Field

Pattern and colour

A draft is a periodic plane pattern with two states, so the symmetry is two-coloured. Colour order, and the things the eye reliably gets wrong.

A draft is a two-coloured pattern. For each weave, the symmetries that leave warp-up as warp-up and those that exchange the two. Only a balanced weave has any of the second kind, because exchanging warp and weft is only a symmetry when there is as much of one on the face as the other.

Weaves as plane patterns

A draft is a periodic pattern with two states, so its symmetry is two-coloured. Some operations leave warp-up as warp-up and others exchange the two, and only a balanced weave has any of the second kind.

Colour and weave

Thread the same weave with a different order of coloured ends and a pattern appears that is nowhere in the draft. Houndstooth is an ordinary two-and-two twill, and nothing about its interlacement is unusual at all.

The 2/2 twill. The 2/2 twill on point paper, a filled square meaning the warp is on the face. Its longest float, its interlacing count and the number of separable cloths it describes were all counted from the matrix that drew it.

The diagonal is not a thread

A twill's diagonal is the most looked-at feature in weaving and the most misdescribed. No thread in the cloth runs diagonally. What runs diagonally is a pattern of which thread is on top, which is a different sort of object.

The twelve groups a draft can have. Every four-by-four draft in which each thread interlaces, classified by its plane symmetry group. Twelve of the seventeen groups occur; the five that do not are the ones needing a three-fold rotation, which no grid of warp and weft admits at any size.

The seventeen groups a draft can have

Twelve of them, in fact. The five missing ones all need a three-fold rotation, and no grid of warp and weft admits one at any size — which makes it a theorem rather than an artefact of the census.

Three different weaves, one cloth. Drafts drawn from one collision class, with the blind intersections marked, and the single surface all of them produce. Where the two crossing threads are the same colour the weave leaves no trace, so the cloth cannot report what it is.

Colour and weave as a two-colour problem

Where the two threads crossing are the same colour, the intersection looks identical whichever is on top. Half of them are, so 22,874 drafts collapse onto 256 surfaces — eighty-nine weaves apiece, and no way to tell them apart.

How many four-by-four weaves there are. The same census counted four ways. A draft is a notation; shifting the repeat's origin, turning the cloth over and turning it end for end all change the matrix and not the fabric. Each bar is the number of distinct objects left once those identifications are made, counted by canonical form and checked against Burnside's lemma.

How many cloths are there

Twenty-two thousand eight hundred and seventy-four four-by-four drafts. Four hundred and twenty-six four-by-four cloths. And the site's own separation rate — the fraction of drafts that look like fabric and are not — more than doubles when fabrics are counted instead of notations, which is the opposite of what was expected.

A beat at 19.1 mm from grids at 0.5 mm. Two grids at 0.5 and 0.5 mm pitch, the second turned by 1.5°, over a 30 mm window. The dashed rules are one predicted beat period apart. The pattern between them is 38 times the pitch of either grid and neither grid has anything at that scale.

Watered silk is a beat

Fold a ribbed cloth on itself and press it, and a figure appears at a scale neither ply has — fifty times the rib pitch, wandering across the piece. It is the difference of two wavevectors, it is enormous because the angle is tiny, and no two pieces match because no two are folded at the same angle.

The reed leaves its own mark

A reed does not space a warp evenly. It groups it, several ends to a dent, and the grouping beats against the weave repeat — so a denting that shares a factor with the repeat treats every repeat identically and shows as a stripe, and one that does not is invisible.

A stripe of a satin stripe on a plain ground. a satin stripe on a plain ground, written as one draft of 8 picks and 20 ends. Each band is generated from its own rule and the whole matrix is measured as one: 10 shafts, which is the number of distinct columns in the union of the two bands, against 8 if every column were shared and 10 if none were. The bands have 0 columns in common.

A stripe is a partition of the warp

A striped cloth is not a weave with decoration on it — it is one matrix in which different bands of ends obey different rules. The shaft count is a set union rather than a sum, and of forty-five pairs of standard weaves only three share a single column.

A tartan sett in 2/2 twill. A pivoted sett of 70 threads used in both systems, which is what makes a tartan a tartan rather than a check. On the left, the cloth at block scale: the squares on the diagonal have one colour in both systems and the rest are mixtures. On the right, one mixture rectangle at thread scale, where the weave decides. In 2/2 twill the reflection survives on 50.0 per cent of a mixture's intersections, against a ceiling of 75 per cent that no weave can reach. The three tints stand for the three colours of the sett and carry no other meaning here.

A check is two stripes and a tartan is one

A tartan is quoted as one sett of thread counts because the warp and the weft carry the same order. That is said to make it symmetric about its diagonal, and the arithmetic says it cannot be — the ceiling is three quarters, and a 2/2 twill reaches one half.

