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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

17 September 2026

11 essays on pattern and colour, compound and figured cloths, after the loom, weaves, knits and other structures and setting and geometry

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. Pattern and colour

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.

6 figures
The crossed shed's three spans at an eye friction of 0.3. A leno's crossing end in the crossed shed on an ordinary broad loom, back standard 4 shafts behind the doup, with both heddle eyes gripping at a coefficient of 0.3 and a background tension of 0.5 N. The end turns 69 degrees at the doup and 64 at the back standard, capstans of 1.44 and 1.40. From fell to doup it settles at 6.95 N, a strain of 5.73%; from doup to back standard it settles at 9.48 N, a strain of 7.97%; from back standard to back rest it settles at 6.78 N, a strain of 5.57%. With frictionless eyes every span would take 7.08 N; an ordinary end at the doup takes 1.02 N and the yarn breaks at 3.74 N on its initial modulus. Vertical scale exaggerated 3 times. What the drawing cannot show is the tension's fall round each eye, which happens over the eye's own few millimetres. Compound and figured cloths

A heddle eye lets the kink through

A leno's crossing end is pulled up at its doup and held down at its back standard, and the length that costs was priced as if both heddle eyes were frictionless. They grip, and gripping ought to trap the kink between them at nearly a hundred per cent strain. It does not come close. A capstan bounds a ratio of tensions, not a difference, and the spans either side are already stretched, so at a coefficient of 0.3 the span between the eyes takes 9.48 newtons against 7.08 with no friction at all — a third more, not fifteen times more — and an easer has to give back 67.9 millimetres rather than 64.4. What friction changes more is when the length is wanted.

8 figures
A three-direction net over a square voile, averaged each way. The light passing through a 1.55 mm net of three thread directions in front of a 0.3 mm voile of two, 50 mm apart and seen from 3.0 m, sampled on a fine grid over 160 mm, averaged along one direction and smoothed over two net pitches. Down the voile the profile rises and falls with the 6.1 mm family one net set makes; across it, with the 27.4 mm family two sets together make; the vertical rules are the predicted spacings. Each panel is scaled to its own range, and the second family is roughly a tenth the strength of the first. What the profiles cannot show is the fringes' look in two dimensions, where both families cross. Pattern and colour

A net of three directions beats a voile one way at a time

A tulle's threads run three ways at sixty degrees and a voile's run two ways at ninety, so a net hung over a voile could show one family of fringes, three, or a lattice of them. It shows two, at right angles, and they are nothing alike. Along the voile threads that lie parallel to one of the net's, the net beats exactly as a one-directional net does: six-millimetre fringes from three metres. Across them no set of the net lies anywhere near, and the only slow beat comes from a line of points two sets make together at √3 over the net's pitch — fringes four times wider and a fifteenth as strong, with a null at 8.8 metres where the strong family has none.

6 figures
A 2/2 twill with warp 1/1 and weft 2/2, as drawn and woven across. A 2/2 twill coloured 1/1 in the warp and 2/2 in the weft, drawn as the face of the cloth, and the same cloth turned through a right angle, which is what the loom makes if the two colour orders are exchanged and the weave turned with them. As drawn the weft order is 2/2 and needs boxes at one side; woven across the weft order is 1/1 and needs a loom picking at will. What the drawings cannot show is whether the cloth's two systems can be exchanged, which depends on their yarns and setts. Pattern and colour

A colour-and-weave look costs its cheaper order

The finest colour-and-weave effects need the rarest loom because a weft order is thrown pick by pick and a warp order is laid out once, so an effect's price was said to be its weft. That is true of a construction and false of a cloth. The same cloth can be woven lying across the loom, with its warp order thrown as weft and its weft order laid in the warp, and then it costs the other order. Over every two-colour look twelve small weaves make with orders up to six threads — 4,036 of them — 55 per cent need a loom picking at will as drawn and 31 per cent need one either way round. The looks turning rescues are the ones fine in one direction only, and not one of the trade's named effects is among them.

6 figures
The beams a disc 12 blocks across needs, column by column. A disc 12 blocks across, 12 blocks by 12, figure shaded, with the beam feeding each column of ends written above it, for a figure floating 8 on a ground floating 1, whose warp crimps are 14.15 per cent apart, at 4 mm blocks over a 100 m piece with 1.2 mm of slack. It has 4 distinct columns and 4 distinct shares of the repeat in figure, and needs 4 beams. What the grid cannot show is the cloth's own crimp interchange, which would move tension between the columns instead. Compound and figured cloths

A figured warp needs a beam for every share of its figure

Figure and ground take up warp at different rates, so a figured cloth on one beam is bounded in how long its figure may run, and a second beam is the obvious escape. It escapes only for ends that live wholly in one region. An end that crosses the figure for part of the repeat consumes warp at its own rate, and two ends can share a beam only if they spend the same share of the repeat in the figure and never drift a slack apart inside it. So a round figure twelve blocks across needs four beams, ninety-six blocks across needs twenty-nine, and any crimp difference at all — a third of a per cent will do — costs every one of them over a piece.

