A six-end shading can be even or have a plain centre, not both

A tone step does not need a satin found that what a shading actually requires is a Latin square — a split of the repeat into parts, each with one mark in every end and every pick — and that at six ends, where there is no satin at all, a repeat still has 1,128,960 of them. It counted every decomposition at four ends and at five, and stopped before six because holding the list would have cost a gigabyte. Two six-end squares were built by name instead: the cyclic one a twill writes, whose best chain floats 5, 2, 2, 2, 5, and the table of the symmetric group on three letters, which floats 5, 3, 2, 2, 5.

The list never needed holding. Each square can be measured and let go — its sixty-two tone steps as six bits a row, its 720 chains walked, a handful of counts kept — and the whole census then takes seconds. It agrees with the held census at four and five ends, profile for profile, and at six it answers what two named squares could not.

Two thousand eight hundred and sixteen of the 1,128,960 hold every tone between the extremes to a float of two. Set aside where the repeat starts and which way it is read, and those are sixty-four different shadings, the twill’s among them. Seven in ten decompositions cannot bring their worst middle tone below four. And the one thing no six-end shading can do is the thing a designer might most want — an even ramp with a plain weave at its centre — because a plain weave costs every other tone in its chain a float of at least three.

One decomposition in four hundred holds every tone to two #

A six-end shading has five tone steps, with one to five marks in every end and every pick. The two outer steps are forced: one mark in six floats five in the weft and five marks in six float five in the warp, whatever the decomposition. Everything a choice of square can change lies in the three tones between, and the useful question about a square is how short the worst of those three can be made by choosing the order its parts are added in.

Across the census the answer is two for 2,816 squares, three for 325,486 and four for 800,658. No square does better than two in the middle and none does worse than four. So seventy-one per cent of the ways to build a six-end shading leave at least one middle tone floating four, and one in 401 keeps all three at two.

The bars above are drawn to one scale, and the first is a sliver. That is the finding in the form an eye takes it: the shading anybody would want is almost none of the ways of making one. Designing to a float limit ranks weaves by their longest float, and by that ranking a decomposition picked at random has a seventy-one per cent chance of a ramp whose loosest middle tone floats twice as far as the best ramp’s.

The square above is not special among the 2,816. It is the first of the largest classes, and there is nothing about it — no diagonal, no group structure — that would make a designer reach for it. That is the practical meaning of the count: the even ramps are not a family with a recognisable look, and they have to be found by asking every square.

Sixty-four shadings, and the twill’s is a class of two #

A decomposition is written as a square with its first row in order, and that convention counts one shading many times. Start the repeat at a different pick or a different end, or read the picks from the bottom, and every float of every tone step is unchanged — the cloth is the same — but the square written down is a different square. Grouping the squares that differ only in that way turns the 2,816 into sixty-four classes: twenty-four of seventy-two squares, twenty-four of thirty-six, eleven of eighteen, four of six, and one of two.

The class of two is the twill’s. The cyclic square written from any corner is the cyclic square again, and the only other form it has is its mirror: the same diagonal running the other way, the difference between an S twill and a Z. So the construction every account of shading reaches for is one even shading in sixty-four, and it is the one with the fewest ways of being written down, which is another way of saying it is the most symmetric of them.

The other sixty-three are not twills, and none is written by a satin, since six ends has no satin to write one. And the twill’s diagonal is what holds a tone ramp’s surface level, so what the other sixty-three do to a cloth’s thickness is a different question from what they do to its floats — the census answers the second and leaves the first open.

A plain weave costs its neighbours a float of three #

Five hundred and seventy-six squares, in thirty-one classes, can put a plain weave at the midtone. A plain weave is the firmest cloth there is — the most interlaced a repeat can be — and a shading with one at its centre is at its most matt exactly where its tone is middling, which is the shape the account of what a shading changes found in the eight-end spread chain. None of the 576 can put it between twos.

The reason takes two sentences and no census. A plain weave on six ends is the checkerboard: every pick has a mark on every other end. The tone one part fewer takes one mark out of every pick, and the gap it leaves joins the unmarked ends on either side into a weft float of three. The tone one part more puts a mark into one gap on every end, and joins the marks above and below it into a warp float of three.

Nothing in that depends on which part is taken out or put in. There are thirty-six ways to take a part out of a six-end plain weave and keep every end and pick uniform, and thirty-six ways to add one; all seventy-two float exactly three, and so do all 1,152 at eight ends. In the census, 41,472 chains pass through a plain weave, and every one of them floats three on both sides of it.

