Textile Structure
how cloth is put together
Generator

Lustre against the tolerance that defines it

Lustre against the tolerance that defines it
Lustre against the tolerance that defines it. The specular area of plain, 2/2 twill, satin 5, satin 8 in sheeting, against how near the mirror direction has to be to count. Every curve rises with the tolerance, because a wider acceptance takes in more of the arc either side of each crown, and the curves are very nearly proportional to it — the width is b·sin ε and a sine is its angle at these sizes. The ordering does not change anywhere in the range, which is the point of drawing it: the ratio between a satin 8 and a plain is 22.4 at two degrees and it stays there, so nothing that follows depends on where the line is drawn. What the tolerance does decide is the absolute number, and no absolute number is quoted anywhere without it.

The specular area of plain, 2/2 twill, satin 5, satin 8 in sheeting, against how near the mirror direction has to be to count. Every curve rises with the tolerance, because a wider acceptance takes in more of the arc either side of each crown, and the curves are very nearly proportional to it — the width is b·sin ε and a sine is its angle at these sizes. The ordering does not change anywhere in the range, which is the point of drawing it: the ratio between a satin 8 and a plain is 22.4 at two degrees and it stays there, so nothing that follows depends on where the line is drawn. What the tolerance *does* decide is the absolute number, and no absolute number is quoted anywhere without it.

8 essays call shine-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the weave its essay argues about.

Where it is called

Every figure listed here is drawn from this one rule.

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. Pattern and colour

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.

 How much every four-by-four draft can shine. All 22,874 four-by-four drafts in which every end and every pick interlaces, at sheeting's construction and a tolerance of 2°, counted by specular area. The range runs from 0.02% to 0.93%, a factor of 60.0, and the distribution is not smooth — it clusters, because the quantity behind it is a count of whole crossings and takes only certain values. The dullest drafts in the catalogue are the plain weaves, which have no plateau at all and shine only from the crowns of their turns; the brightest carry the most float on the face, with the fewest turns interrupting it. Lustre over the catalogue is a length census, and nothing about the yarn enters it. Weaves

Lustre is a length times a width

The specular area of a cloth factors exactly: a length of crown line, which the draft supplies, times a width of section within the tolerance, which the yarn and the finish supply. Neither factor knows anything about the other, and over the four-by-four catalogue the first alone spans a factor of sixty.

 A calender multiplies the highlight by 32, and all of it is width. A 2/2 twill in sheeting pressed at increasing force, with the specular area recomputed at each state from the standard compression model. It rises from 0.91% of the plan to 28.9%, a factor of 32, while the cloth thins from 381.6 µm to 186.1 µm. The gain is not in the length of the crowns: that moves by 5 per cent. It is in their width, which moves by 31.8 times, because pressing puts a flat top on the section and a flat top has one normal rather than a fan of them. The finish does not polish the thread. It changes the dimension of the highlight, from a line to a band, and the arithmetic says so by refusing to put any of the gain in the other factor. Weaves

A calender buys the width

Press a cloth and its lustre multiplies by twenty-four. None of that comes from the length of its crowns, which moves by six per cent; all of it comes from their width, because a flattened section has a plane on top of it and a plane has one normal rather than a fan of them. The arithmetic refuses to put any of the gain in the other factor.

 Turn the cloth and the highlight changes hands. A 2/2 twill in sheeting turned under a light, with the specular area of each system counted separately at each angle. The warp peaks at 0° and the weft at 83°, a quarter turn apart, and neither returns anything worth seeing where the other peaks. The reason needs no dye and no interference: a warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. A cloth woven with one colour in the warp and another in the weft therefore shows one colour at one angle and the other a quarter turn away, which is the whole of shot silk — a geometric effect that has been sold as a mysterious one for three hundred years. Weaves

Turn the cloth and the shine changes hands

A warp crown's normals all lie in the plane across the warp and have no component along it, so a warp float can only mirror light that arrives from across the warp. Turn the cloth a quarter turn and the weft takes over. That is the whole of shot silk — a geometric effect with no dye that changes and no interference in it.

