Textile Structure
how cloth is put together
Concept

Latin square — where it appears

An n by n array in which every symbol appears once in every row and once in every column. A chain of tone steps writes one, with each intersection numbered by the step it joins at, so a shading needs a Latin square rather than a satin — and a four-end or six-end repeat has thousands of them and no satin at all.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The cyclic decomposition of a 4-end repeat. A 4-end repeat split into 4 parts, each with exactly one warp mark in every end and every pick, drawn above with each intersection numbered by the part it belongs to. That object is a Latin square, and it is what a shading actually requires: a tone step is a union of parts, so it has exactly k marks in every end and pick and its tone is k over 4 exactly. This is the cyclic square, which is what a satin's cosets write — and at four and six ends there is no satin, so the same square has to be reached through a twill instead. The drafts below are the tone steps in the best order this square admits, whose longest floats run 3, 1, 3. What the drawing cannot show is that the numbering is arbitrary: relabelling the parts gives the same square and a different chain, which is exactly the freedom the order is chosen out of.

A tone step does not need a satin

Every account of shading builds its tone steps out of satin cosets, and at four ends and at six there is no satin to build them from. The construction was never about satins: what a tone step actually needs is that every end and every pick carry the same number of marks, which makes a chain of them a Latin square. A four-end repeat has twenty-four of those and a six-end repeat 1,128,960.

pattern · Shading One tone of an 8-end cell, arranged three ways. The same 32 marks in the same 8 × 8 cell, placed three ways, with the number of marks in each end printed beneath it. On the left the clustered dot a halftone screen makes: 4 of its threads carry every mark or none, so they never leave a face, and the criterion reports 16 separable layers rather than one cloth. In the middle a cloth built from the tone step by moving 2 marks sideways within their own picks — every pick still carries 4, the ends run 3 to 5, and that difference is a warp stripe of 25.0% contrast the design did not draw. On the right the tone step: every end and every pick at exactly 4. What the drawing cannot show is how visible the middle one's stripe is, which depends on the sett and on the viewing distance and is not computed here.

A weave is a halftone screen with n greys

An eight-by-eight cell of dots gives a printer sixty-five levels of grey. The same cell in cloth gives seven. Sixteen of the missing fifty-eight go to the requirement that every thread reach both faces and forty-two go to the requirement that every thread carry the same number of marks — so evenness, not interlacing, is what a weave pays for its tone scale.

pattern · Shading Every 6-end decomposition, by the float its best chain holds the middle tones to. All 1,128,960 Latin squares of order 6 with their first row in order — every way of splitting a 6-end repeat into 6 parts with one mark in every end and every pick — each asked for its chains of tone steps. 2,816 can hold every tone between the extremes to a float of 2, and they fall into 64 classes once the repeat's starting corner and reading direction are set aside, the cyclic square a twill writes among them; 800,658 cannot do better than 4. 576 can put a plain weave at the midtone, and every chain that does floats three on either side of it. Every one of the 434,540 distinct tone steps met is one cloth. What the rows cannot show is which of the classes a designer would choose, since the float profile is one criterion among several.

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

A six-end repeat can be split into the parts a shading is built from in 1,128,960 ways, and every one of them has now been walked. Only 2,816 — sixty-four distinct shadings — hold every tone between the extremes to a float of two, and seven in ten cannot do better than four. Five hundred and seventy-six can put a plain weave at the midtone, and not one of those can keep twos beside it: taking a part out of a plain weave, or adding one, always leaves a float of three.

pattern · Shading The surface of every even six-end shading, against the twill's. The height of the cloth's surface at each tone of a six-end shading on a sheeting at 0.50 N, for every chain of the 64 even classes, drawn once per distinct shape — 5 of them — and for the cyclic square's twill read one step at a time. The twill is level to the micrometre and floats 5, 4, 3, 4, 5. Every even chain sinks to exactly 43.3 µm at its midtone; the shapes differ only at the second and fourth tones. What the plot cannot show is which of these a raking light would reveal, which depends on the finish.

