A blind set hides less the thinner it is spread

Colour and weave as a two-colour problem found the fact the whole of colour-and-weave rests on: where the warp and the weft crossing at an intersection are the same colour, the intersection looks the same whichever thread is on top, so the draft there is invisible. The colour order that hides least swept every two-colour order against every other over the catalogue of 22,874 four-by-four drafts and found that the arrangement of the colours never matters, only their counts — and that the number of blind intersections is not the statistic: three pairs of orders with eight blind intersections each separate the catalogue three different ways.

It left two questions. What a third colour does, and — the one underneath it — if not the blind count, then what.

The answer to the second is the shape of the blind set. With more than two colours the blind intersections can be arranged in ways two colours cannot reach, and across all of them one rule holds: at a fixed number of blind intersections, the catalogue keeps more of its cloths apart the more threads those intersections are spread over.

A blind set is a union of rectangles #

Colour an order in any number of colours and look at where it is blind. An intersection is blind when its pick and its end are one colour, so the blind intersections of one colour are every pick of that colour against every end of it — a rectangle, once the picks and ends of each colour are gathered together. Two colours never share a thread, so the rectangles of different colours sit on different picks and different ends.

So a colour-order pair’s blind set is a union of rectangles, one per colour used in both systems, with dimensions (picks of that colour) × (ends of that colour). Two colours make two rectangles, three make three, four make four.

The arrangement of the colours cannot matter, in any number of colours. The catalogue of four-by-four drafts is closed under permuting its picks and permuting its ends — shuffle the rows of a draft that interlaces and it still interlaces — so any two colour orders that give the same rectangles up to a shuffle confuse exactly the same number of cloths. The two-colour essay found that arrangement never matters by sweeping every order; here it follows from one symmetry, and it holds at three colours and at four. What is left is the rectangles’ dimensions, and that is small enough to take whole.

Three colours fill in the counts two colours skip #

With two colours, $a$ dark ends of four and $b$ dark picks, the blind count is $ab + (4-a)(4-b)$, which can only be 0, 4, 6, 8, 10, 12 or 16. A single blind intersection is impossible, and so is any odd number.

A third colour makes one blind intersection possible — one end of a colour, one pick of it, and every other thread in a colour the other system does not use — and then every count from nought to ten. Eleven, thirteen, fourteen and fifteen cannot be made in any number of colours, because the blind count is a sum of products of two ways of splitting four, and those sums skip them.

A fourth colour adds no count at all. What it adds is shapes. Four blind intersections can be one strip of four, a square, two strips of two, a strip of three with a single beside it, or — only with four colours, each used once in each system — four singles, one to a pick and one to an end. A fifth colour adds exactly one more shape, three scattered singles, and after that there is nothing left to add: twenty-nine shapes of blind set exist at four by four, and every one of them is in the census.

The first blind intersection is the dearest #

Before the shapes, the plainest reading of the census is the cost of blinding one intersection at a time, spread as thinly as possible.

One blind intersection takes the catalogue from 22,874 surfaces to 14,414 — thirty-seven per cent of the distinctions gone for one cell of sixteen. Two scattered cells leave 9,096, three leave 5,744, four leave 3,632. Each additional scattered cell removes a little over a third of what is left, so the fractions fall roughly as a power: 0.63, 0.40, 0.25, 0.16.

That is steeper than the count alone suggests. A blind cell hides one bit, and if every bit were independent, one blind cell would at most halve the catalogue; it removes only thirty-seven per cent because for many drafts the draft with that one cell flipped does not interlace, so there is nothing for it to be confused with. The fraction per cell is constant to within a few parts in a thousand — 0.630, 0.631, 0.632 — because thinly spread blind cells barely interact. Gather them and they interact: the square’s four cells keep 2,402 where four independent-acting cells would keep 3,632.

The same four, seven ways #

The hero figure is those seven shapes of four blind intersections, ranked by how many distinct surfaces the 22,874 drafts collapse onto.

