A rectangular block is not half a rule

A figure is not a stripe proved a theorem and recorded a hole in it.

The theorem: a figured cloth whose block is a whole repeat of both weaves in both directions cannot separate. Inside one column of the profile every block is a complete piece of its own weave; blocks stacked in a column share all of that column’s ends; columns share all of the picks; so the whole is strongly connected. It is a stripe of stripes, and stripes of sound weaves are sound.

The hole, in that essay’s own words: the block rule is stated for aligned square blocks and nothing else. A block that is a repeat wide and half a repeat deep satisfies half of the condition, and the proof needs both halves. Whether the weaker condition is enough is not known here and the census has not been run.

It has been run.

The claim #

Two of them, and the second is the one a square census could never have found.

Half the condition rules out the invisible failure. Over every pair and every shape run, a block that is a whole repeat of both weaves in either direction — not both — produces no figure that separates without a thread lying loose on the face. It produces plenty of failures; every one of them has a float a weaver would reject on sight.

A block is not its own transpose. An eight-end satin figured on a 2/2 twill at a block eight picks deep and two ends across fails on 55,536 of the 65,536 profiles. Turn the block through a right angle — two picks deep and eight ends across — and it fails on none of them. Same two weaves, same area, same number of blocks.

Why the second claim is invisible at square blocks #

A square block is its own transpose. That is the whole reason the previous rung, which ran every census at a square block, could not see any of this: the asymmetry it would have had to detect is an asymmetry between a shape and its turn, and its shapes had no turns.

Nor is the asymmetry visible in any other measure the site takes. Floats, interlacings, balance, shafts and layers are all preserved when a cloth is turned through a right angle, because turning a cloth exchanges warp and weft and every one of those quantities is symmetric under the exchange — which is the same symmetry the catalogue of 426 cloths is already quotiented by. This is a property of the boundary rather than of either weave, and it needed a boundary that has a direction.

The identity that explains it #

There is a clean reason for the asymmetry and it turns an observation into a theorem.

Turning a figured cloth through a right angle turns its block through a right angle and turns both of its weaves through a right angle with it. The set of profiles is closed under transposition, a loose thread stays a loose thread, and a stripe stays a stripe. So every one of the four counts is preserved:

census(F, G, a×b) = census(Fᵀ, Gᵀ, b×a)

That is the whole explanation. A pair of weaves carried to itself by transposition gives the same census at a block and at its transpose, because the turned pair is the same pair. A 2/2 twill is its own transpose; so is a hopsack; so is plain weave. An eight-end satin is not — its transpose is a different weave — so for a pair containing one there is no reason at all for the two answers to agree, and they do not.

The identity is checked at four counts rather than one, at six shape-and-pair combinations. Checking only the total would miss exactly the failure the check exists for, which is a classifier that shuffles the kinds while preserving how many there are.

What the half-condition result says, and what it does not #

The census covers five pairs of weaves at nine block shapes apiece — 45 exhaustive censuses, 2.9 million figured cloths — and in every one of the twenty half-coarse shapes the invisible failure count is nought.

That is a strong statement and it is worth being exact about its status. It is an exhaustive result at a stated scope, not a proved theorem. The full condition has a proof: the stripe argument runs twice, once down the profile’s columns and once across its rows. The half condition has no proof here. What it has is a census that has not found a counterexample in three million cloths, and a shape that suggests one exists.

The suggestion is this. If the block is a whole repeat of both weaves across, then each block column contains a whole number of repeats of each weave in the end direction, so the ends within a block column are related to each other exactly as they are in the weaves themselves. That is enough to make each block column strongly connected on its own ends and its own picks — but only if no pick in that column is left without an interlacing, which is precisely the condition that a thread is not lying loose.

So the half condition and the loose-thread condition are plausibly two halves of one statement, and a proof would be a matter of showing that a half-coarse figure with no loose thread is connected. That has not been done and is recorded as owed.

The practical form #

For a designer the result is worth having in a shape that does not mention transposition.