A figure is not a stripe

Two sound weaves side by side always make sound cloth — that is a theorem, and it was proved here. Put one of them inside a *region* instead of a band and it stops being true, and whether it is true or not turns out to depend on a number that appears nowhere in either draft: where the two weaves start relative to each other.

What a figure costs in shafts. Two families of block designs in 8-end satin on 8-end sateen at blocks of 8. One repeats three block-columns however wide it gets and costs 24 shafts at every width from 24 to 120 ends. The other gives every block-column a different pattern and costs 16, 24, 32, 40, 48 shafts as it grows — exactly 8 more per new column. Shafts are counted as distinct columns of the composite matrix.

What a figure costs the loom

A block design's shaft count does not depend on how wide the figure is, or how deep, or how many blocks it has. It depends on how many *different* block-columns it has, and each new one costs exactly one repeat of the ground weave — 8 shafts, then 16, then 24, in a straight line with no slope to fit.

The staircase a woven outline is. Four straight edges on a block grid, stepping 1 across in 1, 1 across in 2, 2 across in 1, 1 across in 4. Each tread is one repeat of 8-end satin, which at 24 by 22 threads per centimetre is 3.33 mm across and 3.64 mm up. A jacquard hook at 140 cm width is 1.17 mm, so the machine resolves 2.9 times finer than the cloth can use.

A woven outline is a staircase

A jacquard's resolution is quoted as its hook count, and on a 1,200-hook machine 140 cm wide that is a step of 1.17 mm. The cloth cannot use it. A figure's smallest feature is one repeat of the ground weave, which on an eight-end satin is 3.33 mm — so the machine resolves nearly three times finer than the fabric can hold, and a finer machine buys nothing at all.

The block condition, one half at a time. Every block shape from 8×8 down, for 8-end satin figured on 8-end sateen, each an exhaustive census of all 65,536 four-by-four profiles. The bar is split: the dark part is figures that separate with a thread left loose on the face, and the light part is figures that separate with nothing visible wrong. Shapes satisfying the block condition in one direction only are marked, and none of them has a light part at all. What the chart cannot show is why: the census is exhaustive and the theorem behind it is not proved here.

A rectangular block is not half a rule

The block rule was proved for square blocks and the rectangular case was recorded as not run, on the grounds that a block a repeat wide and half a repeat deep satisfies only half the condition. Running it turns up two things: half the condition rules out the failure nobody can see, and a block turned through a right angle is a different design — which a square census cannot notice, because a square block is its own transpose.

$Everything the relative origin decides. Each of six quantities computed from a weave matrix, swept over all 64 relative origins of 8-end satin figured on 8-end sateen. A quantity that takes one value across the sweep is a gauge freedom of the pair; one that takes several is something a designer chooses without knowing it. longest float takes 2 values; interlacing rate takes 2 values; weft-face fraction takes 3 values. The origins fall into 3 different partitions, so the moving quantities are not all decided by the same number. What the table cannot show is the census over all 65,536 profiles, which is a seventh column, and the one that showed the origin decides whether the cloth holds together.$

What else the relative origin decides

Where two weaves start relative to each other decides whether the cloth holds together, and it has a second consequence beside that. Two is what a search finds when it looks twice. Sweeping every origin against every measure of the cloth turns up a third — the longest float, which is the one property a weaver actually looks at — and turns up an earlier figure caption that says the opposite.

The step between a figure and its ground. Three figures on a plain ground, all in one sheeting's threads at its own setts. A thread presses on the thread it crosses only where it turns, and it turns at its interlacings — so the pressing a region receives per unit area is the contact force at one turn times the turns per unit area, and the second factor is a property of the matrix exactly. A five-end satin turns two fifths as often as a plain weave, is pressed two fifths as hard, flattens less, and stands 55 µm proud of it. What the rows cannot show is that both regions are given a plain weave's weave angle: a satin's crimp is genuinely smaller and its turns genuinely gentler, so the real step is larger than this, by an amount not computed here.

A figured cloth has a step in its surface

A damask is one cloth in one set of threads at one sett, and it is not flat. A thread presses on the thread it crosses only where it turns, so a region that turns less often is pressed less often, flattens less and stands thicker — and the step is a ratio of interlacing rates, read off the matrix with no yarn property in it.

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

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.

A 40-by-40 motif on a poplin, drawn and finished. A motif 40 ends wide and 40 picks tall, at the setts the reed and the take-up were set to, and the same motif measured on the finished cloth. Point paper has one cell per end and per pick, so a motif's proportions are the ratio of the two setts — and both setts move in the finishing, in opposite directions. On the poplin the aspect changes by a factor of 0.8429, so a circle drawn as a circle at the loom's numbers comes back 15.7% out of round and a designer who wants a circle must draw an ellipse of 1.1863. What the drawing cannot show is that the correction is not a property of the design: it belongs to the cloth, so the same card woven on a different construction is a different shape.