Every row hides four intersections of sixteen, and the best keeps 51 per cent more of the catalogue apart than the worst. The two-colour essay could only reach the strip of four — the one shape of four blind cells two colours can make, with one light end against a light weft — and it sits fifth of seven.

The ordering is not arbitrary. Read the second column: the separation rises with the number of picks and ends the blind set touches, with every tie at six broken by a few surfaces either way. Four singles touch eight threads; the square touches four.

Why spreading helps #

A draft is admitted to the catalogue only if every end and every pick interlaces — no thread lies entirely on one face. That constraint is what makes blind cells cost different amounts depending on where they are.

A blind cell hides one bit of the draft, but the bits it hides are not free. Two drafts that differ only in blind cells are confused, and they are both in the catalogue only if both interlace. A thread with one blind cell and three visible ones has its visible cells deciding almost everything about whether it interlaces, so the blind bit is often forced; a thread whose cells are all blind is free to take any of its fourteen interlacing patterns and every one is invisible. Concentrating the blind cells on a few threads gives those threads whole patterns to hide; spreading them gives each thread one cell, most of whose values are already determined by what is seen.

The largest-class column shows the same thing from the other end. A strip of four is one pick entirely blind, and its largest class is fourteen — exactly the fourteen interlacing patterns of a four-cell pick. The square’s largest class is sixteen, every pattern of a two-by-two block, because each of its threads keeps two visible cells that already guarantee interlacing and the block is left completely free.

The rule holds at every count #

At every blind count the line rises: two blind cells in one strip keep 8,390 surfaces and the same two scattered keep 9,096; three in a strip keep 4,802 and a pair with a single 5,304. At five, three shapes run from 1,520 to 1,958 as the spread goes from six threads to eight.

Six is where the third colour stops helping. The best shape of six blind intersections is two strips of three at right angles — one end of a colour against three picks of it, three ends of the other against one pick — and that is a two-colour order, one dark end against three dark picks, at 1,022 surfaces. Every shape a third or fourth colour can make at six blind intersections gathers two of them into a square somewhere, touches fewer threads or the same number less evenly, and keeps fewer: 962, 890, 686.

So a third colour is not an improvement as such. It is a finer dial and a larger set of shapes, and whether a particular extra colour helps depends on whether it lets the blind cells be spread further or forces them together.

The two-colour table, read by shape #

The two-colour essay’s own anomaly is the same rule. At eight blind intersections its three pairs kept 256, 254 and 196 surfaces, which looked like the blind count failing to be a statistic.

Read as shapes they are in order. Two two-by-two squares touch all eight threads and keep 256; a strip of two with a three-by-two block also touches eight and keeps 254; a two-by-four block touches six and keeps 196. The two at eight threads differ by two surfaces, which is the size of the tie-breaks everywhere in the census. The blind count was never the statistic because it is the area of the blind set, and what decides how much a blind set hides is how it lies across the threads — its spread first, and its exact arrangement after that.

A tartan spreads its blind cells as far as they can go #

The shape rule says something pointed about one family of cloths. A tartan is a check whose warp and weft carry the same colour order, so every colour’s blind rectangle has as many picks as ends — a tartan’s blind set is a set of squares on its diagonal — and, because every colour it uses is in both systems, those squares touch every thread in the cloth.

So a tartan is never badly spread. Whatever its colours, its blind set covers all eight threads, which is as far as a blind set can reach. What a tartan pays in instead is the count: its blind intersections number the sum of the squares of its colour counts, which is least when the colours are used equally. Two colours two-and-two make eight blind intersections, the best eight-blind shape in the census at 256 surfaces; a colour used twice and two used once make six, keeping 962 — a little under the 1,022 of the non-tartan pair with the same spread, the difference being the few-per-cent tie-break between squares and strips.

The tartan with every colour used once is the extreme — four colours, one end and one pick of each — whose squares are all one by one. It is blind at only four intersections, those four touch every thread, and it is the best four-blind cloth in the whole census. A tartan hides least when it repeats no colour, and it hides more with every repeat, not because the blind cells gather but because there are more of them.