An even shading cannot keep its surface level

Sixty-four six-end shadings hold every middle tone to a float of two, and the question left over was whether any of them keeps a tone ramp's surface level the way a twill read in order does. None does, and all of them sink by exactly the same depth — 43.3 micrometres on a sheeting, a sixth of the cloth. The reason is a counting argument a recording engineer would recognise: a limit on how long a thread may float is a limit on run length, and a run-length limit forces a floor on how often the thread changes face. A level ramp needs every tone at the extremes' rate, which needs a float of half the repeat at the midtone and more beside it — exactly the floats the twill in order has.

pattern · Shading The depths an even eight-end shading can sink. Eight-end shading chains with every middle tone held to a float of three and the centre to two, sorted by the interlacing rate of their worst tone, with the depth that tone sinks below the extremes on a sheeting pressed at 0.50 N: 0.5000, 42.8 µm, 27,904 chains; 0.5625, 50.1 µm, 7,296 chains; 0.6250, 56.7 µm, 35,680 chains; 0.6875, 62.6 µm, 8,960 chains; 0.7500, 68.1 µm, 18,624 chains; 0.8125, 72.7 µm, 1,536 chains. The shallowest is the float argument's floor. The walk stopped after 100,000 chains, so the list is a lower bound on the depths the family has.

An eight-end shading is pinned at three everywhere but its centre

At six ends every even shading holds its middle tones to a float of two, and every one sinks to the same depth. The question left was whether eight ends does the same. It cannot even start: at eight ends no shading can hold its second tone to two, a float of three there needs the second mark of every pick exactly opposite the first, and then the third mark cannot undo it. Tones two, three, five and six are pinned at a float of three, only the centre is free, and a family with a free centre does not sink to one depth — it sinks to at least six.

pattern · Shading Every even eight-end shading, by the depth it sinks. All 1,001,574,400 eight-end shading chains whose middle tones float at most three and whose centre floats at most two, counted exactly and sorted by the interlacing rate of their worst tone, with the depth that tone sinks below the extremes on a sheeting at 0.50 N. 0.50000: 10.6 per cent, 42.8 µm; 0.53125: 0.9 per cent, 46.4 µm; 0.56250: 4.3 per cent, 50.1 µm; 0.59375: 1.0 per cent, 53.3 µm; 0.62500: 17.1 per cent, 56.7 µm; 0.65625: 2.3 per cent, 59.7 µm; 0.68750: 13.3 per cent, 62.6 µm; 0.71875: 2.3 per cent, 65.2 µm; 0.75000: 37.8 per cent, 68.1 µm; 0.78125: 0.2 per cent, 70.4 µm; 0.81250: 3.0 per cent, 72.7 µm; 0.87500: 3.0 per cent, 76.7 µm; 0.93750: 2.1 per cent, 80.5 µm; 1.00000: 2.1 per cent, 83.6 µm. The largest class is 0.7500, with 37.8 per cent of the family. Beside each share is the share the walk's first hundred thousand chains gave it, which reached 6 of the fourteen classes and put 27.9 per cent at the floor against a true 10.6.

Only one start in six can reach the shallowest shading

The even eight-end shadings — middle tones floating three, centre floating two — were too many to walk, and a walk stopped at a hundred thousand chains found six depths. Split at the centre, the family can be counted whole: 1,001,574,400 chains in fourteen depths, every one of them decided by the centre alone. Only 816 of the 5,040 ways to begin a shading can still reach the shallowest ramp, and a designer who wants it has to choose the first two parts before anything else.

pattern · Shading

Named alongside it

The objects these essays reach for when they reach for this one.

ShadingToneCensusFloatFloat lengthCloth thicknessEnumerationIntegrityFirmnessDamaskJacquardMove number

All concepts