cloth, and across those rows the marks are further apart, by about twelve per cent.

That is the diagonal. It is not made of adjacent marks, which is what a twill’s diagonal is made of, and the satin condition rules those out. It is made of marks two ends and two picks apart, and the satin condition says nothing whatever about those.

Ten ends ties and eight ends cannot #

The ten-end satin on three reduces to u of one end and three picks and v of three ends and one pick back. Both are √10, and they are perpendicular. Its marks lie on a square lattice, turned about eighteen degrees off the grid, and every mark has four neighbours at the closest spacing, pointing four ways at right angles. There is a row at the closest spacing and another crossing it at exactly the same spacing, and an eye looking for the diagonal has two candidates of identical strength.

That is what “no diagonal” can mean for a regular satin, and it is the only thing it can mean. A lattice with a single shortest step always shows a row along it. How strongly depends on how much longer the next step is, and the number that measures it is the ratio of the two lengths: 1.00 for ten ends, 1.12 for eight.

Twelve per cent is small. It is also not zero, and eight ends has no move that does better. Its only satin moves are three and five, which are mirror images of each other and have the same lattice shape, so whichever is threaded, the row is there.

The square of the move decides the shape #

A draft read as a periodic plane pattern has a plane group — the turns and reflections that carry it to itself — and a satin’s can be read straight off its move. Turning the draft a quarter turn carries the satin on m to the satin on minus the inverse of m. Reflecting it across the diagonal of the grid carries it to the satin on the inverse of m. So the satin keeps a quarter turn exactly when $m^2 \equiv -1$ modulo n, keeps a mirror exactly when $m^2 \equiv 1$, and otherwise keeps only the half turn every lattice has. Computed for every satin from five to sixteen ends, the groups come out p4, then cmm or pmm, then p2, in exactly that correspondence.

Each symmetry forces something on the shortest step, because a symmetry carries shortest steps to shortest steps.

A quarter turn forces a tie at right angles. The turned copy of u is as short as u and perpendicular to it, so the lattice is square. From five to forty ends the square satins are exactly the ones whose move squared is minus one, at five, ten, thirteen, seventeen, twenty-five, twenty-six, twenty-nine, thirty-four and thirty-seven ends.

A mirror forces the row onto forty-five degrees, or a tie. The reflected copy of u is also a shortest step. Either it lies along the same line as u, which puts u on one of the grid’s diagonals, or it points somewhere else and the lattice has two shortest steps of equal length. The eight-end satin, the twelve-end satin and the sixteen-end satin on seven take the first way; fifteen on four takes the second, as do twenty-four on five and thirty-five on six. Every mirror satin up to forty ends does one or the other.

With neither, nothing forces a tie, and none occurs. Across the 356 satin moves between five and forty ends whose square is neither one nor minus one, every one has a single shortest step. These are also the satins that are chiral in the plane, since no reflection carries them to themselves, and the handedness shows as the lean of that row: the satin on the complementary move is the mirror image, and its row leans the other way, as a twill’s direction does.

Twelve and sixteen ends keep a stronger one #

Twelve ends admits only the moves five and seven, and both have a mirror. So the twelve-end satin’s closest marks lie on forty-five degrees at √8, and its next step is √18, perpendicular to the first. The ratio is 1.50, and there is no other twelve-end satin with a move number to choose instead.

Sixteen ends has six moves. Seven and nine are mirror satins whose closest spacing is only √8, along forty-five degrees, with the next step at √34. Three, five, eleven and thirteen spread further, to √10, and they are the ones the spread ranking picks. Their lattice has no symmetry to force a tie, and the next step is √26. The ratio is 1.61, along a row that climbs three picks for every end.

So the choice at sixteen ends is between a better-spread satin with a steep row of ratio 1.61 and a closer-packed one whose row lies where a twill’s would and is stronger still, at 2.06. Neither escapes. That is worth knowing before a sixteen-end satin is specified in the hope of losing a line an eight-end one showed: whichever move is threaded, the row comes back stronger.

The best satin at each order, read for its row #

Laid out order by order, the ratios never settle into a trend. Five ends ties. Seven gives 1.41, eight 1.12, and nine 1.84 — the strongest row up to sixteen ends, on a satin whose closest marks sit only √5 apart. Ten ties again, eleven gives 1.14, twelve 1.50, thirteen ties, fourteen gives 1.41, fifteen ties, and sixteen gives 1.61.

Nothing in that sequence tracks the order, because nothing about the lattice does. What decides is whether the order admits a move whose square is minus one, which needs every odd prime factor of the order to leave a remainder of one on division by four and allows at most one factor of two, or a mirror move that happens to tie.

