How many lamps make one lamp

The lamp’s size over its distance, and nothing else treated the lamp as one continuous strip, 36 cm across, and everything in it — the degeneracy, the two-card recovery, the conditioning — depended on the source actually being that: one extended thing, imaging evenly through an edge. A great many real fixtures are not built that way. A strip light is often a row of individual bulbs or diodes; a softbox is a diffuser in front of a smaller, harder source; a window is a wall with several panes. The question this essay asks is what happens when the “one lamp” of the last four essays is really several point lamps standing in for it.

A row of point sources leaves a shadow that is not a soft-edged band but a staircase, because a place on the floor either can or cannot see each individual lamp and there is nothing in between — and the size of the steps, not merely their presence, is what says how good the counterfeit is. Measuring that size, and the rate at which more lamps buys a smaller one, is the whole of what follows.

Every figure below keeps the floor flat and the card a straight edge, which is not an incidental simplification. A soft shadow on a curved floor is not the lamp’s image already showed that even a genuinely continuous source’s penumbra stops being a clean record of the source the moment the receiver curves, and an array of point lamps would inherit every one of those distortions on top of its own quantisation — so a flat floor is what isolates the question this essay actually asks, which is what discreteness alone costs, independent of anything the receiver contributes.

The control: a single lamp is not a coarse array #

Before asking how fast the staircase smooths out, it is worth being exact about where the sequence starts, because the obvious guess about the starting point is wrong.

The natural way to picture “an array standing in for a continuous lamp” is a blurrier, lower-resolution version of the same ramp — as if going from many lamps to few simply coarsened an existing curve. That picture is wrong at the near end of the sequence in a way worth stating precisely: a single point source has no ramp to coarsen. It has a hard edge, in the same sense a single occluding edge does, because a point either is or is not visible from a given place, with nothing graded about the transition. The step from “one lamp” to “two lamps” is therefore not one step in a smoothly narrowing sequence; it is the introduction of gradation where none existed, and only once there are at least two lamps does “how coarse is the staircase” become a meaningful question to ask of a curve that has an actual staircase to describe. Without this control the falling sequence below would read as a single smooth trend; with it, the claim is sharper — an array becomes a lamp, starting from an object, the point source, that resembles one hardly at all.

This is a different two-lamp question from the one two lamps and one map asks, and the difference is worth being explicit about since both essays draw two lamps and both quote a number near a quarter. There, two lamps are two independent, genuinely separate sources, each casting its own complete shadow, and the finding is a map — a single scaling — that carries one shadow exactly onto the other. Here, the “two lamps” at 0.250 wrong are not separate objects being compared with each other at all; they are two candidate stand-ins for one single object, and the number describes how far their joint shadow falls short of that one object’s shadow. A reader who conflated the two would expect the 0.250 here to shrink to zero as the two point lamps are moved apart, exactly as the homothety in that essay’s map does when its two lamps coincide — and it would not, because moving these two apart is not a degenerate case of this construction, it is the whole of what this construction is testing.

The staircase, read directly off itself #

With the starting point settled, the shape of the approximation at a moderate lamp count is the next thing to make exact, and it is exact in a way that follows from counting rather than from any property of light.

That the worst departure is close to half a step rather than a whole one follows from where the staircase sits relative to the ramp it approximates: each tread is centred on the ramp’s own value at that point, rather than lagging or leading it, so the largest gap between the two curves is reached exactly halfway between one riser and the next, where the staircase is still flat and the ramp has moved half a step’s worth of height above or below it. A staircase built the other way — each tread aligned with the ramp’s value at its leading edge rather than its centre — would be wrong by a whole step at its worst, twice as badly, for the same sixteen lamps. The half-step figure is not an incidental fact about this particular array; it is the best a piecewise-constant curve can do against a straight ramp it is built to straddle.

The rate, and the single lamp excluded from it on purpose #

Sixteen lamps is one point on a curve, and the curve is what says whether adding lamps is a fast or a slow way to buy fidelity.

Excluding the single lamp from the fit is not a convenience; it is the same finding as the control above, stated as a number rather than shown as a picture. If a single point source really were the natural zeroth term of the same convergent sequence, it would sit on the fitted line, and it does not — it sits at twice the error the -1 power law would predict for one lamp, which is the fit’s own way of confirming that “one lamp” is a qualitatively different object rather than merely the worst-resolved member of the family. The -1 exponent itself is not an empirical curiosity either: a step is one over the lamp count by construction, the worst-case error is half a step by the previous reading, and half of one-over-N is exactly proportional to one-over-N — so the measured -1.0075 is the arithmetic confirming itself through an entirely independent route, casting real rays at real counts rather than substituting the formula into itself.

The 0.0075 by which the fitted exponent misses −1 exactly is itself worth accounting for rather than waved past as noise. The fit is a straight line drawn through the logarithm of the worst error against the logarithm of the count, over the range of counts this measurement actually casts rays at, and a genuinely exact −1 law would need the half-a-step argument to hold at every one of those counts with no rounding at either end of the band — the row of lamp positions is finite and their bands are read off a floor sampled at finitely many points, so the last few parts in a thousand of the exponent are the same kind of arithmetic residue the flat floor’s own straight-line fit carried in the previous essay, where a genuinely proportional relationship still reported a departure of 8.9e-16 metres rather than a mathematical zero. A −1.0075 next to a predicted −1 is agreement, not a discrepancy needing its own explanation.

What the even spacing assumes #

Every array in this essay places its point lamps evenly across the width a continuous lamp would occupy, and the −1 law depends on that choice in a way worth stating rather than leaving implicit. A staircase whose steps are all the same size follows directly from lamps that are all the same distance apart; a real fixture whose individual emitters are not evenly spaced — bunched toward the centre, say, or deliberately staggered — would leave a staircase with treads of different widths, and the worst-case error would no longer be simply half of one over the count, because “one step” would no longer be one well-defined size. Nothing computed here rules that case in or out; it says only that the clean −1 power law belongs to the evenly-spaced array specifically, and an uneven one would need its own version of the rate figure rather than inheriting this one.

