Lustre is a length times a width

The specular area of a woven cloth — the fraction of its plan whose normal points within a stated tolerance of the mirror direction — has a closed form, and the closed form is a product of two things that come from opposite ends of the fabric.

The claim #

Specular area = crown length × section width, exactly, and the two factors are independent.

The length is the total horizontal crown line in the repeat, plus a small arc contribution at each turn: it is a property of the float map and has no yarn in it.

The width is what the thread’s section presents inside the tolerance — the flat top of the section, plus b·sin ε of arc either side of it: it is a property of the section and has no draft in it.

Two consequences follow and both are practical.

The draft is worth a factor of sixty and the finish is worth a factor of twenty-four, and because they multiply, a calendered satin is worth the product of the two.

And neither can substitute for the other. A cloth with no float has no length to multiply; a cloth with a round section has almost no width. Buying one does not compensate for lacking the other.

The decomposition, derived #

At the specular geometry an element reflects when both its slopes are inside the tolerance, and the two slopes are independent because one runs along the thread and the other across it.

Along the thread the slope is exactly zero over a plateau and sweeps to the weave angle over a transition. So a plateau contributes its whole length, and a transition contributes only the arc near its crown where the tangent is still within tolerance — a length of D·ε for an arc of radius D/2 turning through ε either side.

Across the thread the slope is zero over the section’s flat top, if it has one, and reaches the tolerance an arc of b·sin ε either side of the highest line. So the width is flat + b·sin ε.

Multiplying, and dividing by the area of the repeat:

$$A_{\text{spec}} = \frac{(L + N D \varepsilon)(F + b \sin \varepsilon)}{\text{repeat area}}$$

with L the crown line, N the number of turns, F the section’s flat. Every symbol in it comes from somewhere else in this collection, and none of them is fitted.

What was counted, and how #

Twice, and the two routes share nothing.

The closed form multiplies the length off the float map by the width off the section, as above.

The sampled route builds the height field, computes the exact normal at every sample, and counts the samples whose normal is inside the tolerance cone.

They agree to about a quarter, and both reasons for the residual are known and are stated rather than hidden: the sampled route counts a cell whose centre qualifies, which over-counts by roughly one cell across a strip only a few cells wide, and the closed form ignores the occlusion of one system’s crown by the other’s, which over-counts in the opposite direction and dominates for a plain weave. No result in this ladder turns on the difference, because every one of them is a ratio between weaves and both errors are common to the two sides.

The named weaves in one sheeting at two degrees come out as: plain 0.062 per cent, two-and-two twill 0.911, three-and-one twill 0.944, five-end satin 1.120, eight-end satin 1.385, twelve-end satin 1.532. An eight-end satin returns twenty-two times a plain weave’s specular area on one yarn at one sett.

The census, and where it differs from the contact census #

The four-by-four catalogue answered for specular area rather than for contact gives a different ordering from the same catalogue answered for bearing, and the reason is a single sentence.

A bearing curve cares which crown is higher and a reflection does not. The lower thread system of an unbalanced cloth is out of contact entirely until something has sunk past the step between the crowns; it reflects exactly as well as the higher one, because a reflection is about a normal and not about a height.

So the contact census counts one system and the shine census counts both. A one-and-three twill, which touches at points because its warp crowns higher, shines like the floated cloth it looks like — its weft’s long plateaux are three micrometres too low to bear and exactly the right shape to reflect.

The range over the catalogue is a factor of sixty, against a factor of about four for the contact census once the plain weaves are set aside. Reflection discriminates between drafts far more sharply than contact does, and it does so because it counts both systems and because its floor is genuinely small rather than zero.

The shape of the census, which is not a smooth distribution #

The histogram clusters, and the clustering is informative rather than a sampling artefact.

The crown line of a draft is a sum of (run length − 1) over its face runs, times a thread spacing. Those are integers times two fixed lengths, so the quantity takes a discrete set of values, and a four-by-four repeat has few enough runs that the set is small. Twenty-two thousand drafts land on a few dozen distinct values.

