A turned block is a moved origin

A rectangular block is not half a rule ended on a number that looked like a design rule. An eight-end satin figured on a 2/2 twill, in blocks eight picks deep and two ends across, separates on 55,536 of the 65,536 four-by-four profiles. Turn the block through a right angle, two picks deep and eight ends across, and it separates on none of them. Same two weaves, same area, same number of blocks. The essay read this as the block’s shape, and explained it by the satin: an eight-end satin turned on the table is its own sateen, a different weave, so nothing obliges the two shapes to agree.

Every one of those censuses was run with the twill started where the satin starts. What else the relative origin decides had already shown that where the two weaves start relative to each other decides whether a figured cloth holds together at all. The two sweeps were never crossed: shapes were swept at one origin, and origins at square blocks, which are their own transposes.

Crossing them is sixteen origins, two shapes and 65,536 profiles at each — just over two million figured cloths for this one pair — and it changes what the 55,536 means.

The claim #

For an eight-end satin figured on a 2/2 twill, the difference between a block and its transpose is the relative origin in disguise. At every one of the sixteen origins exactly one of eight-by-two and two-by-eight fails. Eight-by-two fails at eight origins, two-by-eight at the other eight, and summed over all sixteen they fail on the same 444,288 profiles. The same holds at four by two against two by four: eight origins each, 358,400 each.

A genuine shape asymmetry does exist, and it needs a different satin. A seven-end satin on move 2, figured on the same twill, fails on 27,120 profiles at seven by two and on 37,312 at two by seven — at every origin, with no relabelling that could reconcile them. The test that separates the two cases is a line of modular arithmetic about the satin’s move.

The turn a census cannot see #

The earlier essay’s identity was built on the turn a person makes with a swatch on a table: rotate it a quarter turn. Warp becomes weft, and since the face that was showing warp now shows the same threads as weft, the matrix is transposed and complemented. Under that turn an eight-end satin becomes its sateen, which is a different weave, and the identity it gives is true.

It is not the only turn available, and it is not the one the census is blind to. Whether a cloth hangs together is a statement about its connectivity: which threads hold which. A thread lying loose on the face is lying loose on the back as well, and two cloths are two cloths from either side. The verdict cannot see which face is up. So any turn that differs from the table turn only by turning the cloth over is equally a symmetry of the census.

How many cloths are there spelled out what turning a cloth over does to its draft: it is a reflection composed with a complement, because the thread on the face is now underneath. Compose that with the quarter turn and the two complements cancel. What is left is the plain transpose — the swatch flipped over about the line joining two of its corners. That is a physical motion of a piece of cloth, and it carries a weave somewhere quite different from where the quarter turn carries it.

Why the eight-end satin comes back #

A regular satin on n ends with move m places its one interlacing in each pick at an end that advances by m from pick to pick. Transposing exchanges the roles of pick and end, so the transposed satin advances by whatever number undoes m: the move m′ with m·m′ equal to one, counting modulo n.

On eight ends with move 3 that number is 3 again, because three threes are nine, and nine is one more than eight. The eight-end satin is its own transpose, unshifted, and the drawing above confirms it cell for cell.

A 2/2 twill is simpler. Its interlacings lie along lines parallel to the diagonal the flip turns about, and a flip leaves such a line parallel to where it was; the transposed twill is the twill, started one end along. A 3/1 twill does the same and lands two ends along.

The rectangular-block essay’s rule of thumb said that an unbalanced weave is never its own transpose. That is true of the table turn, which exchanges warp face and weft face. It is not true of the flip, which exchanges neither, and the 3/1 twill — as unbalanced as a regular twill can be — comes back as itself.

The identity, with the origin written in #

With both weaves carried to themselves the argument is short. Take a figured cloth at blocks a by b with the twill started p picks and e ends along, and flip it about its diagonal. Every intersection moves to its mirror across the diagonal, so the result is a figured cloth: the profile transposed, the blocks b by a, the satin exactly where it was, and the twill started e picks and p ends along — and then one more end, because that is where the flipped twill starts.

census(satin, twill, a×b, origin (p, e)) = census(satin, twill, b×a, origin (e, p + 1))

The set of profiles is closed under transposition, a loose thread stays loose and two cloths stay two. So the two censuses are one census read at relabelled origins, and that is what the letters in the first figure record.

It is checked twice over. Matrix for matrix: at all sixteen origins and three unrelated profiles, the flipped figured cloth is compared intersection by intersection with the cloth built directly at the relabelled origin, and they are required to be identical. And count for count: at every origin the sound, visible, stripe and invisible totals at eight by two must equal the four totals at two by eight at the relabelled origin. Checking one total would pass a classifier that shuffled figures between kinds, which is exactly the fault such a check exists to catch.

