Only one start in six can reach the shallowest shading

An eight-end shading is pinned at three everywhere but its centre proved that no eight-end shading can hold its second or third tone to a float of two, so the tightest even family floats three at tones 2/8, 3/8, 5/8 and 6/8 and leaves only the centre, 4/8, free. With the centre held to two, it found the family reaching the rate floor — 42.8 micrometres of sag on a sheeting — and then sinking further: six depths among the first hundred thousand chains a search reached, up to 72.7 micrometres.

That search had to stop. Latin squares of order eight number about 10²⁰, and the essay said plainly that its six depths were six that exist rather than all there are, and that its counts described the walk’s order rather than the family’s proportions. It ended on the question the stopped walk could not reach: what decides the centre’s rate, and whether the shallowest ramp can be reached from every start.

Both have exact answers, because the family, though far too large to walk, is not too large to count.

A chain splits at its centre #

A shading chain is eight parts — disjoint permutation matrices, one mark in every end and every pick — added one at a time, so that tone k is the first k of them. The family in question holds tones 2/8 to 6/8 to face floats of 3, 3, 2, 3, 3: no weft floating over more than that many ends along a pick, no warp over more than that many picks down an end.

Look at what those bounds actually constrain, tone by tone.

Tone 2/8 is nearly fixed. Two marks in a pick of eight leave six unmarked places, and a float of three means they fall as two runs of three — so the second mark sits opposite the first, four ends along, in every pick. Each end then carries two marks, so each of the four opposite pairs of ends, {0, 4}, {1, 5}, {2, 6}, {3, 7}, is carried by exactly two picks. There are 8!/2⁴ = 2,520 such second tones, and nothing else can be one.

Those 2,520 second tones do not all interlace alike. A pick always changes face four times, but an end’s two marks can sit together or apart, and the tone’s rate runs from 0.375 — when every end’s two marks are neighbours, which 48 second tones manage — to 0.5, when none are. The stopped walk’s essay gave 0.375 as the floor the float bound allows at the second tone; it is attained, by exactly those 48.

Tone 3/8 is free. A third mark in a pick splits one of its two runs of three and leaves the other, so it floats three whatever the third part is. Down an end, three marks cannot float more than three.

Tones 5/8 and 6/8 are the same, turned. Read from the back, the unmarked cells of the later tones are a chain of their own, with picks and ends exchanged — so the upper half of a chain is the lower half’s arithmetic applied to the centre’s complement, turned through a right angle.

So a chain is a centre with two independent halves hanging off it. The lower half is any second tone inside the centre, any split of it into two parts, and any split of what is left into two more; the upper half is the same question asked of the complement. The family is the sum, over every centre, of the lower half’s count times the upper half’s. A walk of a billion chains becomes a list of centres and two small searches beside each.

Fourteen depths, counted #

There are 19,639,930 tones of four marks per thread on eight ends whose face floats are no longer than two. Each was tested for both halves; the counts were multiplied and added; and the whole was divided by eight to match the stopped walk’s normal form, which fixes where the repeat starts.

The family is 1,001,574,400 chains, and it sinks to fourteen depths: every worst rate from 0.5 to 1.0 in steps of a thirty-second, except 0.84375 and 0.90625, which no chain has. The shallowest is the floor, 42.8 micrometres on a sheeting; the deepest is a plain centre, 83.6. Eight of the fourteen are depths the walk never saw: five fill the thirty-seconds between its sixteenths, and three — 76.7, 80.5 and 83.6 micrometres — lie beyond the deepest it reached.

The distribution has one heavy class. Thirty-eight per cent of every even eight-end shading sinks 68.1 micrometres, at a worst rate of 0.75, and more than two-thirds of the family sinks between 56.7 and 68.1. A shading chosen at random from the family is most likely to be half again as deep as the floor.

The walk’s first hundred thousand were not a sample #

The stopped walk’s counts are drawn beside the census, and the difference is the caution that essay gave made quantitative. It found 27.9 per cent of its chains on the floor; the family has 10.6. It found 18.6 per cent at 0.75; the family has 37.8. It found nothing at all in eight of the fourteen classes.

None of that is a defect in the walk; it did what it said. A depth-first search visits chains in an order fixed by how it builds them, and its first parts are the ones it tries first — which, for this search, happen to be the starting parts that lead to the floor unusually often. The census makes the point that essay could only state: a stopped walk describes its own order. The one thing it established that survives is the one it claimed, that the floor exists and that the family does not stop there.