7 figures
6 tapered panels laid across a 150 cm cloth three ways. 6 panels 14 cm across the top, 32 cm across the bottom and 75 cm long, laid across a cloth 150 cm wide, drawn to scale. Turned end for end alternately, 6 fit side by side and the 6 take 75 cm of cloth. Laid all one way in lanes, 4 fit and they take 150 cm. Laid all one way with alternate columns shifted half a length, 5 columns fit and they take 150 cm. What the drawing cannot show is a real marker's other pieces, which fill the gaps these leave. After the loom

A nap is paid for by the taper of the pattern

A raised cloth's fibres lean, so a panel turned end for end shows a different amount of fibre and every piece of a garment has to lie the same way along the bolt. What that costs is not a property of the cloth. A rectangle costs nothing laid one way; a tapered panel costs (1 − r)/(1 + r) of extra cloth in lanes, where r is its narrow width over its wide one; and the best any one-way lay can do is exactly half of that, because a trapezoid's difference body is a hexagon and hexagons tile. On a real width it arrives in whole panel lengths: six skirt gores take 75 centimetres two ways and 150 one way.

7 figures
8/3 at 1.00 and 8/3 at 1.29, drawn in cloth. Satin marks drawn over two repeats at the spacings of a cloth, with the pick direction stretched by the sett ratio — ends per centimetre over picks per centimetre — and every closest pair of marks joined. 8-end satin, move 3 at a sett ratio of 1.000: closest pairs point one way: a row; 8-end satin, move 3 at a sett ratio of 1.291: closest pairs point 2 ways: no row. What the drawing cannot show is the float each mark interrupts, which is what a reader of the cloth actually sees. Weaves

A satin's row belongs to its sett

Point paper draws an end and a pick as equal squares, and every result about which satins have a row was taken there. A cloth is not square: it is set at so many ends and so many picks a centimetre, and in cloth the ties that made twelve satin orders row-free are ties between steps of different shape, which break at the first per cent of unequal sett. The reverse happens too. An eight-end satin, rowed on paper, has no row at exactly 1.291 ends per pick; an eleven-end at 1.265 and 1.528. And the only scatter that survives a range of setts is an irregular satin whose closest steps are mirror images — which nine ends has from the first per cent and ten only by 1.3.

6 figures
Imbalance against the torque a tuck carries, for six named structures. For six named two-bed structures, the imbalance of front-bed loops against back-bed loops in the worst fabric, as the share of a knit loop's torque a tuck carries runs from nought to one: single jersey from 1.000 to 1.000; a one-by-one rib from 0.000 to 0.000; a half-cardigan from 0.333 to 0.000; a full cardigan from 0.000 to 0.000; a rib that both tucks and floats from 0.333 to 0.143; a half-milano from 0.333 to 0.333. Structures without tucks are flat lines; a half-cardigan falls from a third to nought and a rib that tucks and floats from a third to a seventh. What the lines cannot show is where along them a real tuck sits, which is not measured. Knits and other structures

A tuck decides whether a third of two-bed fabrics lean

A jersey leans because every loop is on one bed and a rib does not because its loops are mirrored across two, and the count that says so treated a tuck as nothing. A tuck is a loop of the same lively yarn wrapped round a needle of one bed, and nobody here has measured how much of a knit loop's lean it carries. It matters. Of the 1,135 two-bed fabrics a two-needle, two-course frame can make, 135 are balanced whatever a tuck carries and 612 lean whatever it carries — and 388, a third, are balanced under one answer and lean under the other. A half-cardigan leans a third of a jersey if a tuck carries nothing and not at all if it carries a full loop's torque, which makes it the instrument that would settle the question.

6 figures
8×2 and 2×8 over every origin. Two grids of the 16 relative origins of 2/2 twill under 8-end satin, the row being how many picks the ground is started along and the column how many ends. The left grid is the census at a block 8×2, the right at 2×8; a square is filled where some of the 65,536 profiles separate. 8×2 fails at 8 origins and 2×8 at 8; both fail at 0 and neither at 0. Turning the cloth over and through a right angle sends each origin to another, and the letters mark where: every letter lands on a square with the same answer, so the two shapes are one census read at relabelled origins. Pattern and colour

A turned block is a moved origin

An eight-end satin figured on a 2/2 twill fails on 55,536 profiles at a block eight picks by two ends and on none at two by eight, and that was read as a property of the block's shape. Sweep the relative origin as well and the two shapes trade places: at every one of the sixteen origins exactly one of them fails, and over the sixteen they fail equally often. A shape asymmetry that no origin removes exists, and it needs a satin whose move squared is not one.