A figure boundary stepped at a whole repeat in one direction is safe from the failure nothing can see. That matters because the resolution of a woven outline is set by the block, and a designer buying safety with a coarser block is paying in exactly the currency the design is about. The floats it produces are visible and can be dealt with by inspection; the two-piece cloth that looks perfectly sound cannot occur. That is the useful half of the block rule at half the cost, and the cost matters — a block eight picks by eight ends on an eight-end satin is a coarse figure, and a block eight picks by two ends is four times finer across.

And the direction of the stepping is not free. A damask is its own complement, which is why the damask pair behaves symmetrically here and is the one case where a designer can ignore all of this. If the two weaves are not transposes of each other, stepping the boundary coarsely in the warp direction and finely in the weft is a different design from the reverse, and one of the two may be unweavable while the other is safe. Nothing in a manual mentions this, because the manuals’ rule is stated in terms of the ground weave’s repeat and is symmetric on its face.

Where the failures actually are #

The table for a satin on its own reverse, which is the pair the blocks ladder has used throughout:

Two things stand out. The first is the one the criterion exists for and the second is about the rows that look harmless. The half-coarse rows have failures and none of them is invisible — those two are the degenerate profiles that are really stripes, and the stripe case is settled elsewhere. And the 4 × 4 row, which satisfies neither half, has no invisible failures either — so the half condition is sufficient and not necessary, which is the ordinary shape for a result of this kind and is worth stating so that nobody reads the census as a characterisation.

The float is what a weaver sees, and the visible failures are all floats. The invisible failures appear when both dimensions are below half a repeat, and they appear in force: at a block of two by two, 1,032 of the 65,536 profiles are genuine two-dimensional figures with no loose thread, no long float, both weaves sound, and the cloth in two pieces.

Which pairs can show it, decided before any census is run #

The transposition identity does more than explain the asymmetry. It says in advance which pairs of weaves are capable of showing one, and the test is cheaper than a census by every measure.

The identity is census(F, G, a×b) = census(Fᵀ, Gᵀ, b×a). So the two shapes agree whenever the pair is carried to itself by transposition — which happens two ways, and it is worth keeping them apart because one of them is not obvious.

Both weaves are their own transposes. A basket against a balanced twill, plain against a hopsack: turning either through a right angle gives it back, so the turned pair is the same pair and the two censuses are the same census.

Or the two weaves are each other’s transposes. This is the damask case and it is the reason a damask behaves symmetrically here despite neither of its weaves being self-transpose: a satin’s transpose is its own reverse, so transposing the pair swaps its members, and swapping the members of a figure-and-ground pair gives the same set of cloths with the figure and the ground exchanged.

Everything else can show the asymmetry, and there is a necessary condition on a weave being its own transpose that a designer can check by eye. Transposing exchanges warp and weft, so a self-transpose weave must be balanced — equal warp and weft on the face. An unbalanced weave is never its own transpose.

That gives the rule in the form worth carrying:

If either weave in the pair is unbalanced, and the two are not each other’s reverse, then stepping the boundary coarsely down the cloth is a different design from stepping it coarsely across.

Applied to the weaves this site names, it sorts them without running anything. Plain, the baskets and the balanced twills are self-transpose and safe in any combination. The satins and the unbalanced twills are not, and a pair drawn from those two groups — an eight-end satin figured on a 2/2 twill, which is the pair the essay above finds failing on 55,536 profiles one way round and none the other — is exactly the case the rule flags.

It also halves the work. A sweep over every ordered pair at every rectangular block computes each answer twice for the self-transpose pairs and once for the rest; knowing which is which lets a census skip the duplicates rather than recompute and compare them. The identity was found here as an explanation and it is more useful as a schedule, which is the usual fate of a symmetry once somebody writes it down.

What was counted, and how #

Every census is exhaustive over all 2^16 profiles on a four-by-four grid of blocks. Nothing is sampled anywhere.