 Specular area against the size of the source. The share of a sheeting's face that reflects, for plain, 2/2 twill, 5-end satin, 8-end satin, against the angular half-width of whatever is lighting it. Plain weave's line has twice the slope of the others, because a specular area is a length times a width and both are proportional to the angle for a weave with no float — while a float contributes a crown line that does not shrink with the source at all. Weaves

No cloth shines under a sky

Every specular figure on this account is drawn at a tolerance of two degrees, which is a stand-in for how wide the source is. Sweeping it says something the account has not: the contrast between a floated weave and a plain one is inversely proportional to the source's angular size, exactly, because a float keeps a crown line as the source narrows and a plain weave has only turns. A satin outshines a plain weave by 177 under the sun's disc, by 22 under a small lamp, and by exactly nothing at sixty degrees.

 A lamp in front of a window: the contrast against the lamp's share of the light. The contrast an eight-end satin shows over a plain weave when a 1° lamp and a 16° window both lie in the direction the cloth reflects to the viewer, against the lamp's share of the light: 0.0% gives 3.01, 0.5% gives 3.22, 1.0% gives 3.43, 2.0% gives 3.85, 5.0% gives 5.11, 10.0% gives 7.21, 20.0% gives 11.39, 30.0% gives 15.57, 50.0% gives 23.88, 70.0% gives 32.16, 100.0% gives 44.48. The points lie on a straight line between the window alone and the lamp alone, to within 0.8 per cent: the room's contrast is its sources' contrasts averaged by power. The dashed curve is a single source as wide as the power-weighted average width, which gives only 3.30 at a tenth of the light where the room gives 7.21. Weaves

A room lights a satin at the harmonic mean of its lamps

A satin outshines a plain weave by a factor inversely proportional to the width of the light, which is a clean law for one source and says nothing about a room. A room has a lamp and a window at once. Sum each source's highlight and the average falls out by itself: the room behaves as a single source whose width is the harmonic mean of its sources' widths, weighted by the power each supplies. The harmonic mean is ruled by the narrowest source, so a small lamp carrying a tenth of the light more than doubles the contrast a large window gives.

 A satin's contrast as the lamp moves off the mirror. The specular area of an eight-end satin over a plain weave, both as a sheeting, for a source 2° wide moved away from the direction the viewer sees mirrored in the cloth. Moved across the satin's floats, the contrast is 22.4 on the mirror and 21.7 forty degrees off. Moved along them it is 22.4 on the mirror and 0.25 once the source is more than twice its own width away: the satin is darker than plain. A calendered satin falls to 1.83 across its floats as well. What the plot cannot show is shadowing, which a lamp far enough off the mirror brings in. Weaves

A lamp off the mirror lights a satin only across its floats

Every source in the account of a room was a source the viewer sees mirrored in the cloth. A lamp off to one side lights the cloth too, and which facets it reaches depends on which way it is off. Displaced across a satin's floats it is caught by the float's own curve and adds contrast like any lamp on the mirror; displaced along them by more than twice its own width it reaches only the turns, and the satin comes out a quarter as bright as plain weave.

 Floated weaves against plain under an overcast sky. The sky's radiance each weave's facets mirror into the eye, over the plain weave's, as the viewer's elevation above a sheeting falls from overhead to thirty degrees, under an overcast sky with a ground of albedo 0.2. 2/2 twill: 1.25 overhead and 0.99 at 30°; 5-end satin: 1.30 overhead and 1.01 at 30°; 8-end satin: 1.37 overhead and 1.01 at 30°. What the plot cannot show is the diffuse reflection, the same for every weave of one fibre, which a real cloth's sheen sits on top of. Weaves

A satin mirrors the top of the sky

A uniform source sixty degrees wide leaves every weave reflecting the same share of its face, and that was read as saying no cloth shines under a sky. A real sky is not uniform. An overcast one is three times as bright overhead as at the horizon, and it stands over a ground darker than itself. Every facet of a cloth mirrors one direction, and a satin's facets mirror the top of the dome while a plain weave's mirror the horizon and the ground — so seen from above, under cloud, an eight-end satin still sends back a third more sky.

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