How the census was taken #

The catalogue is the site’s four-by-four one: every 0/1 matrix of four picks and four ends in which every pick and every end interlaces, 22,874 of them.

The blind shapes are enumerated directly, since arrangement cannot matter: every list of rectangles on disjoint picks and disjoint ends with the picks and the ends each summing to four or less, 29 of them up to transposition, each with the fewest colours that can make it — one per rectangle, and one more for any ends and any picks left matching nothing. For each, the blind set is built and the catalogue is reduced to the drafts’ bits on the visible cells; two drafts with the same visible bits are one surface, so the number of distinct surfaces is the number of distinct reduced masks, and the largest class is the largest number of drafts sharing one. No colour is carried through the count, because none is needed.

A shape and its transpose are required to separate the catalogue alike, because transposing a draft is the catalogue’s other symmetry, and a sample of real colour-order pairs is shuffled thread by thread and required to separate it exactly as before. Shapes are keyed with their orientation — a strip along a pick and a strip along an end are different once there are two of them — and a first version that ignored orientation was caught by the same check: two strips along picks and one along each direction had been filed together, and they differ by two surfaces.

What a designer can do with a third colour #

The census turns into three plain instructions for anyone laying out a colour order.

Blind cells cost most when they share threads. An end that matches three picks in colour hides three cells on one thread; three different ends each matching one pick hide three cells on three threads and keep a fifth more of the catalogue, 5,744 surfaces against 4,802. The instruction is to spread matches across threads rather than concentrate them.

A colour used in one system only is a colour that blinds nothing. The corner of the two-colour grid — a solid warp against a solid weft of another colour — is the limiting case, and every intermediate is available with a third colour: an end in a colour the weft never carries is an end whose whole length is visible. A colour order beats the weave it is threaded on found the colour order’s repeat interacting with the weave’s; this is the other half of that interaction, the part that decides which weaves the eye could in principle tell apart.

A tartan cannot be improved by rearranging, only by recolouring. Its blind set already touches every thread, so the only lever left is the number of colours it repeats — and the finest colour-and-weave effects need the rarest loom is the reminder that every extra colour in the weft is a shuttle or a box the loom has to provide, and a colour-and-weave look costs its cheaper order prices exactly that choice.

None of this touches whether the effect looks good. It says which cloths could be told apart by looking, which is the prior question a pattern book never asks: how many cloths there are is 22,874 at this size, and a colour order decides how many of them anyone will ever see.

What the census cannot say #

It says what the cloth’s surface separates, not what an eye does. Two surfaces that differ in one intersection of sixteen are distinct here and may be indistinguishable to anyone at arm’s length; a scattered blind set is best for the catalogue and may produce a surface so busy that nothing in it is read. That half of the question the essay before this one named is a question about vision, and it stays open.

Four by four is small. The spread rule is measured on one catalogue at one size, where eight threads is everything. Whether it holds as a rule at eight by eight, where the same number of blind cells can be spread over many more threads and the interlacing constraint bites less on each, is a larger census than this one.

Who found which part #

Colour-and-weave is old, and so is the observation that some colour orders make different weaves look alike; pattern books group their effects by appearance for exactly that reason. The two-colour essays here put numbers on it and found the arrangement irrelevant.

The rectangles, the reach and the spread rule are new here. They follow from two symmetries of the catalogue — permuting threads, and transposing — and from the interlacing rule, which is the only thing that makes one blind cell cost a different amount from another.

Still open: whether the rule survives a bigger repeat #

At four by four the census can take every shape, and the rule it finds is that spreading beats gathering. At eight by eight a blind count can be spread much further, and the catalogue is too large to take whole, but the question does not need the whole catalogue: the separation of a blind set is a count of distinct visible masks, and for a scattered set it can be estimated by sampling drafts and counting collisions. Whether the separation of a fixed blind count keeps rising as it is spread over sixteen threads instead of eight, or levels off once no thread carries more than one blind cell, is the measurement that would turn this into a rule a designer of colour orders could use at a real repeat.