Carried to forty ends, the best move ties at twelve of the thirty-five orders that have a satin, and at the other twenty-three it keeps a single row. Most satins still have a diagonal, and so does the best satin at most orders. The ties do not thin out as the orders rise, but they do not become the rule either, and strong rows keep turning up between them: twenty ends at a ratio of exactly two, and thirty at 1.58.

The ideal is hexagonal, and fifteen ends nearly reaches it #

A square is not the best a lattice can do. Among all lattices with one point to every n units of area, the one whose closest points sit furthest apart is the hexagonal one, with three shortest steps of equal length at sixty degrees, and its shortest step squared is 2n/√3, about 1.155 times n. The reduction that finds the shortest steps also proves the bound, since no reduced pair can be longer.

A grid of ends and picks cannot hold a hexagonal lattice exactly, because no three points on a square grid form an equilateral triangle — the same fact that removes five of the seventeen plane groups from every weave. It can come close, though, and fifteen ends does. The fifteen-end satin on four has shortest steps of four ends and one pick, and of one end and four picks, both √17, and a third of three ends and three picks the other way at √18. Three nearly equal directions, sixty-two and fifty-nine degrees apart: a slightly squashed hexagon.

Its closest spacing squared is 17 on an order of 15, which is 1.13 times the order, against the ceiling of 1.155. No satin from five to forty ends does better relative to its order. At every square order the best spacing squared is exactly the order, and nothing else at those orders beats the square.

An irregular satin can scatter where a regular one cannot #

Everything so far concerns satins with a move number, and those are a small minority of the satins that exist. Keep one mark per end and per pick with no two touching, drop the move, and the marks no longer have to lie on any lattice. There is then no shortest step, only closest pairs, and nothing requires the closest pairs to point the same way.

At seven ends every one of the ten satins, regular or not, keeps its closest marks √5 apart. The four regular ones line all their closest pairs up along a single direction. The least orderly irregular one has six closest pairs pointing four different ways, with no direction holding more than two of them, and one of its seven marks has no neighbour at that distance at all. By closest spacing it is exactly as good as the best regular satin, and by the row test it has no row.

Nine ends does the same with more room. All 350 satins tie at √5, and the most scattered irregular one points its eight closest pairs four ways, with no direction holding more than a quarter of them. Nine is also the order whose best regular satin has the strongest row of any up to sixteen. At nine ends, then, an irregular satin gives up nothing in spacing and removes a row of ratio 1.84.

At eight, ten and eleven ends only a regular satin spreads furthest #

That trade is not on offer at every order, and the census says where it is. At eight ends there are forty-seven satins, counted up to where the repeat is started, and only two reach the best closest spacing of √8: the regular ones on three and five. At ten ends two of 3,005 reach √10, and both are regular. At eleven, four of 28,722 reach √10, and all four are regular.

At those orders an irregular satin pays for its scatter in spacing, and spacing is what the satin was chosen for. At eight ends the regular satin’s row has a ratio of only 1.12, so going irregular to lose it gives up the closest spacing to remove the weakest row on the table. At ten ends the regular satin already ties, and there is nothing to remove.

So the orders sort into three kinds. Where a square or tied lattice exists — five, ten, thirteen, fifteen — the best regular satin already has no row. Where every satin ties on spacing — seven and nine — an irregular satin removes the row for nothing on the cloth, and its only cost is a lifting plan written out in place of a stepped treadling. And where only the regular satins reach the best spacing — eight and eleven — the row is the price of the spacing and cannot be bargained away.

The counter is a guess at the lattice, and it misses the squares #

The trade’s rule for picking a move is the counter: take the move nearest the square root of the order. Read against the lattice it is a sensible guess. A step of one end and m picks has a length of about m; the lattice has one mark in every n cells; and a square cell of area n has a side of √n. So a move near √n makes the construction’s own step about as long as a balanced lattice’s shortest.

It fails wherever the balanced lattice’s shortest step is not the construction’s step. Thirteen ends is the first case. Its square satin is on five, and the shortest step of that satin is not one end and five picks but three ends and two picks, at √13; the counter takes four instead, at √10. Thirteen, twenty-five and twenty-nine are three of the four orders up to thirty at which the counter is beaten, and all three are square orders whose square move sits far from the root. At seventeen ends the square move is four, which is the root, and the counter finds it. The fourth failure, twenty-one ends, belongs to a mirror satin on eight with its row on forty-five degrees — again a lattice whose short steps have nothing to do with the size of the move.