6 figures
How many derivation orbits each measure separates. For the 426 four-by-four cloths in 157 derivation orbits, the number of classes each invariant measure cuts the orbits into: marks, up to exchanging face and back, 5; interlacings, 8; layers, 2; plane group, 12; floats of both faces, both systems, 60; changes of face per end and per pick, 12; distinct ends and distinct picks, 4; all seven familiar measures, 120; census of two-by-two patches, 127; all seven, and the two-by-two patches, 153; census of three-by-three patches, 157. Only the census of three-by-three patches reaches 157. What the bars cannot show is which orbits a measure confuses, which the pair figure draws for the seven measures together. Weaves

A cloth's derivation class is its census of small patches

The manuals' derivations cut the 426 four-by-four cloths into 157 orbits, and an orbit is found by searching a group of 256 operations. The question left was whether a short list of numbers read off a draft could do the same job. The familiar ones cannot. Marks, interlacings, layers, plane group, float lengths, crossings per thread and distinct ends and picks are all invariant, and all seven together tell 120 of the orbits apart. A census of the two-by-two patches a draft contains tells 127 apart. A census of its three-by-three patches tells all 157 apart — a complete invariant of derivation, computed by counting windows rather than by searching operations.

5 figures
What the count moves at 150 grams. Plain cotton cloths that all weigh 150 g/m², from 20 tex to 200 tex, with sett and crimp solved together. Thickness, which is two yarn diameters, rises 3.16 times across the line; the cover factor of each thread system falls 2.75 times; their product with one plus the crimp is the same number at every count, because the weight has fixed the volume of fibre and the count only decides whether it is laid out flat or stacked up. Setting and geometry

A weight fixes the fibre and not the drape

Every plain cotton cloth of 150 grams a square metre contains the same fibre, and the count decides only how it is arranged. Across the counts that can make that weight, thickness rises threefold and cover falls in step, so their product holds still. The bending length does something stranger: at the bound a woven yarn actually sits near, it depends on neither the count nor the weight, only on the fibre.

6 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

13 September 2026

9 essays on pattern and colour, compound and figured cloths, what cloth is, cloth doing a job and knits and other structures

12 September 2026

15 essays on pattern and colour, compound and figured cloths, what cloth is and weaves

5 September 2026

50 essays on pattern and colour, compound and figured cloths, what cloth is, weaves, knits and other structures, setting and geometry, after the loom, cloth doing a job and mechanics and drape

3 September 2026

20 essays on knits and other structures, what cloth is, after the loom, setting and geometry, cloth doing a job and mechanics and drape

1 September 2026

20 essays on knits and other structures, after the loom, mechanics and drape, cloth doing a job and setting and geometry

31 August 2026

20 essays on pattern and colour, knits and other structures, what cloth is, compound and figured cloths, after the loom, cloth doing a job, mechanics and drape, weaves and setting and geometry

30 August 2026

20 essays on what cloth is, pattern and colour, knits and other structures, compound and figured cloths, after the loom, cloth doing a job, weaves, mechanics and drape and setting and geometry

28 August 2026

20 essays on what cloth is, pattern and colour, knits and other structures, compound and figured cloths, after the loom, cloth doing a job, mechanics and drape, setting and geometry and weaves

27 August 2026

15 essays on what cloth is, pattern and colour, knits and other structures, compound and figured cloths, after the loom, cloth doing a job, mechanics and drape, setting and geometry and weaves

26 August 2026

15 essays on what cloth is, pattern and colour, knits and other structures, compound and figured cloths, cloth doing a job, weaves, setting and geometry, after the loom and mechanics and drape

24 August 2026

15 essays on what cloth is, pattern and colour, knits and other structures, compound and figured cloths, cloth doing a job, mechanics and drape, after the loom, weaves and setting and geometry

22 August 2026

15 essays on what cloth is, knits and other structures, pattern and colour, compound and figured cloths, cloth doing a job, mechanics and drape, after the loom, setting and geometry and weaves

21 August 2026

15 essays on what cloth is, pattern and colour, knits and other structures, mechanics and drape, setting and geometry, compound and figured cloths, after the loom, cloth doing a job and weaves

20 August 2026

15 essays on what cloth is, pattern and colour, knits and other structures, mechanics and drape, setting and geometry, after the loom, compound and figured cloths, cloth doing a job and weaves

19 August 2026

15 essays on what cloth is, pattern and colour, mechanics and drape, knits and other structures, setting and geometry, after the loom, compound and figured cloths, cloth doing a job and weaves

18 August 2026

15 essays on what cloth is, mechanics and drape, pattern and colour, knits and other structures, setting and geometry, compound and figured cloths, cloth doing a job, after the loom and weaves

16 August 2026

15 essays on mechanics and drape, pattern and colour, what cloth is, knits and other structures, setting and geometry, weaves and compound and figured cloths

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11 August 2026

15 essays on compound and figured cloths, what cloth is, pattern and colour and weaves

10 August 2026

12 essays on mechanics and drape, what cloth is, setting and geometry, knits and other structures and weaves

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