Textile Structure
how cloth is put together
Series

Shading — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A spread shading on 8 ends. The 7 tone steps of a spread shading on an 8-end repeat, each built by adding one more coset of the 8-end satin with move 3. Every coset has exactly one mark in every end and every pick, so the fraction of warp on the face is k over 8 exactly at every step, with no averaging in it. The numbers beneath are the longest float, and they run 7, 3, 3, 1, 3, 3, 7 — so the lustre scale is not the tone scale. The midtone is a plain weave and the extremes are satins, which is why a spread shading is at its most matt exactly in the middle. What the drafts cannot show is the surface: a long float stands proud of a short one, so an evenly toned shading is not an even surface either.

    A shading changes two things at once

    The tone steps of a shaded damask are exactly even — each adds one satin coset, so the fraction of warp on the face is k over n with no averaging in it. The lustre steps are not even at all: on eight ends the longest float runs 7, 3, 3, 1, 3, 3, 7, so a series that grades smoothly in tone is at its most matt exactly in the middle.

    part 1 · pattern
  2. The surface of an 8-end shading. The height of the cloth's own surface at each tone of an 8-end shading, for the two chains, on a sheeting at 0.50 N in the end. A thread presses on the thread it crosses only where it turns, so a region that turns more often is pressed more often and finishes thinner — and firmness rises towards the midtone of a shading. The spread chain therefore sinks 84 µm between its ends and its midtone and the consecutive chain 43 µm, with 2 of its steps at exactly one thickness against the spread chain's 0. The tone scale is level by construction and the surface under it is not. What the plot cannot show is the light: a step in the surface reads as a line under a raking beam whatever the tone is doing, which is why a relief nobody specified is visible at all.

    A tone ramp is a valley, and the satin digs it

    A shading's tone is exact and its lustre is measured; its thickness is neither, and nobody specifies it. A firmer weave is pressed harder at every crossing and finishes thinner, so an eight-end shading sinks eighty-four micrometres between its ends and its midtone — about a third of the cloth's whole thickness, on every cloth tried. Build the same chain on a twill instead of a satin and the sag is exactly nothing.

    part 2 · pattern
  3. 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.

    part 3 · pattern
  4. 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.

    part 4 · pattern
  5. The warp floats across a tone edge on 8 ends. 2 strips of point paper, each one repeat of a tone on either side of a straight edge between picks, with the edge ruled and every warp float of the greatest length that crosses it drawn along its thread. Ground the exact complement: tones of 7 and 1 marks per end (cosets 1 to 7 against coset 0), not nested, and the longest warp float across the edge is 8 against 7 inside either tone. Ground one pick along: tones of 1 and 7 marks per end (coset 1 against cosets 1 to 7), nested, and the longest warp float across the edge is 7 against 7 inside either tone. A float drawn in the warning colour is longer than anything either tone has on its own. What the strips cannot show is the cloth: the edge here is one intersection wide, and in a woven piece the two tones take up yarn differently, so the change is spread over threads that point paper draws as belonging wholly to one side or the other.

    A damask's edge floats further than its figure

    Figure and ground in a damask carry the same longest float, which is true of both areas and false along the line between them. A float can cross the edge where two tones meet, and when one tone's marks lie inside the other's it can never be longer than a float either tone already has. A damask built as an exact complement is the one place in n that its ground can start which breaks this, and it floats n picks at its edge against n − 1 inside.

    part 5 · pattern
  6. 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.

    part 6 · pattern
  7. 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.

    part 7 · pattern
  8. 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.

    part 8 · pattern
  9. 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.

    part 9 · pattern

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