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

No cloth shines under a sky swept the width of the light and found the contrast between a floated weave and a plain one inversely proportional to it. An eight-end satin outshines a plain weave by 177 under the sun’s disc, by 44 under a small lamp across a room, by 3 under a large window close to, and by nothing at all once the source is sixty degrees wide.

Each of those is one source. It ended on the question a real room asks, which has a lamp and a window and walls at once. Its guess was that a room’s contrast would lie between its sources’ answers, weighted by how bright each is — and so that a weak spotlight in a bright room would buy less than its own number suggests.

The sum has now been done, and the guess survives it exactly: the room’s contrast is its sources’ contrasts averaged by power, to within one per cent. What the guess did not say is how large that makes a small lamp’s part. A spotlight’s own number is enormous — 44 against a window’s 3 — so a tenth of it is more than the window’s whole contribution, and the same statement written in widths rather than contrasts is a harmonic mean, which is ruled by its narrowest term.

Each source adds its own highlight #

A specular highlight is a piece of the cloth whose surface faces exactly between the viewer and a source, within the source’s own angular half-width. A float reflects into a line, and the area that reflects is a length times a width: crown line from the draft, times a width of section from the yarn, the finish and the source.

In a room each source that lies in that direction adds its own highlight, at its own width, and each highlight is as bright as its source’s radiance. So a weave sends the viewer

$$\sum_i L_i\,A(\varepsilon_i)$$

— the sum over sources of radiance times specular area at that source’s width. The room’s contrast between two weaves is the ratio of their two sums. No average has to be chosen in advance; the sum decides it.

The average that falls out #

The two areas scale differently with the width, and that is what makes the sum interesting.

A floated weave has crown line that shines however narrow the source, so its area grows in proportion to the width. A plain weave has no crown line at all, only turns, so its area grows as the width’s square. And a source’s total power is its radiance times its solid angle, which goes as its width squared.

Put those together and the ratio of the two sums is the single-source contrast at one width:

$$\varepsilon_{\text{eff}} = \frac{\sum_i P_i}{\sum_i P_i/\varepsilon_i},$$

the harmonic mean of the sources’ widths, weighted by the power each supplies. The room behaves exactly as one source of that width would.

The derivation uses the narrow-source forms of both areas, and the arithmetic does not rely on it: the room’s contrast is computed from the full closed-form areas at each source’s width, the single width that reproduces it is found by bisection, and that width is required to equal the harmonic mean to within two per cent. It does, to within one.

The narrowest source rules #

A harmonic mean is dominated by its smallest terms. A width of one degree enters it as one over one; a width of sixteen enters as one over sixteen. So a lamp carrying a small share of the light pulls the room’s width down far out of proportion to its share.

With one per cent of the light from the lamp, the room behaves as 13.9 degrees instead of 16. With five per cent, 9.1. With a tenth, 6.4 — and the contrast has gone from 3.01 to 7.21. A tenth of the light has more than doubled the contrast. By a third of the light the room behaves as a 2.9-degree source and the satin shows 15.6 over plain, a third of the way to what the lamp alone would give.

So the guess was right and its conclusion undersold it. A spotlight in a bright room does buy less than its own number — a tenth of the light buys a tenth of 44 — but it buys far more than its share of the light suggests, by a factor of about its width ratio, because contrast is paid for in power over width and a narrow source has a great deal of power per degree.

Two averages, and which one the cloth takes #

There are two natural ways to average a room, and they disagree by a factor of two at a tenth of the light.

One averages the geometry: the room is about as wide as its sources, weighted by power, so a 16-degree window with a tenth of its light moved to a 1-degree lamp is a 14.5-degree room, and a 14.5-degree source gives the satin 3.30 over plain — barely more than the window alone. The other averages the answers: each source gives its own contrast, and the room’s is those contrasts weighted by power, 7.21. The cloth takes the second, and the reason is the plain weave. Its highlight does not depend on how wide the light is, so the denominator of every contrast is the same whichever source lit it, and a ratio with a fixed denominator averages exactly as its numerators do.