Sixteen origins are four #

The relabelling explains why the totals agree. It does not yet explain the sharper fact in the first figure: that the two shapes never fail at the same origin, and never both succeed.

A twill moved one pick down and one end across is the same twill, because its interlacings lie on that diagonal. So of the two numbers in an origin only their difference means anything: the diagonal offset, ends along minus picks along, counted modulo four. Sixteen origins fall into four classes of four, and every origin in a class gets the same verdict — the collapse from sixteen to four the origin sweep found for a twill pair, reappearing here at a rectangular block.

In offsets the relabelling reads x → 1 − x. It sends the offsets where eight-by-two fails, 0 and 3, onto 1 and 2, which is precisely where two-by-eight fails. And because the failing set at one shape happens to be exactly half of the four offsets, its image is exactly the other half. That is why exactly one shape fails at every origin: not because the two are opposed in some deep way, but because the turn is an involution on four offsets with no fixed point, and the failures occupy one of its two orbits.

For a designer this is a usable statement. Eight-end satin on a 2/2 twill at a fine step across: start the twill at offset 1 or 2 and no profile separates. At a fine step down the cloth: offset 0 or 3. The shape is not the bad choice. The combination is.

What fails, and where on the cloth #

Every failure in this sweep is a visible one: a thread left loose across the whole repeat. None separates without one. That makes the mechanism easy to see, and worth seeing, because the origin’s effect is otherwise an abstraction.

Take an end in the two-by-eight drawing. It crosses four blocks of two picks each, eight picks in all, which is exactly one repeat of the satin. In a satin block the end spends seven picks in eight on one side and interlaces on the eighth, so if that eighth pick falls inside a satin block the end is held. If it falls inside a twill block, the end is held only if the twill takes it to the other side on one of those two picks.

A 2/2 twill shows each end two picks up and two down, so on some pairs of picks it holds the end steady on the satin’s side and on others it does not. Which pairs, relative to the satin’s single interlacing, is exactly what the offset sets. At one offset the twill blocks lie across the satin’s interlacing with the end on the same side for both picks, and the end runs loose; one end along, the twill flips it, and the cloth holds. Swap picks and ends and the same account covers the loose picks at eight by two.

This also says why the rectangular-block essay’s number was not an accident of one profile. At eight by two and offset 0, a row of the profile leaves a pick loose when it holds exactly one twill block, or two twill blocks with a satin block between them — six of the sixteen possible rows — and a profile survives only if all four of its rows are among the other ten. That is $10^4 = 10{,}000$ sound profiles, and 65,536 less 10,000 is the 55,536.

A shape asymmetry no origin removes #

Whether a block’s shape can matter beyond its origin is now a question about weaves rather than about censuses. It can only matter if one of the two weaves fails to come back as itself when flipped, and for a regular satin that is the question whether its move, multiplied by itself, is one.

On seven ends the satin moves are 2, 3, 4 and 5. Two twos are four, which is neither one nor six. So a seven-end satin on move 2 flips to the satin on move 4, and move 4 is not move 2 started anywhere else. The census is under no obligation to agree, and it does not.

Two things differ from the eight-end case and both are visible in the grid.

The origin has stopped mattering. A block seven picks deep is not a whole number of twill repeats, so going down the profile the pick the twill starts on at the top of each block advances by three. Four blocks down, every starting pick the twill has has been tried inside a single cloth, and moving the origin only reorders them. The origin is averaged away inside the fabric.

And the shape has started mattering, for real. At every origin two-by-seven fails on 10,192 more profiles than seven-by-two. There is no relabelling that could turn one into the other, because the flipped cloth is a figure in a different satin.

The asymmetry has a partner that makes it concrete. The move-3 satin on seven ends gives 37,312 at seven by two and 27,120 at two by seven: the same pair of numbers with the shapes exchanged. So on seven ends the choice a designer controls is not the origin but the move. Thread the move-2 satin and the fine step belongs across the cloth; thread move 3 and it belongs down. Which satins are worth weaving ranks moves by how scattered their interlacings look, and by that measure move 2 and move 3 on seven ends score exactly the same, since each is the other flipped and mirrored. In a figured cloth they are not interchangeable.

A pair the identity does not cover, which agrees anyway #

The flip test is a sufficient condition for agreement, not a necessary one, and a five-end satin shows it.

On five ends move 2 squared is four, which is minus one rather than one. The flipped satin is move 3, the mirror of move 2, and no shift makes it the original — so the identity says nothing. Yet at five by two and two by five the census fails on 22,336 profiles at every origin at both shapes. The two agree for a reason this essay has not found.