The centre decides every depth #

A chain’s depth is set by its worst tone, and the worst could in principle be a shoulder — tone 3/8 or 5/8 interlacing more than the centre. Across the whole family, it never is.

In every one of the 1,001,574,400 chains, no shoulder interlaces more than the centre. In seven chains of eight the centre is strictly the worst, and in the rest a shoulder only ties it. That is a counted fact about the whole family rather than a proof — the census checked every chain and did not find a reason — but it settles the practical question exactly: the depth an even eight-end shading sinks is the depth of its centre, and the shoulders, however they are chosen, cost nothing.

So the lead the stopped walk ended on — what decides the centre’s rate — is the whole of the question. A designer who wants a given depth chooses a centre at that rate and has only to find a chain through it.

One centre in a hundred and seventy-five can be reached #

Most centres cannot be found a chain through at all.

A centre is reachable exactly when two things hold. It must contain a second tone — an opposite pair of ends in every pick, each pair in exactly two picks — because the first two parts of any chain through it are such a tone. And its unmarked cells must contain the same shape turned: an opposite pair of picks down every end, each pair down exactly two ends, because the last two parts are that shape read from the back. Everything else about the chain follows, since the parts in between can always be split two ways.

Only 112,218 centres, one in 175, meet both. The rest float two perfectly well, and no even chain passes through any of them: their four marks per thread are arranged so that no two parts of them can be a second tone, or so that their complement’s cannot. A designer who draws a centre first and hopes to build a shading round it is drawing, a hundred and seventy-four times in a hundred and seventy-five, a centre that belongs to no even shading.

The exclusion is not spread evenly, and where it falls matters. At the floor every one of the 112 float-two centres is reachable, and so are all 512 at the next rate up; the unreachable centres crowd the middle of the range, where the family’s commonest depths are. So the shallowest centres are never the obstacle to a shallow shading. Whatever stands between a designer and the floor is somewhere else.

A depth the walk never saw #

The classes the census adds are not curiosities of counting; each is a real shading.

It sits between the floor and the walk’s second class, and it is the second-shallowest shading the family has. A designer who could not have the floor — because the first two parts were already chosen — might still have this, and a list of depths that stopped at sixteenths would have said nothing between 42.8 and 50.1 micrometres existed.

Only one start in six reaches the floor #

The stopped walk’s second question was whether the shallowest ramp is open from every starting part, or only from some. The census answers it for every start at once, because a start fixes its second tone and the second tone is where the lower half of the count begins.

Only 408 of the 2,520 second tones can reach the floor, and in the walk’s own normal form only 816 of the 5,040 starting parts can: one in six. Every start has chains of some depth; none is a dead end. But a designer who takes a first part at random and then chooses the centre carefully has five chances in six of having already given up the shallowest ramp.

The second tone’s own rate says which way to lean. The flattest second tones, at 0.375 — the rate the float floor allowed and the stopped walk could only bound — lead to the floor most often, and 32 of the 48 of them can reach it. At 0.46875 only 64 of 960 can.

An order of work for a shallow shading #

Put together, the census gives a designer at eight ends a procedure rather than a search.

Choose the second tone first, and choose it flat: a pair of opposite ends in every pick, the pairs arranged so that the tone interlaces at 0.375 — 48 of the 2,520 — and among those one of the 32 that can still reach the floor. Then choose a centre containing it, at the depth wanted: at the floor, a centre that changes face only four times along every thread, as a 2/2 twill does, and every such centre is reachable. Then take any shoulders, since no shoulder in the family is ever worse than its centre. The float bounds do the rest.

That inverts the order a Latin square suggests, which is to build tones one mark at a time from the lightest. The census says the lightest tone barely matters, the second decides whether the floor is still available, and the centre decides everything else.

What a shallow centre does for the greys and the edges #

A weave is a halftone screen with n greys counted seven greys in an eight-end cell and found the float bounds fixing six of them; the census says what the seventh costs in relief. Every grey from 1/8 to 7/8 has its tone fixed by the count of marks, but the middle grey’s surface can sit anywhere from 42.8 to 83.6 micrometres below the ends, and which it sits at is chosen, in effect, when the second grey is.