The quantity behind the numerator is radiance — brightness per degree of source — because a specular facet shows the viewer the source’s own surface, not its total output. A window is a large source of modest radiance; a small lamp of the same power is a small source of enormous radiance. Averaging the widths gives the window most of the say. Averaging what the cloth actually reflects gives the lamp most of it.

The harmonic mean is what that difference looks like once the two weaves’ areas are put in. A floated weave’s highlight is its area, which grows with the source’s width, times the source’s radiance, which falls as the width squared for a given power — so a source adds to a satin’s highlight in proportion to its power over its width. Contrast is bought in power per degree, and a narrow source has a great deal of it.

A plain weave does not care how wide the light is #

The same arithmetic run on the plain weave gives a result worth stating by itself. Its highlight is all turns, and its area grows as the width squared — exactly as fast as the radiance of a fixed power falls. The two cancel. A plain weave’s highlight depends on how much light falls on it and not at all on how that light is spread: a lamp and a window of equal power give it the same shine.

That is what makes the plain weave the natural reference for every contrast of a floated weave. It is the one cloth that reads a room as a total, with no sense of where the light comes from or how concentrated it is, so every change in a floated weave’s contrast is a change in the floated weave alone. A satin measured against a plain weave under two different rooms is being measured against the same yardstick twice, provided the two rooms carry the same power.

A tenth of the light, from lamps of four sizes #

The narrower the lamp, the more its tenth is worth.

A bare bulb close to, four degrees wide, with a tenth of the light, lifts the contrast only to 3.85. A 0.25-degree point of the same power — a glint of sun through a gap in a curtain, a halogen spot across a room — lifts it to 20.6, nearly seven times the window’s. The power is the same in every case; what changes is how much of it arrives per degree.

That is a practical statement about showing a satin, and it is the reverse of the instinct to light a cloth generously. More light of the same kind changes nothing about the contrast, since every term in the harmonic mean scales together; only a change in the mix of widths moves it, and the mix is moved most cheaply by adding a small, bright, narrow source rather than by brightening the broad ones. A satin shows its float best under one small bright source against dim broad ones, and the broad ones are not neutral, whatever their colour or their direction on the ceiling: every watt they add enters the harmonic mean at its own wide width and pulls the room’s contrast down.

Every floated weave gains by much the same factor #

The effect is not a satin’s alone.

Under the window alone a 2/2 twill shows 2.28 over plain, a five-end satin 2.61, an eight-end satin 3.01. Add a tenth of the light from a 1° lamp and they become 4.97, 5.97 and 7.21 — multiplied by 2.2, 2.3 and 2.4. The ordering is the weaves’ own and the lamp does not change it; what it changes is how far apart they read. Under the window the twill and the satin differ by 0.7; under the room they differ by 2.2.

Walls are broad sources and they cost contrast #

A room’s walls are sources too — very broad ones, lit by everything else — and a broad source reflects from every weave alike.

At sixty degrees every weave reflects the same share of its face, so light from the walls adds equally to the satin’s luminance and the plain weave’s, and pulls their ratio towards one. Thirty per cent of the light from the walls takes the eight-end satin’s contrast from 7.21 to 6.30. Because a wall’s power is spread over so wide an angle, its radiance is low, and it costs less than its share would suggest — the mirror image of the lamp.

How a satin has always been shown #

None of this would surprise anybody who has sold silk. Damask and satin are shown under a single point of light held low, or turned in the hand towards one bright window pane, never laid flat under an overcast skylight — and the arithmetic says why the habit works as well as it does. A satin’s value is in its float, a float’s value is a crown line that shines at any source width, and a crown line is only distinguishable from a plain weave’s turns when the source is narrow.

The same holds for the defects a satin is judged by. A satin’s row belongs to its sett: the diagonal a regular satin shows is visible or not depending on how the sett stretches the move, and it is seen by the highlight the floats make. Under broad light the row and its absence are equally invisible; under one narrow source the row stands out. So a buyer inspecting a satin under a showroom’s panel lighting is inspecting it in the light that hides both its virtue and its fault, and the satins worth weaving can only be told from the others under a lamp.