A motif is drawn at the wrong shape on purpose

Point paper has one cell per end and per pick, so a design's proportions on the cloth are the ratio of the two setts. Both setts move in the finishing and they move in opposite directions, so a circle drawn as a circle comes back out of round — by three per cent on a balanced cloth and by sixteen on a warp-dense one.

Two layers of one cloth, against how they happen to lie. Two identical muslins laid over one another and slid across each other by one thread spacing. In register the pair passes 37.9 per cent — as much as one cloth, because every hole is over a hole — and it falls linearly to 12.5 per cent before rising again at the next thread. The rule everybody uses is that two layers pass the product of their open areas, which is 14.3 per cent. That number is the average of this curve over all offsets, exactly — an identity, not a fit — and it is the answer at two points on it and nowhere else. Nothing about a real pair of layers is at its average, and the openness varying from place to place across a folded cloth is what a moiré is.

Two layers are the product on average and nowhere

Everybody knows what two layers of a cloth pass: the product of their open areas. That figure is exactly right — it is the mean of the true answer over every way the two layers can lie — and it is the answer at two registrations out of a continuum. In register a doubled cloth is as open as a single one; half a thread out it can be shut completely. The variation across a folded curtain is what a moiré is.

How open a batiste is, against where the light is. One minus the cover is a cloth's openness to a light directly behind it, and for this batiste it is 40.1 per cent. Move off the normal and a line of sight has to clear the hole at the top of the cloth and the same hole one thickness below, so what it can use falls as t·tan α is taken off each side — reaching nothing at 39.0° across the ends. Averaged over the whole sky with Lambert's cosine weighting, the cloth is 6.42 per cent open — 6.3 times less than the covering rule says, and the whole of the difference is the thickness. One minus the cover is the openness of a cloth with no thickness, and no cloth has none.

Opacity is not cover

The covering rule counts a thread as a bar that stops everything, and a single fine cotton thread held to a window plainly does not. What a cloth transmits is the open area plus whatever comes through the threads, so the covered fraction is a lever rather than a barrier — and the lever is longest exactly where the rule says the cloth is most closed. Nothing here computes a thread's transmittance, and saying why is the useful half.

A random error hides and a periodic one shows

A warp whose threads vary by fifteen per cent looks perfectly even. One dent of the reed a tenth of a millimetre wide makes a streak that gets the piece rejected. The same amount of error, arranged two ways — and the ratio between them is √(2n/π), with nothing fitted in it.

$One weft fault at three periods, a fraction of a millimetre apart. A 260 mm slice of a 1500 mm cloth at 22 picks per centimetre, 220 picks deep, with a thick place recurring along the weft. Each pick takes a whole width of yarn, so the marks land 35 to a pick across the full width and step sideways by the remainder of the width divided by the fault's period. On the left that remainder is zero and every thick place in the piece falls in the same columns — a warp-way stripe made entirely by a weft fault. In the middle the period is five hundredths of a millimetre longer and the same fault draws steep diagonals. On the right it is a third of a millimetre longer again, the marks land nowhere near each other, and the fault reads as texture. Nothing about the yarn distinguishes the three; the cloth's width does. The slice is drawn rather than the whole width because thirty-five marks a pick fill a panel solid at any step but zero, which is a true picture of a dense pattern and a useless one of its structure.$

A slub finds the width of the cloth

A thick place recurring along a weft yarn does not make a bar. It makes diagonals — and when the cloth's width happens to be a whole number of fault periods, it makes stripes down the piece instead, from a fault that is entirely in the weft.

$How much of a 20 tex yarn's unevenness survives being woven. Weaving averages, and the averaging is a square root. A patch of cloth 3 mm across contains 7.2 warp threads and as many picks, each contributing its own mass independently, so the patch's coefficient of variation is 3.53% against the yarn's 13.4% — a reduction of 3.8-fold. The rule at the top is the yarn's own figure. The curve steepens past the staple length, where a patch starts to contain independent samples along each thread as well as across them, and the second regime is the one a large area of cloth is judged in.$

A cloth cannot be more even than its yarn

It can be very much more even than its yarn, and by exactly the square root of the threads in view. Which raises a question the fineness argument left open — and the answer is that at a fixed cloth weight the count cancels out entirely.

A damask's figure and ground trade places when the cloth is turned. A satin 8 figure on a sateen 8 ground in sheeting — one cloth, one set of threads, one sett, and the ground is the figure's own complement. Their total specular areas are within a few per cent of one another, so neither is intrinsically the brighter. What differs is the direction: the figure's crowns run with the warp and the ground's with the weft. So the contrast between them is 2.0-to-one with the light coming from 8° and 0.47-to-one from 90° — it reverses, exactly, a quarter turn apart. That is what makes a damask visible in one colour, and it is not the step in its surface: the step is fifty micrometres and returns no light at all under a diffuse illumination, while this contrast is a factor of 2.0 and is present whenever there is a direction in the light.