That means there is no such thing as fine control of lustre through the draft at this repeat size. A designer can have one of about forty specular areas, and the gaps between adjacent ones are several per cent. Continuous control requires a larger repeat, which is what a satin of eight or twelve ends supplies — and which is one more reason, beyond the familiar ones, that a lustrous cloth is woven on a long repeat.

The two ends of the distribution are worth naming. The maximum, 0.93 per cent, is reached by drafts that concentrate the face in one system with the fewest turns; the minimum, 0.015 per cent, by the plain weaves, whose entire specular area is the arcs at their sixteen crowns. The ratio is sixty and there is nothing in between the plain weaves and the rest — the gap below the second-lowest value is the largest gap in the whole distribution.

Why the two factors cannot substitute #

The product form makes a practical statement that is easy to get wrong.

A mill wanting a lustrous cloth has two levers: weave a longer float, or press the cloth harder. The arithmetic says both work, that the second is worth more, and that using one does not reduce the need for the other.

Take the extreme cases. A plain weave has a crown line of zero: its whole specular area comes from the arcs at its turns, so calendering it multiplies a very small number. A satin with a round section has 18 millimetres of crown line per repeat and a width of six micrometres: the length is there and there is nothing to multiply it by.

A calendered satin has both, and its specular area is the product — which is why every fabric sold on its lustre is both floated and finished, and has been for as long as there have been calenders.

What the census says about the shading of a damask #

This collection already has a result that the surface reading sharpens. A shading changes two things at once: the tone steps of a shaded damask are exactly even, because each adds one satin coset, while 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.

That essay used the longest float as the lustre proxy. The specular area is the better quantity and it says the same thing more strongly: the crown line of those eight steps follows the same non-monotone path, because a crown line is a sum over runs and the runs are what the cosets are rearranging.

So a shaded damask is a series in which one property is linear and another is symmetric about the middle, and the second is now computable rather than inferred from the float length. What a reader sees running across such a cloth is a tone ramp with a lustre valley in it, and the valley is where the two halves of the series meet.

Which satins are worth weaving, answered again #

Which satins are worth weaving settled the question by float length and by the regularity of the move: a satin whose move number shares a factor with its repeat is not a satin at all, and among those that exist some scatter their interlacings better than others.

The specular arithmetic gives a second reading of the same list, and the two agree in ordering and disagree in spacing.

Over the satins this collection draws — five, eight and twelve ends — the specular area at two degrees runs 1.120, 1.385 and 1.532 per cent. The gains are diminishing sharply. Going from five ends to eight buys 24 per cent; going from eight to twelve buys 11 per cent — and a six-end satin does not exist at all; and the limit as the repeat grows without bound is the case where the crown line is the whole thread length and the turns contribute nothing, which is about 1.8 per cent for this cloth — a twenty-four-end satin has reached 1.679 and a sixteen-end 1.605.

So the lustre available from lengthening a satin’s repeat is bounded, and most of it is already spent by eight ends. That is a genuinely useful thing to know, because a long-repeat satin costs shafts, costs float length that snags, and costs strength — and past eight ends it is buying very little of the property it is being woven for.

How the satins approach their limit #

The gains from lengthening a satin’s repeat are called sharply diminishing above, and the diminution has a form. Reading it off the numbers gives a designer the exchange rate directly, in shafts.

The specular areas run 1.120 at five ends, 1.385 at eight, 1.532 at twelve, 1.605 at sixteen and 1.679 at twenty-four, against a limit near 1.8. Subtract each from the limit and multiply by the order:

The last column is nearly constant, so the shortfall is inversely proportional to the order. That is what the factorisation predicts: a satin of order n turns twice per repeat, so the fraction of its thread that is turning rather than plateau goes as one over n, and the specular area approaches its limit at that rate.

Inverting gives the design statement. To come within a stated fraction of the limit needs an order of about 3.3 divided by that fraction times the limit:

within ten per cent of the limit needs eighteen ends; within five per cent, thirty-seven.