The argument reaches further than the neighbours. A chain is nested, so every tone below its plain weave is a subset of the plain weave and every tone above is a superset. A subset keeps its marks on alternate ends and leaves a gap of at least three wherever it has dropped one; a superset keeps every mark of the plain weave and joins at least three wherever it has added one. In any chain through a plain weave, every other tone floats at least three, on any even repeat whatever.

The twill offers two shadings and cannot combine them #

The cyclic square reaches a plain weave. Its best chain is 5, 2, 2, 2, 5 and it also admits 5, 3, 1, 3, 5, and the account that found both called the choice between them a matter of what a designer wants rather than a fact about the square. The census makes it a fact about every square: 5, 2, 1, 2, 5 — the chain that would combine the two — is exactly the chain a plain weave forbids, and no decomposition of six ends has it.

The two chains are different cloths in the one respect a shading exists for. The even chain changes tone at a steady rate and keeps its longest float constant through the middle: three tones with floats of two. The plain-centred chain puts a sudden firm band into the middle of the ramp, between two tones looser than any middle tone of the even chain. A designer choosing the twill’s square is choosing between those two ramps, and there is no third one to be had from any other square.

At six ends the plain-centred chain has no freedom at all. Its extremes float five because the repeat forces it, its midtone floats one because it is the plain weave, and its other two tones float three because the plain weave forces it. Every plain-centred six-end shading, from any of the thirty-one classes, has the profile 5, 3, 1, 3, 5, and what the classes differ in is everything the float does not measure.

One other class offers the same choice #

Of the sixty-four even shadings, two can also reach a plain weave: the twill’s class of two, and one class of six. The class of six is not a twill, a satin or a group table, and it offers exactly the choice the twill offers, from a square no coset construction writes.

Drawn in its other order, the same square passes through its plain weave, and pays the three either side of it that every plain weave charges.

Eight squares are both even and plain-reaching, and those two classes hold all of them. The other 568 plain-reaching squares cannot hold their middle tones to two in any order; for them the plain-centred chain gives up nothing they could otherwise have had, since their best alternative has a three in it anyway.

Being good at each tone is not being good in one chain #

The census can also ask each middle tone on its own: of all the unions a square has at that tone, how short is the shortest float. The second tone can be held to two in 61,440 squares. The midtone can be a plain weave in 576 and held to two in 428,058. Those are large numbers beside 2,816, and the gap between them is the whole cost of nesting.

A chain needs the second tone inside the midtone and the midtone inside the fourth. A square can own a good two-part union and a good three-part union that do not contain one another, and then no chain uses both. Of the 61,440 squares whose second tone can float two, only one in twenty-two can go on to hold two in a nested chain.

That is the same rule the edge between two tones turned on, arriving from the other side. There nesting was what kept an edge from floating further than its tones; here it is what stops a square from choosing its best union at every tone independently. The rule that makes a shading clean at its boundaries is the rule that makes a good one rare.

A group table is likelier to be even, and still rarely is #

Eighty of the six-end squares are the multiplication tables of a group — the cyclic group of order six and the symmetric group on three letters, in every numbering that puts the first row in order. At four ends being a group’s table did not decide whether a square could reach a plain weave. At six the census gives the proportions.

Four of the eighty hold every middle tone to two, twenty-eight are held to three and forty-eight to four. Five per cent of group tables are even against a quarter of one per cent of all squares, so a group table is twenty times likelier than a random square to make an even shading — and ninety-five per cent of group tables still do not, and sixty per cent are among the worst. The algebra that produces the coset construction tilts the odds and settles nothing.

Every uniform tone step is one cloth, at every repeat #

The census builds 434,540 distinct tone steps — every six-by-six draft with the same number of marks, from one to five, in every end and every pick, since every such draft is a union of some decomposition’s parts. Every one of them hangs together: not one comes apart into two cloths.

The account that counted four and five ends found the same and called it measured rather than proved. It can be proved, for every repeat, by counting marks twice. Suppose a uniform draft with k marks in every end and pick, k between one and n − 1, came apart. Then some set of its threads would form a layer nothing else lies above: every end that lies over one of its picks belongs to it, and every pick that lies over one of its ends belongs to it. Say it holds a ends and b picks.