A figure shows by its shine, not its step

A damask is one cloth in one colour and its pattern is plainly visible. This collection attributed that to the step in its surface — fifty micrometres of relief, computed from the interlacing rates. The step is real and returns almost no light. What makes the figure visible is that its crowns run at right angles to the ground's, so the two trade places when the cloth is turned.

$A yarn's diameter is a contour, not a length. The fraction of the space beside a 20 tex cotton yarn that is occupied by hair, against height, grossed up from the modelled population by the measured split between the long and short populations. At the yarn's own surface it is 8.9% — the layer is almost entirely gap — and it falls away exponentially from there. Every instrument that reports a yarn diameter is picking a contour of this curve, and the contours are far apart: a threshold that needs half the space filled is never met at all, one that needs a twentieth is met at 355 µm, and a hair counter triggers out at 2353 µm. The earlier model gave the layer a single thickness of 25 µm, which is a fair description of where most of the material is and wrong about its extent by more than a decade. What the curve cannot say is which contour any particular instrument uses, which is a fact about the instrument.$

A cloth is more opaque than it is closed

Opacity is not cover — this collection established that already and left the discrepancy attributed to the thickness of the threads. Part of it is not in the threads at all. A hair standing in a hole blocks light exactly as well as a thread does and costs nothing in air.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 1400 turns a metre. Its own torque is 2.060 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 0.91 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

A crepe is a yarn that will not lie still

A crepe cloth's pebbled surface is the yarn's own torque acting on a cloth that cannot resist it. The instability that makes a slack yarn snarl says how large the pebbles should be, and the answer is a millimetre — which is what a crepe looks like.

Twist is not torsion: a straight rod, twisted. A rod drawn with a cross painted along it, and the cross turned through 2 full turns from one end to the other. The rod is straight, so it has no Frenet frame at all — a straight line has no osculating plane to turn. The two discs above are the rod's ends seen down its own axis, which is where the twist is an angle rather than a foreshortened wiggle. The twist is the rotation of the painted cross about the rod's own tangent, and it is 2 turns whatever the centre line does. That is the quantity a torsional rigidity resists, and it is why a collection that solved a plane curve had not thereby dealt with torsion.

A shadow stripe is two twists

A cloth striped in one colour, where the stripe is visible only because alternate bands are spun the other way round. The pattern is carried entirely by which way the fibres lie on the yarn's surface, and it disappears when the light moves.

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.

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.

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.

A colour order against a 2/2 twill. The visible face of a 2/2 twill under 2 colour orders, drawn at the repeat the divisor arithmetic allows and outlined at the repeat the surface has. A filled cell is a dark thread on the face, which is the warp's colour where the warp is up and the weft's where it is not — so none of these patterns is in the draft, and the draft is the same in all of them. The colour period and the weave repeat beat exactly as a reed's grouping beats against a weave: the surface repeats on the least common multiple of the two, which here is 8×8 and 4×4. What the panels cannot show is colour: the two threads are drawn as filled and empty, and two colours of similar value make a pattern far weaker than this.

A colour order beats the weave it is threaded on

The reed's grouping beats against the weave repeat and the arithmetic is a least common multiple. A colour order is a second grouping of the same warp and the arithmetic is identical — but where the reed's beat is a fault to be dented out of a cloth, the colour order's beat is the pattern the cloth is sold for. Across 472 colour orders on four weaves the divisor bound is the surface's exact repeat in 470 or more, and the handful that beat it have no pattern left at all.

The magnification a cloth will carry. The largest magnification a moiré can be read at, against the irregularity of the cloth making it. The points are measured: a grating whose spacings are drawn from a seeded lognormal is laid against a perfect one, the fringes are found from the phase difference, and the gain is recorded at which their count first departs from what the ideal beat predicts. The product of the irregularity and that gain comes out at 0.82 to 0.84 across every irregularity tried, so the ceiling is 0.84 divided by the coefficient of variation — the solid curve. The dashed curve is the accumulated-wander model, in which position errors random-walk and the ceiling goes as the inverse square; it is wrong by a factor of 42 at a two per cent irregularity. What the plot cannot show is what a cloth's spacing irregularity actually is: it has not been measured, and the curve is therefore a prediction with an unmeasured input.

A moiré is a vernier, and it magnifies the error too

Two gratings a per cent apart in pitch beat at a hundred pitches, so a moiré reads a pitch difference at a hundred times — which is what a vernier is. The magnification is free and its ceiling is not: the fringes split when one thread's own spacing error reaches 0.84 of the pitch difference the beat is built on, so the usable gain is 0.84 divided by the cloth's coefficient of variation, inversely and not inverse-squarely. At an ordinary yarn's spacing irregularity, a moiré carries a magnification of five.

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.