The practical rule is one line: show a satin under one small bright source and keep everything else dim, because every broad source in the room enters the harmonic mean at its own wide width and pulls the room towards a sky.

And a calender adds to what the lamp gives #

A calender buys the width: pressing a cloth flattens the top of every thread into a plateau whose width does not depend on the source at all. That is a third term in a floated weave’s highlight — crown line times flat top — and it too is untouched by the source’s width, so it strengthens exactly the part of the highlight a narrow source rewards. A calendered satin under a lamp is the arrangement every result on lustre points to, and the cover a calender best suits is the other half of the recipe.

What a photograph of a mirror would measure #

The essay that asked the question proposed the measurement that would settle it for a real room: a flat mirror where the cloth will be, photographed from where the viewer will stand, showing every source at its true size and brightness.

The harmonic mean says exactly what to do with that photograph. Each bright region of the mirror image is a source; its power is its brightness summed over its area in the image, and its width is its angular size. The room’s effective width is the total power divided by the sum of each source’s power over its width. That is one pass over the image, and it turns every contrast here into a number for a particular room.

It also says what to look for in the image without computing anything. The room’s contrast is set by its smallest bright spot, not by its biggest bright area. A cloth that looks flat in a showroom lit by ceiling panels will come alive under one spotlight, and a photograph of a mirror will show which of the two a room has.

The model named #

The cloth is a sheeting-weight construction woven as a plain weave, a 2/2 twill, and five- and eight-end satins, with the surface built from its threads’ paths and sections. The specular area at a source’s width is the closed form — crown line plus an arc at each turn, times the section’s flat top plus its curved sides within the tolerance — up to sixteen degrees, and counted directly off the surface for a wall at sixty. Each source lies in the specular direction, has a half-width and a power, and a radiance of its power over its width squared. The room’s contrast is the ratio of two weaves’ summed radiance-weighted areas.

What was counted #

A lamp of 1° and a window of 16° at eleven shares of the light from nothing to all; the effective width at each, found by bisection against the single-source contrast; lamps of 0.25°, 1°, 2° and 4° at a tenth of the light; three floated weaves against plain in the room; and the same room with thirty per cent of its light from walls at 60°.

What the model cannot show #

Every source is placed in the specular direction. A real room’s sources lie at different angles, and a source away from the mirror direction lights only the facets tilted towards it. A satin’s plateaus face straight up, so an off-axis lamp lights its turns and its threads’ rounded sides, much as it lights a plain weave’s — and the contrast from that lamp is lower than its width alone would give. The harmonic mean is the answer for sources the viewer sees reflected, which is the arrangement a person holding a cloth up to a lamp makes without thinking.

Radiance is taken as uniform across each source. A lamp with a bright filament inside a frosted bulb is two sources, and the harmonic mean says the filament wins.

And the hair layer is ignored. A hair layer veils a highlight: standing fibres scatter light from above the crowns into every direction, which acts like a broad source attached to the cloth itself, and on a raised cloth it would pull every contrast here towards one.

Who found it, and when #

That a narrow source makes surfaces look glossier is a photographer’s and a lighting designer’s common knowledge, and gloss measurement standardises its source geometry for exactly that reason. The inverse law for a floated weave against a plain one is the sky essay’s.

What is new here is the weighting: that summing sources in the specular direction makes the room a single source at the power-weighted harmonic mean of their widths, so that a small share of light from a narrow source dominates the contrast, where the other natural guess — a room as wide as its sources’ average width — would have it matter little.

Still open: the sources that are not in the mirror #

Every source here is one the viewer sees reflected in the cloth’s general plane. A lamp off to one side lights the cloth too, and it lights different facets: the turns and the rounded sides of threads, which a plain weave and a satin both have, rather than the plateaus only a float has.

So an off-axis source should behave like a broad one — adding to both weaves and diluting the contrast — by an amount set by how far off the mirror direction it lies and by which way the cloth is turned. Whether a lamp twenty degrees off the mirror still raises a satin’s contrast, or only raises its brightness, is the calculation the facet counts could make next.