The continued fraction sees the circuit and not the row #

The finer rule reads the move’s continued fraction and prefers the move whose expansion has no large term. A large term means some multiple of the move lands close to a multiple of the order, so a few steps of the construction nearly close the circuit and the marks bunch along that nearly closed path.

That catches one way a lattice can have a short step and misses another. At sixteen ends the reading prefers seven and nine, whose expansions have no term above three, over three and five, whose expansions carry a five. But two steps of seven land fourteen picks up, which is two picks short of the repeat, two ends along — a short step of √8 made without any circuit nearly closing. The spread is √8 against √10, and the reading picks the worse move, first at sixteen ends and then at eleven of the fifteen orders from sixteen to thirty. The reduction sees both kinds of short step because it looks at every step at once, which is why it is the measurement and the expansion only a reading of it.

What was counted, and how #

Every regular satin from five to forty ends was reduced to its two shortest steps, and at sixteen ends and below the shortest step was compared with a direct search over every pair of marks on the repeat. The square satins were confirmed to be exactly the ones whose move squared is minus one; every mirror satin was confirmed to lay its row on forty-five degrees or to tie; every satin with neither congruence was confirmed to have a single shortest direction; and a satin and its mirror image were confirmed to have the same lattice shape and, up to sixteen ends, the same plane group.

That last comparison found a fault in how plane groups were being named, and the fault is worth recording. Whether a mirror comes with a centred lattice was decided by looking only at lattice steps written inside one repeat, which missed every step that points backwards. The eight-end satin on five came out pmm while its mirror image, on three, came out cmm. No four-by-four draft is large enough to show the difference, so the count of twelve groups among the 22,874 four-by-four drafts is unchanged by the repair, and the satins that exposed it are now compared by name.

The hexagonal ceiling was confirmed at every order to forty, and at every order with a square satin the best spacing squared was confirmed to equal the order exactly. The irregular satins came from the same exhaustive search that found the six-end satin, which runs to eleven ends; each was scored by its closest spacing, measured round both edges of the repeat, and by how its closest pairs divide among directions.

Where the model stops #

A mark is not a point. In cloth an interlacing is a short interruption in a float, and in a warp-faced satin the floats give the surface a grain along the warp that the lattice knows nothing about. A row seventy-two degrees off the weft runs within eighteen degrees of that grain, and a row at forty-five crosses it obliquely. There is no reason to expect the two to show equally.

The yarn has a direction too. The fibres on a float lie at the twist angle, and the rule weavers use for twills is a subtraction between that angle and the line’s. A satin’s row has an angle, so the same subtraction applies to it, and a satin whose row runs with the fibres should hide it better than one whose row runs across them. Nothing here computes that.

Closest spacing is one number. The irregular satins that remove the row at seven and nine ends were chosen by where their closest pairs point. Their next-closest pairs may still line up, and a regular satin’s longer steps make rows of their own. A full account would weigh every distance, and the natural instrument is the pattern’s Fourier transform, which for a regular satin is another lattice and for an irregular one is the repeat’s lattice with uneven weights.

And nothing here says what ratio an eye can see. The eight-end satin’s 1.12 is weak and the sixteen-end satin’s 1.61 is strong only in the sense that one number is larger. Whether 1.12 shows at a fine sett, with the lustre of the floats lying across it, is a question for a set of cloths and a set of observers.

Still open: which of these rows a weaver actually sees #

The arithmetic says where the rows are and how strong each is against the others. It predicts that a sixteen-end satin on three shows a steeper and stronger row than an eight-end one, that a ten-end or thirteen-end satin shows none, and that a nine-end satin shows the strongest row below sixteen ends unless it is woven irregularly. Every one of those can be woven and looked at.

The measurement is simple to state. Weave a set of satins at one sett in one yarn, photograph them under raking light, and read the strength of the photograph’s Fourier transform along each lattice direction. That would say whether the lattice is the right model of the surface, and at what ratio a row appears. Until it is made, the best satin means the best-spread satin, and the row is a prediction.

The order itself is capped from the other side. The float limit decides how long a float a yarn and a finish will tolerate, which decides how many of these orders can be woven at all — and on a yarn that stops at eight or twelve ends, a satin with no row was never on the table.

Who worked it out #

The reduction of a two-dimensional lattice to its shortest steps is Lagrange’s, from 1773, in the same work that showed the hexagonal lattice to be the densest arrangement of points in the plane; Gauss gave the reduction its lasting form in 1801 as a theory of binary quadratic forms. The symmetry of satins as periodic patterns was set out by Grünbaum and Shephard around 1980, in their geometry of woven fabrics. The census of rows and ties satin by satin, and the comparison against the irregular satins, was computed directly for this essay.