Two layers in depth, and the beat perspective makes. An eye, a near grid and a far grid of the same pitch, with a ray from the eye to every bar of the far grid and a dot where each ray crosses the near one. The far bars land on the near layer at 8 to every 9, so the two grids drift out of register and back into it every 8 bars: in register the gaps line up and light comes through, half-way between them the far bars sit in the near gaps and block it. That spacing is the distance divided by the gap, times the pitch, and nothing about the threads or the angle between the layers enters it. The gap here is drawn at one eighth of the distance so the bars can be counted; two sheers 50 mm apart seen from 3 m are at a gain of 60. What the drawing cannot show is a real layer's thickness and its own irregular spacing, both of which the arithmetic treats as absent.

Two sheers make a moiré that walks with the viewer

Hang two identical sheer curtains a few centimetres apart and a moiré appears with no angle between them and no difference in their threads. Perspective alone makes the far one look finer. The fringes are p·V/D apart, which means they cover the same angle from every distance; they move one for one with a person walking past, which is the parallax of the horizon; and a far layer stretched by one per cent makes them vanish at exactly one distance, which says which layer is coarser and by how much.

Seeing through a white voile, from the street and from the room. How much of a scene's contrast survives a white voile, whose open area is 49%, looked through from the street into the room and from the room out to the street, against how much brighter the street is than the room. The scene comes through the clear lines of sight and nothing else; everywhere else the viewer sees thread lit from the viewer's own side, which returns light with no image in it. On a bright day, a hundred times brighter outside, 0.5% of the room's contrast reaches the street and 65% of the street's reaches the room. At equal light both are 28% — the curves cross at a ratio of one whatever the cloth — and with the lamps on after dark the room is the side on show, at 63%. What the plot cannot show is the threads' own optics: their reflectance and transmittance are assumed values for a white sheer rather than measurements, and only the crossing point is independent of them.

A sheer hides whichever side is darker

A net curtain hides a room by day and shows it at night, and the cloth has nothing to do with which. What a viewer sees through a sheer is an image through its clear lines of sight against a veil of lit thread, and the only thing that decides their balance is how much brighter one side is than the other. At equal light the two views are identical for every cloth there is. And a black sheer of the same openness shows twelve times more of the room by day than a white one.

Two weft colour orders thrown on a loom with boxes at one side. Weft colour orders thrown pick by pick on a shuttle loom that picks alternately from the two sides. For an order with runs of four, two, two and four, every throw finds a shuttle of its colour on the side it leaves from, so the order can be woven. For an order with runs of three and three, pick 4 has to be thrown from the right in a colour whose shuttle is on the other side, and the order cannot be woven. On a loom with boxes at one side the box opposite holds only the shuttle just thrown, which the next pick must throw straight back, so colour can change only between pairs of picks. What the drawing cannot show is the mechanism that drops the boxes, which decides how fast a change can be made but not which changes are possible.

A weft stripe is counted in pairs of picks

A warp's colour order is laid out once at warping and the loom never has to think about it. A weft's is thrown, one pick at a time, by shuttles that cross the cloth and stay where they land. On a loom with boxes at one side that makes every coloured band an even number of picks; with boxes at both sides it admits odd bands and pays for them in shuttles; and a tartan, which uses one order in both directions, is designed for its weft whether its designer knew it or not.

$A warp pinstripe in a 2/2 twill, 1, 2, 3 threads wide. A light stripe of ends in a dark 2/2 twill, drawn as the face a reader sees at 1, 2, 3 threads wide over 3 repeats. A single stripe thread is on the face at 50% of the crossings and goes under for up to 2 at a time, so it draws a broken line. Adjacent threads of the same colour cover one another's gaps, and the line becomes unbroken at 3 — the fewest neighbours for which, at every crossing, at least one is on the face — though an unbroken line is not a solid one, and beneath each panel is how much of its width is light, which varies along it until the line is a whole repeat wide. What the drawing cannot show is distance: a broken line whose gaps are a fraction of a millimetre reads as a fainter unbroken one from arm's length, and how far that is depends on the sett and on the eye.$

No weave draws an unbroken line one thread wide

A pinstripe is drawn on point paper as a single coloured column, and in cloth a single end is on the face only where it is up — so in every weave that interlaces, a line one thread wide has gaps in it. Neighbours of the same colour fill each other's gaps, and the fewest that leave no gap is a property of the weave: two in a plain weave, three in a 2/2 twill, and in a warp-faced sateen two across the warp and eight across the weft. Unbroken is not solid either — an eight-pick bar in that sateen has no gap and is an eighth light.

A figured sheer is a negative from one side

A net curtain with a pattern in it carries two patterns, one for each side, and by day they are opposites. A more open figure in a white voile is a dark figure a quarter below its ground from the street and a light one five per cent above it from the room, and after dark the two views trade places exactly. The figure vanishes from the street at one light ratio and from the room at another. And a figure can be made that the room cannot see at all while the street sees it nearly black — along one line of openness and thread tone, and never from both sides at once.