The block is a pair rather than a number now, everywhere in the machinery, with a bare number meaning a square block — so every call written before this rung means exactly what it meant. That change touched the composite builder, the coarseness test, the fast classifier and the census itself, and the one place it could have gone wrong is the fast classifier, whose bit masks previously assumed the ends and the picks were the same population size. They are not any more.

The inner loop is still written twice and the two are still required to agree. The fast path represents the digraph as a bit mask per node and decides strong connectivity by forward and backward reachability from one vertex; the slow path builds a validated matrix and runs the site’s own connectivity computation. A census only ever asks the slow path about profiles the fast one has already rejected, so a fast path that wrongly called something sound would undercount in silence — which is why the two are checked against each other in both directions on a smaller grid rather than only where they meet.

Three assertions guard the results. A coarse block that ever produced a separation would falsify the block rule, and the census refuses to return if one does. The half-coarse shapes must produce no invisible failures, which is the finding stated so that a single row could refute it. And they must produce some visible failures, because a claim that the half condition rules out one kind of failure is worth nothing if it turns out to rule out all of them.

Where the model stops #

The half condition is a census result and not a theorem, as above, and the sketch of a proof is the obvious next piece of work.

The scope is five pairs. They span the range the ladder has used — satins against their reverses, baskets against twills, a basket against a satin — and five is five. A pair chosen adversarially might behave differently, and nothing here rules that out.

The grid is four by four. A profile on a larger grid has more room for a boundary to do something awkward, and the censuses at that size are 2^25 and beyond.

And the boundary is still on a lattice. A real jacquard figure is drawn freehand on squared paper and its staircase has treads of different sizes. Everything here is about a boundary whose treads are all one shape; a boundary that mixes shapes is a superset of these cases and can only be worse.

The generalisation #

The transposition identity is the piece worth carrying out of the subject.

When an object is built from a shape and some ingredients, a symmetry of the construction acts on both at once — and a question about the shape alone is only well posed if the ingredients are fixed by that symmetry. Here the symmetry is transposition, the shape is the block, and the ingredients are the two weaves. Asking “does a rectangular block behave differently from its transpose” has no answer until it is said whether the weaves are being turned too, and the reason the answer looks surprising is that intuition silently turns the block and leaves the weaves alone.

The second lesson is about experimental design and is more uncomfortable. A parameter swept only at its symmetric values cannot reveal an asymmetry. The previous rung swept block size thoroughly — one, two, four, eight — and every value it tried was a square, so the whole of this rung’s second finding was invisible to it by construction rather than by oversight. There is no way to detect that from inside the sweep; the only defence is to notice that a parameter has more degrees of freedom than the sweep is using.

Who found it, and when #

Block design is old and thoroughly worked out in practice, and the trade’s rule for stepping a figure boundary is stated in terms of the ground weave’s repeat, in one number, with no direction attached to it. Summer and winter, huckaback, M’s and O’s and the crackle weaves are all designed on profile drafts, and none of the manuals distinguishes a step down the cloth from a step across it.

The connectivity criterion is the periodic-fabric work of the 1980s, Grünbaum and Shephard’s, and a blend of two fabrics over a region is not an object in that theory — so the question does not arise there and there is nothing to have missed.

The transposition identity is elementary once written down, and its use here is this site’s: it converts an observation about two numbers into a statement about which pairs of weaves can possibly show the effect, which is a much stronger thing to have.

Where the ladder goes next #

The immediate next thing is a proof of the half condition, and the shape of it is sketched above.

Sideways, the relative origin decides more than the previous rung found — including the longest float, which is the one property a weaver looks at — and the two results are about the same figured cloths seen from two directions: one asks what the block’s shape decides and the other what the weaves’ offset decides.

Further out, the same census on a rectangular grid of blocks rather than a rectangular block is untouched, and so is the question of what a figure boundary does physically: two weaves with different interlacing rates take up thread differently, so a stepped boundary is a stepped ridge, and that needs two reed pitches and a model of take-up this site does not have.