The basket on a 3/1 twill is the third kind of case and the tidiest. A basket is its own transpose, the 3/1 twill flips to itself two ends along, and the relabelling on offsets is x → 2 − x. That fixes offsets 1 and 3 and swaps 0 with 2, and the failures, at every offset except 1, are carried onto themselves. So both shapes fail at the same twelve origins. The rectangular-block essay found this pair’s two shapes equal at its one origin while its rule counted the unbalanced 3/1 twill against it; the flip is the reason the rule was wrong about it.

What the rectangular-block rule becomes #

The rule the rectangular-block essay proposed for a designer was that a pair containing an unbalanced weave, unless the two are each other’s reverse, makes stepping the boundary down the cloth a different design from stepping it across. With the origin swept, that needs rewriting in three places.

Balance is the wrong test. For whether a figure holds together the relevant turn is the flip, and the flip does not care which face shows warp. Satins with a move whose square is one, and all regular twills, come back as themselves.

A fixed-origin asymmetry is usually an origin. If both weaves come back as themselves, a block and its transpose are the same census at relabelled origins, and any difference seen at one origin is a statement about where the ground was started.

A real asymmetry is a property of the move. It needs a weave that comes back as a different weave, and a seven-end satin on move 2 or 3 is the smallest satin that provides one. Even then it is not guaranteed, as five ends shows.

The earlier identity stays true. The table turn does carry the census of a satin on a twill at eight by two to the census of a sateen on a twill at two by eight. What changes is the conclusion drawn from it, because a second symmetry was sitting beside the first.

What was counted #

Five pairings of weaves and block, each a rectangular block against its transpose, at every relative origin of the ground: sixteen origins apiece, since every ground here has a four-by-four repeat. Each census is exhaustive over the 65,536 profiles on a four-by-four grid of blocks. That is 160 censuses and 10.5 million figured cloths, nothing sampled.

The classifier is the same pair of implementations every census of figured cloths here has used. A bit-mask reachability test screens each profile, and every profile it rejects is rebuilt as a matrix and put through the full connectivity computation, which must agree that it is not sound. Four checks guard this essay’s claims specifically. The flipped cloth equals the relabelled cloth intersection for intersection. The four counts agree at relabelled origins. Exactly one shape fails at each origin for the eight-end pair, at offsets 0 and 3 for eight by two. And the seven-end satin on move 2 has no relabelling and fails on different numbers at its two shapes. The last is there so that the identity is seen to do work: an identity that held for every pair would be one nobody had checked.

Where the model stops #

Only integrity is face-blind. The flip is a symmetry of whether a cloth hangs together and of almost nothing else a weaver looks at. A figure shows by its shine, and shine depends on which threads are on the face, so the appearance of a figured cloth at eight by two and at two by eight is not relabelled by anything here. The longest float is face-blind only if back floats are counted with front ones.

Every failure found is a loose thread. None of the shapes swept is fine enough in both directions to produce the invisible failures, which are the ones that make figured cloths hard; whether they obey the same relabelling is a question the identity answers — yes, since it preserves all four counts — but no census here exercised it.

The grid is four by four and the pairs are five. The seven-end asymmetry is exhaustive at that scope and stated at that scope.

The five-end agreement is unexplained. It is a coincidence of 22,336 and 22,336 at every origin, and a coincidence that exact usually has a reason.

The generalisation #

When a question is posed about an object with a symmetry, the symmetries that matter are the object’s, not the ones the question’s vocabulary suggests. A block “turned through a right angle” suggested the table turn, because that is what the words describe. But the property being asked about — holding together — is blind to which face is up, and that blindness doubles the group. Inside the larger group the satin was symmetric all along.

The second lesson is the one a figure is not a stripe started from, arriving from a new side. A difference measured with one parameter held fixed may be a difference in that parameter. The block’s shape and the relative origin looked like two design choices, each swept on its own. For most pairs they are one choice, and the only way to find that out was to sweep them together.

Who found it #

The symmetry groups of periodic fabrics — including turning a fabric over, and the observation that doing so complements the pattern — are Grünbaum and Shephard’s, from their work on isonemal fabrics in the 1980s. Figured cloths, two weaves in a region, are not objects in it, and the trade’s own block rules give the ground’s repeat as one number with no direction and no origin. The crossing of the two sweeps, the flip identity with the origin written in, and the seven-end counterexample are this site’s.

What this leaves open #

The immediate question is the one left open above: why a five-end satin, whose flip is a mirror image rather than itself, gives equal counts at both shapes. The likely route is that the census is also blind to mirroring the whole cloth, which would enlarge the group again — but a mirror changes the twill’s hand, and a proof has to deal with that.

Sideways, the proof of the half-coarse condition that the rectangular-block essay left owed is now a smaller job: the flip identity halves the cases it has to cover.

Further out, the float — which the origin sweep found moving — is face-dependent on a figured cloth, so the flip does not relabel it. A float census across shape and origin together is the sweep that would show whether a designer who picks the safe origin for integrity has picked a long float along with it.