A 1.55 mm net 50 mm behind a 0.3 mm voile, from 0.6 m, 1.5 m, 4 m. A 1.55 mm net 50 mm behind a 0.3 mm voile, drawn across 40 mm of the near layer at true pitch as seen from 0.6 m, 1.5 m, 4 m. Two grids this different beat through a harmonic: the net's k-th against the voile's first, for the k nearest the ratio of their pitches as the eye sees them. From 0.6 m that is the fifth, in register every 6.2 mm; From 1.5 m that is the fifth, exactly in register, with no fringe; From 4 m that is the fifth, in register every 14.9 mm. What the strips cannot show is how strong each family is, which falls with the harmonic, nor the net's second family of threads at right angles.

A net over a voile beats through a harmonic

Two identical sheers hung apart make a moiré by perspective alone. A net in front of a voile is not two identical sheers — its mesh is five times the voile's pitch — and it beats anyway, through the net's fifth harmonic, which is a grid 3.3 per cent coarser than the voile. With the net behind, that is a pair of sheers with its coarser layer at the back, and the fringes vanish at exactly 1.5 metres. Closer in, the harmonic changes, the fringes dissolve into a texture twice the net's pitch and re-form, and they vanish again at 17 centimetres.

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.

A white thread 70% open figure behind a net 60% open, from both sides. A lozenge figure in the inner curtain of a pair, drawn as the street sees it and as the room sees it through a plain net 60% open, with the street 30 : 1 as bright as the room. From the street the figure's contrast against its ground is −7.0% where a single curtain would have given −24.6%; from the room it is +6.9% where a single curtain would have given +5.4%. Each panel is shaded relative to its own brighter region, both at one gain, so the two steps are to scale against each other, and the contrast is stretched 9 times to be visible at all — the numbers rather than the depth of the shading are the measurement. What the drawing cannot show is the light between the two curtains, which is 19.80 times the room's illuminance from the street and 0.57 from the room, nor the thread optics, which are assumed.

A net in front gives the figure to the room

The commonest double curtain is a plain net outside a patterned one, and the plain net does not merely dim the pattern. It weakens the figure from the street by a factor of three and a half and strengthens it from the room by a quarter, so a pattern four and a half times stronger outside than in becomes one the room sees slightly better. A figure designed to be invisible from indoors reappears at nearly three per cent, and the day-and-night exchange a single curtain obeyed exactly stops holding at all.

A curtain is gathered so that it is seen edge-on

A curtain is hung with more cloth than window, and the surplus is not decoration. Laid in folds, a length of cloth spans its own length times the cosine of its flank angle, so the fullness is the secant of that angle exactly — and a line of sight through the window meets the cloth at it. A voile's view halves at a fullness of 1.08, its flanks shut completely at 1.48, and at the two and a half times a curtain is actually hung at, every flank passes nothing and the whole of what comes through is the crests.

What a passer-by is looking at, for rooms of four reflectances. The street's view of a window behind a sheer white thread 49% open at 30 : 1, split into the part that is an image of the room, arriving through the holes, and the part that is the cloth's own threads lit by the street. a room reflecting 8% contributes 0.4% of the view; a room reflecting 20% contributes 1.0% of the view; a room reflecting 35% contributes 1.8% of the view; a room reflecting 60% contributes 3.1% of the view. The veil is identical in every row because the cloth and the street are: only the furniture changes. What the chart cannot show is where in each room that reflectance sits, and a dark room with a lit lamp in it is not its own average.

What a sheer hides is decided by the furniture

Every result these essays have produced assumed the room and the street reflect the same three tenths of the light on them. That was never a fact about cloth. A room reflecting a twentieth is hidden two hundred and thirty-eight times better than the street it faces and one reflecting four fifths only fifteen times, with the same curtain at the same window — and the day-and-night exchange they rested on, exact for equal scenes, is out by more than half the glow for an ordinary pair of unequal ones.

The edge of a warp line in a 2/2 twill, at 3 and 4 threads. A light line in a dark 2/2 twill, drawn at 3 and 4 threads wide over 3 repeats with both of its boundaries traced crossing by crossing. The line has no gap at either width, and neither boundary is straight: at the crossings where the outermost thread of the band is under the ground, the edge retreats to the next thread in. At 3 it swings 2 threads with a period of 4; At 4 it swings 2 threads with a period of 4. What the drawing cannot show is distance, at which a swing of one thread width is below what an eye separates and a swing of three may not be.

An unbroken line is not a clean one

A line of colour has two boundaries and neither is straight, in any weave there is. The thread at the edge must go under somewhere, and where it does the edge retreats to its neighbour — so the boundary steps, and by exactly one thread less than the narrowest unbroken line the weave draws. Over 22,874 drafts there are three widths and three swings and no draft anywhere else, and widening the line past its narrowest unbroken width leaves the edge precisely where it was.