Each of its picks has k ends lying over it, all in the layer, so its ends carry at least bk marks; they carry exactly ak, so a is at least b. Each of its ends has n − k picks lying over it, all in the layer, so its picks carry at least a(n − k) gaps; they carry exactly b(n − k), so b is at least a. So a equals b, and both counts are exact: the layer’s picks have all their gaps at the layer’s ends as well as all their marks. A pick with every crossing at the layer’s ends means the layer holds every end, and then every pick. The only layers are the whole cloth and nothing, so a uniform tone step is one cloth on any repeat, and the 434,540 are the proof’s six-end instance rather than its evidence.

At eight ends a plain centre costs the same #

The eight-end census is out of reach, at 2.7 × 10¹⁵ squares, but the plain weave’s cost is not a census result and it does not stop at six. At eight ends the extremes float seven and every tone in a chain through a plain weave floats at least three. The spread chain the first account of shading drew floats 7, 3, 3, 1, 3, 3, 7 — three at every tone the plain weave constrains — so no eight-end shading with a plain centre can do better than the spread chain, at any tone, whichever decomposition it is built on.

What eight ends leaves open is the other half of the choice. Whether any eight-end decomposition holds all five middle tones to two, as 2,816 six-end squares hold their three, is a question the plain-weave argument does not touch and no enumeration will reach.

What was counted, and how #

Every six-end Latin square with its first row in order was generated one at a time and not stored, by a backtracking search whose count comes to 1,128,960, the number the tabulated reduced count implies. Each square’s sixty-two tone steps are built as six-bit rows, and a longest float is read from a table of the longest cyclic run in every six-bit value — along each pick for the weft and down each end for the warp. The 720 orderings of the parts are walked in the same order, under the same tie-break, as in the held census, and at four and five ends the two censuses agree profile for profile.

The two counts reported do not depend on the tie-break. The inner float is the least, over every ordering, of the worst of the three middle tones; the total is the least sum of a chain’s floats. A class is the set of squares reached from one another by shifting or reversing the picks or the ends, 144 transformations in all, with the parts renumbered each time so the first row is in order; every tone step’s floats are unchanged by all of them.

Each distinct tone step is asked once whether it is one cloth, through the same criterion every draft in the collection is checked against. The plain weave’s neighbours are checked on every way of taking a part out of or adding a part to a plain weave at four, six and eight ends. And a census at seven or at three ends, a plain weave’s neighbours on an odd repeat, and a chain through a plain weave on a square that has none are all refused.

Where the census stops #

The float is one criterion. Two squares in different classes with the same profile are different cloths — in their crown line, their firmness and the relief a finished ramp takes — and the census ranks them as equal. The sixty-four even classes are sixty-four shadings with the same float profile and nothing is known about how they differ as cloth.

The classes set aside the repeat’s origin and its reading direction, and nothing else. Turning a draft through a right angle exchanges which system carries which float, and turning the cloth over exchanges the tones; neither is counted as the same shading here, so a designer indifferent to those would see fewer than sixty-four.

The tie-break still names a profile. Where two chains on one square have the same total, the one printed is the first found, which is why 5, 3, 2, 2, 5 and its reverse are counted as different best profiles in different numbers. The inner float and the total, which are what the argument uses, do not depend on that choice.

And a float of three beside a plain weave is a count, not a sighting. Whether the looser tone either side of a firm band shows in cloth depends on the sett, the yarn and the light, which is how a figure shows, and none of those is here.

Still open: whether the other sixty-three keep the surface level #

A shading chain built on a twill keeps a tone ramp’s surface level where one built on a satin digs a valley into it, because in the satin’s chain the firmer steps are pressed harder and finish thinner. The sixty-three other even six-end shadings have the twill’s floats without its diagonal, and the pressure model that measured the valley can be run on each of them. Whether an even ramp without a diagonal stays level, sags like a satin or does something neither construction does is the next question the census makes it possible to ask, and it is the one that decides whether the sixty-three are worth a designer’s attention or only a count.

Who worked it out #

The six-by-six Latin squares were counted correctly in the 1930s — the 9,408 reduced squares of order six are usually credited to Fisher and Yates, in 1934 — and König’s theorem on regular bipartite graphs is from 1916. The fact that a shading’s tones are a Latin square’s unions was argued in the account of tone steps without satins. The six-end census of float profiles, the classes, the bound a plain weave puts on its chain and the double count for integrity are elementary once the question is asked, and they were computed and argued directly rather than found in a weaving source.