It also changes the question a shaded damask’s boundaries ask. A damask’s edge floats further than its figure found floats running across the line where two tones meet, longer there than in either. In an even eight-end shading every shoulder already floats three and a floor centre floats two, so the floats that can be joined at a boundary are short ones — but whether two short floats meeting across a boundary stay short depends on where each tone’s marks sit, which is exactly what separates one chain from another inside a depth class. The census can now say which chains share a depth; it cannot yet say which of them also meet quietly at their edges, and since the float decides how loosely a boundary holds its threads, that is the second thing a designer choosing among the floor’s chains would want to know.

Where the satin shading sits #

A shading changes two things at once began from the traditional eight-end shading built from the satin’s cosets, whose floats run 7, 3, 3, 1, 3, 3, 7 — a plain centre with threes beside it — and a tone ramp is a valley, and the satin digs it measured it sinking 84 micrometres.

The census places it exactly. A plain centre is the deepest class the family has, a worst rate of 1.0 and 83.6 micrometres, reached by 2.1 per cent of the family, through just two centres: the two plain weaves. The satin shading is a member of the family’s deepest two per cent, and the ninety-eight per cent above it — including the tenth that sits on the floor at half its depth — have floats that are the same everywhere but the centre.

How the count was made #

Every tone of four marks per pick and per end on an eight-end repeat was listed by a search over picks whose weft floats are at most two, keeping only those whose warp floats down every end are also at most two: 19,639,930 centres from 38 permitted picks. For each centre, every second tone inside it was found by giving each pick one of its opposite pairs, each pair to exactly two picks; every split of the remainder into two parts was listed as a perfect matching, and the third tone’s rate recorded. The same was done for the centre’s complement with picks and ends exchanged. Each second tone splits into its two parts in sixteen ways. A chain’s worst rate is the largest of its second, third, centre, fifth and sixth tones’ rates, and the count for every pair of halves was added to its class.

Four things are required to hold, and they are about different kinds of claim. The arithmetic is a count: the classes add to the family, a whole number. Every class has a real chain: a witness is built for each, checked to be eight disjoint permutations whose tones meet the bounds and whose worst rate is its class’s. The census includes everything the walk found. And the walk agrees start by start: the original depth-first search, restricted to a single starting part and run to exhaustion, must count exactly what the census assigns that start, class by class. From the identity start that is 5,695,744 chains in nine classes, every class agreeing.

What the count cannot say #

Why no shoulder is ever worse than the centre. The census found it in every chain and did not find a reason; it is exhaustive, so it is true, but it is not understood.

What a depth looks like. The depths come from the pressing rule every shading essay has used, which reads only a tone’s interlacing rate. Two chains in one class sink alike on that rule and may not under a raking light, where the arrangement of the centre’s marks — which the census distinguishes and the rule does not — also decides how the surface catches it.

Anything beyond eight ends, or looser bounds. The split at the centre works because the float bound of three pins the second tone to 2,520 shapes; a family with a float of four there would have a far larger lower half, and a sixteen-end repeat would have far more centres. The method may reach them; this census does not.

Who split a search in half first #

Splitting a search at its middle and joining the two halves is an old device of computing, usually called meeting in the middle. Horowitz and Sahni used it in 1974 to solve the subset-sum problem in the square root of the time a direct search takes; Diffie and Hellman showed in 1977 that the same trick halves the security of encrypting twice with two keys, which is why double encryption is not used. Its requirement is always the same: a middle state small enough to list, with two halves that do not constrain each other once it is fixed.

A shading chain happens to have exactly that structure. Its centre is the middle state, and the float bounds make the lower half’s choices independent of the upper’s once the centre is fixed. A tone step does not need a satin established that a chain is a Latin square read in order; the six-end census walked every square because at six ends that was possible. That the eight-end family can be counted whole by meeting at its centre, and what the count says, are this essay’s.

Still open: why the shoulders never set the depth #

The one pattern the census found and cannot explain is also the one that made its answer simple. In a billion chains, tones 3/8 and 5/8 never interlace more than the centre; they tie it in one chain in eight and otherwise fall below.

There is a plausible route to a proof. A third tone is a second tone with one mark added to each pick, and the centre is that third tone with one more; interlacing counts changes of face, and a mark added beside an existing run changes nothing, one that joins two runs removes two changes, and one standing alone adds two. If the float bound of two at the centre forced the fourth part to add at least as many isolated marks as the third part did, the centre could never interlace less than the shoulder below it. Whether that holds pick by pick, or only on the whole, is the question; and an even shading cannot keep its surface level would gain a general statement in place of an eight-end count if it does.