The named colour-and-weave effects, sorted by the loom their weft needs. Each named colour-and-weave effect with the shortest band in its colour order and the cheapest loom that can throw that order in the weft, on a loom with 4 boxes a side. end-and-end, runs of 1 and 1, needs picking at will; hairline, runs of 1 and 1, needs picking at will; log cabin, runs of 1 and 1, needs picking at will; tattersall, runs of 1 and 9, needs picking at will; birdseye, runs of 2 and 2, needs boxes at one side; crow's foot, runs of 2 and 2, needs boxes at one side; step pattern, runs of 2 and 1, needs boxes at both sides; three-and-one, runs of 3 and 1, needs picking at will; houndstooth, runs of 4 and 4, needs boxes at one side; shepherd's check, runs of 6 and 6, needs boxes at one side; gun club, runs of 4 and 4 and 4 and 4, needs boxes at one side; glen check, runs of 4 and 4 and 4 and 4 and 2 and 2 and 2 and 2, needs boxes at one side. The warp costs nothing, because a colour order in the warp is laid out once at warping; the weft is thrown one pick at a time by shuttles that stay where they land, so an effect's price is its weft order alone. What the bars cannot show is the pattern, which is in neither the order nor the weave but in what they make of each other.

The finest colour-and-weave effects need the rarest loom

A houndstooth, a shepherd's check and a gun club check are thrown by the cheapest shuttle loom there is. A hairline, an end-and-end and a log cabin are thrown by none — a colour on every other pick is thrown from the same side every time, so its shuttles pile up at the far end of the loom and never come back. The dividing line is the parity of the bands and nothing else, which is why it survives the fact that no two sources agree about how wide a shepherd's check is.

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.

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.

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.

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.

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.

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.

The colour order that hides least

A caption on this account's top essay left a question: whether a warp of long runs against a weft of short ones hides less than either would against itself, and called it a lever nobody uses. Sweeping all sixteen orders against all sixteen says the lever is real and the caption named the wrong variable. Run length has nothing to do with it — the blind count, the separation and the largest confused class all depend on the two orders' colour counts and on nothing else, and what a mixed pair saves is exactly the square of the difference between them.

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.

A blind set hides less the thinner it is spread

Two colours showed that where a cloth's threads match in colour the draft underneath is invisible, and that how many intersections match is not what decides how much is hidden. More colours say what does. A third colour lets the blind count take every value from nought to ten; a fourth lets the blind intersections sit one to a thread. At a fixed count, the catalogue keeps more of its cloths apart the more threads the blind set is spread across — 3,632 surfaces for four scattered blind cells, 2,402 for the same four in a square.

A sheer's privacy is the error it multiplies

Everything a passer-by sees of a room through a sheer is the room's image plus a veil the street lights, and the veil can be known without going in. So one subtraction and one division ought to read the room's reflectance from the pavement. They do not, and the reason is the curtain's whole purpose: every error in the veil arrives in the answer multiplied by the reading over the image — sixty-four through a white voile by day, three and a half at equal light. The privacy a sheer gives and the precision it denies are one number.

What a pane of glass does to a sheer's multiplier. The factor by which an error in reading a window is multiplied on its way into the room's reflectance, against the angle a passer-by looks at the glass from, at 30 : 1, for three sheers behind a pane of window glass. The dashed level beside each is the same cloth with no glass. white voile: 64.0 bare, 59.8 behind glass straight on, 65.7 at 60°, 92.7 at 75°; grey voile: 27.3 bare, 28.7 behind glass straight on, 34.7 at 60°, 61.7 at 75°; black net: 2.0 bare, 5.2 behind glass straight on, 8.8 at 60°, 25.4 at 75°. The white voile is easier to read through glass up to 56.7°, because the pane dims the street on its threads by more than it mirrors; the black net is harder at every angle. The pane mirrors a street reflecting 0.3; a sky in the mirror would be brighter, and nothing here models one.

Glass hides a black net and not a white voile

A window pane in front of a curtain mirrors the street, and a mirror is a veil with no thread in it. It also dims the street's light on the cloth. For a white voile the two nearly cancel, and straight on the dimming wins, so the voile is slightly easier to read through glass than without it. For a black net the mirror is most of what a passer-by sees: its multiplier goes from 2 to 5 head-on and to 25 from along the pavement. The glass makes the net private, not the voile. The angle at which it stops helping does not depend on the time of day, and a polarising filter at Brewster's angle takes the mirror out altogether.