And a satin of order n needs exactly n shafts, with no cheaper threading available. So the exchange rate is stark:

the last ten per cent of a satin’s available lustre costs eighteen shafts and the last five costs thirty-seven.

That prices a decision the trade makes by convention. An eight-end satin is at 77 per cent of the limit on eight shafts; a twelve-end is at 85 on twelve. Half the remaining lustre costs half again the harness, and past twelve ends the curve is flat enough that the shafts are buying float length rather than shine — which is what the float ladder says a designer is spending on snagging and abrasion.

Two cautions on the extrapolation. The limit itself is a property of this cloth — one yarn, one sett, one section — so the 1.8 per cent moves with the fabric while the 1/n approach does not. And the numbers past sixteen ends are computed rather than woven: no satin of twenty-four ends is an ordinary fabric, and the row is there to establish the asymptote rather than to describe a cloth anybody makes.

What survives both is the shape. A satin’s lustre is bounded, most of the bound is reached by eight ends, and every shaft after twelve is buying something other than shine.

Where the model stops #

The occlusion is not computed. One system’s crown can hide part of another’s from either the source or the eye, and neither is in the closed form. It is the dominant error for a plain weave and small for a floated one.

The section’s flat is taken as perfectly flat. A calendered thread is a racetrack in the site’s arithmetic, which is a rectangle with semicircular ends; a real pressed yarn’s top is neither flat nor smooth, being fibres. The width factor is therefore an upper bound.

And the tolerance is a cone. A real instrument’s acceptance is set by an aperture and a source of finite size and is not a cone at all. Every ordering here is checked across the whole range of it for that reason.

Nothing here is brightness. The quantity computed is an area with a stated orientation. Turning it into anything a person would call gloss needs a photometry that this collection does not have and does not want: the moment a question needs a receptor it is somebody else’s.

What it would take to make a matt satin #

The factorisation invites a construction that does not exist and is worth asking about, because the answer says which of the two factors is really in charge.

A cloth with a long float and no lustre would need a section with no flat and no arc within the tolerance — which is to say a section that is rough at a scale below the tolerance, so that its local normals are scattered. That is exactly what a spun yarn with a hair layer is, and it is why a soft-spun cotton satin is so much less lustrous than a filament one of the same construction.

So the width factor has a second term that this arithmetic does not carry: the fibre-scale roughness of the yarn’s own surface, which can destroy the width entirely while leaving the length untouched. Filament yarn has almost none of it; a carded staple yarn has a great deal.

That is where the fibre finally enters a subject this collection has been insisting is about structure. The draft owns the length and it owns it completely; the width is shared between the finish, the section and the fibre, and only the first two are computed here.

The generalisation #

When a property factors into two independent geometric quantities, the two levers multiply and neither is redundant.

That is a general rule about design and it is worth stating because the usual instinct is to treat two routes to a goal as alternatives. Here they are not alternatives: they are factors, one from the pattern and one from the cross-section, and a design that has only one of them has a product with a small term in it.

The second half of the lesson is about censuses. A property that factors can be censused in one of its factors alone, and the census is then valid for every value of the other. The four-by-four catalogue here was censused on crown line, at one yarn and one finish, and the resulting ordering holds for every yarn and every finish — because those enter the other factor.

Who found it, and when #

The factorisation appears to be new, though every part of it is old: the crown line is a reading of the float map, which this collection has drawn since it began, and the section width is Kemp’s racetrack from 1958 asked a question about tangents.

The trade has always known both halves separately. Weavers know that a longer float shines more; finishers know that a calender adds lustre. What neither says is that the two are factors of one product, so that the ratio between two weaves survives any finish and the ratio between two finishes survives any weave.

Where the ladder goes next #

To the second factor on its own: a calender buys the width, where the pressing is computed with the site’s own compression model and the gain turns out to be twenty-four-fold in width and six per cent in length — the arithmetic refusing to put any of it in the wrong factor.

Sideways, the fact that both systems reflect while only one bears is what makes turning the cloth change which one shines possible at all.