The advantage of spreading a blind set, against the repeat. The number of distinct surfaces a blind set keeps when its cells are spread one to a thread, over the number it keeps when they are gathered into a block, against the size of the repeat: 4 blind, 1.512 at 4, 1.254 at 5, 1.129 at 6, 1.064 at 7, 1.032 at 8, 1.008 at 10, 1.002 at 12; 8 blind, 1.080 at 8, 1.020 at 10, 1.005 at 12. Sampled from forty thousand surfaces at each point. The dashed curves are the closed form — each thread the set misses must interlace on its own, which fails once in 2^(n−1) — and the sampled points sit on them from six threads up; at four and five the form overstates, because a thread the set does touch can still fail there. At four by four spreading keeps half as much again; at eight, a few per cent; at twelve, nothing to speak of. What the chart cannot show is whether an eye can tell the surfaces apart.

A blind cell costs half the cloths at any real repeat

At four by four, spreading a blind set over more threads kept half as many cloths apart again as gathering it. At eight by eight the same rule holds, and it is almost nothing: three per cent for four blind cells, eight for eight. The reason is one factor. A blind set's surfaces are the count's bound, 2^(n² − b), times the chance that every thread it misses interlaces on its own, and at eight threads a thread lies all on one face once in 128 tries. So shapes that touch the same number of threads tie exactly, whatever their rectangles; every blind cell costs very nearly half the catalogue; and by twelve threads the arrangement is free.

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.

What an upper window's glass does to a white voile. How many times harder a room is to read through a white voile behind glass than through the bare cloth, against how far the passer-by stands out from the façade, for a window centred 1.5, 4.5 and 7.5 m up in a street twelve metres wide between ten-metre terraces, under an overcast sky, at 30 : 1. Below one the glass helps. ground floor: 0.95 from 2 m, 0.95 from 5 m, 0.95 from 9 m; first floor: 2.34 from 2 m, 1.55 from 5 m, 0.94 from 9 m; second floor: 4.92 from 2 m, 1.86 from 5 m, 1.49 from 9 m. The first floor's curve drops by a step at 6.3 m, where its pane stops mirroring the sky and starts mirroring the terrace opposite. What the chart cannot show is a clear sky, which is brighter again near the horizon.

An upper pane mirrors a sky no curtain can afford

A pane in front of a sheer helps a reading only while what it mirrors is dim: for a white voile, below about half the brightness of a white wall lit like the window. A street-level pane mirrors the terrace across the road and stays under that. An upper pane, seen from the pavement below, mirrors the sky, and an overcast horizon — the darkest sky there is — is already twice the limit. Whether a first-floor pane shows sky depends on how far out from the façade the passer-by stands and on nothing else, so the same window is harder to see into from close up than from across the street.

What tilting a window pane does to a curtain behind it. How many times harder a room is to read through a white voile behind glass than through the bare cloth, against the tilt of the pane — leaned forward to the left, tilted back to the right — for windows centred 1.5, 4.5 and 7.5 m up, seen from 3 m out in a street twelve metres wide between ten-metre terraces, under an overcast sky. Each curve jumps where its pane starts to mirror the sky: the ground floor tilted back past 18.6°, the first floor leaned forward less than 9.7°, the second floor leaned forward less than 25.6°. Upright the three read 0.95, 1.82, 2.89; leaned forward 26°, 0.97, 0.94, 0.93. What the chart cannot show is a canopy or blind over a shop window, which a pane leaned forward would mirror instead of the street.

A pane leaned forward loses the sky and finds no darker mirror

A pane tilted by some angle sends a passer-by's line of sight off at twice that angle more, and the passer-by's place along the street still drops out. Tilted back about twenty degrees, a street-level pane mirrors sky from everywhere in the street and nearly doubles what a white voile hides. Leaned forward ten degrees, a first-floor pane stops mirroring sky from three metres out, and the voile behind it reads better through glass than bare. Leaning further finds nothing darker: under an overcast sky the road is a brighter mirror than the terrace opposite, by a seventh.

A 2/2 twill line's ends where the reed leaves them. A light warp line 3 ends wide in a dark 2/2 twill, with the ends drawn at their true places at a cover of 60% and both boundaries traced. Set evenly the edge swings 2.00 thread spacings. With 3 ends to a dent, closed by 40% of a spacing so that they touch, and the line on the dent's edge, it swings 1.20. One end along in the same reed it swings 2.40, more than if it had been set evenly. The weave and the reed are the same in the last two. Only where the line starts has changed.

A stripe's edge is cleanest inside one dent

No weave can draw a line with a straight edge, since the thread at the boundary must go under somewhere. A cloth has one freedom its weave lacks, which is the spacing of its ends, and it looked as though crowding a stripe's outer end could halve the swing. It cannot, in most cloth: two ends stop at a diameter apart, so the swing falls only as far as the cover. The freedom that costs nothing is another one. A line's narrowest unbroken width is exactly its swing plus one, so it fits one dent of a reed, and the same reed gives 1.2 thread spacings or 2.4 according to which end of the dent the line begins on.

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