The limit: how many lamps this floor cannot tell from one #

The rate says how fast the error falls; it does not by itself say when it has fallen far enough to matter on a particular floor, which needs one more figure at a much larger count.

The stated rule from the previous figure — 100 lamps for a 1 per cent step, 14.4 millimetres apart — and this figure’s 64 lamps at 1.6 per cent and 22 millimetres are two readings of the same law rather than two separate facts, and seeing both is what makes “a stated threshold needs a stated count” concrete rather than abstract. What decides whether 64 lamps or 100 or 4 is “enough” is not supplied by this measurement at all: it is a question about a reader’s eye at a stated viewing distance, and this floor’s own geometry supplies only the physical spacing the eye would have to resolve, in millimetres, at whatever occluder distance is in force. The geometry says how far apart the bands are; whether that gap is visible is the honest limit below.

A different kind of counting entirely #

Everything so far has asked how many point lamps it takes to counterfeit one continuous source. A completely different question, asked elsewhere on this site with the same word “lamps” in its title, is how many distinct, separately real lamps a photograph can tell apart at all.

How many lamps a drawing has is the essay that construction belongs to, and it is worth being exact about how little it shares with this one beyond a title. There, the lamps are genuinely separate, each with its own shadow-producing rays, and the question is whether a bundle of drawn lines can be sorted into the right number of pencils — a question about distinguishing sources that already differ. Here, the point lamps are a deliberate fiction standing in for one lamp that does not, in fact, have parts, and the question is how well the fiction hides its own seams. Counting shadows is not counting lamps sits between the two: it is the caution that a single lamp’s shadow, cast onto an uncooperative floor, can be mistaken for evidence of a second source — the same confusion this essay’s array is built to avoid rather than fall into, since every one of its steps is honestly labelled as a step rather than smoothed into a shape that hides what caused it.

The honest limit #

Nothing in this measurement says anything about what an eye actually resolves. The spacing between bands — 360 millimetres at 4 lamps, 22 at 64, and the 14.4 millimetres this floor needs for a stated 1 per cent step — is a fact about ray geometry on one particular floor at one particular occluder distance, and turning “these bands are so many millimetres apart” into “a reader would or would not see banding” needs an eye’s acuity, a viewing distance and very possibly the ambient light level, none of which this site computes. The geometry supplies the number a perceptual claim would need as an input; it does not supply the perceptual claim.

That caution has a companion on the other side of the fleet’s own vocabulary for “telling things apart.” The distance at which two lamps part measures how far apart two genuinely separate lamps must stand before a photograph can tell there are two rather than one — a question about position, answered in metres of separation between real sources. This essay’s question is about shape rather than position: the point lamps here are not trying to be told apart from each other at all, they are trying to be mistaken, collectively, for a single different object, and the quantity that matters is a contrast in the shadow rather than a distance between sources. The arrangement the count cannot see is the same family of caution again, aimed at a third target: it is possible to count the right number of lamps and still have missed how they are arranged, exactly as it is possible here to know the count of point lamps in an array and still not know whether that count is enough for a given floor and a given eye.

Nor does the model behind every figure here claim anything about how real point sources actually add their light. Each lamp in this array is treated as either fully visible or fully hidden from a given place, with the lit fraction simply the count of visible lamps over the total — which is exactly right for the geometric question of how much of the source a place can see, and says nothing about interference, about a diffuser mixing several small sources into something that looks smoother than its own geometry, or about a fixture whose individual emitters are not equally bright. Those are optical and photometric questions belonging to a different subject; the staircase measured here is the shape a purely geometric count of visible sources produces, which is the shape the softening in a real fixture starts from rather than the shape it ends at.

A related question this essay does not ask is what the array looks like once the card’s edge itself is replaced by something wider — a second array, say, or an occluder with a size of its own rather than a straight edge. Every figure above keeps the occluder a single edge precisely because a card would introduce a second length scale into the geometry, exactly as it did for the degeneracy in the previous essay, and mixing that complication into a measurement about counting lamps would make it impossible to say which of the two effects a given number belonged to. The count-versus-fidelity law measured here is therefore a statement about the source alone, isolated from the occluder, and a reader wanting both effects at once would need to compose this essay’s staircase with the earlier essay’s card-shaped umbra rather than read either number as already including the other.

What is settled is narrower and does not need either caution to stand: an array of point lamps converges to a continuous one at a stated, measured rate — the -1 power of the count — starting from an object, one point lamp, that is not a coarse instance of the same curve but a qualitatively different thing. The lamp’s size over its distance, and nothing else is the essay this one inherits its floor and its lamp’s width from, and everything measured here about staircases and their rate of smoothing is layered on top of a construction that essay already established behaves exactly like a genuine extended source when the receiver is flat.

The two essays are also a matched pair in what each one recovers from a shadow and what it cannot. That essay’s two-card construction recovers a continuous lamp’s size and distance exactly, from two straight-edge measurements, provided the source really is one continuous strip; nothing in it says what to do if the source turns out to be several discrete lamps standing in for one, since a discrete source does not obey the same closed-form relationship between an occluder’s height and its band’s width that the two-card recovery is built from — its “band” is a staircase rather than a clean edge, and the same formula fitted to a staircase’s outer corners would recover a slightly wrong height and width, by an amount this essay’s own worst-case figures already bound. So the two constructions are not simply stackable: knowing how many point lamps counterfeit a continuous one, from this essay, does not by itself say how badly the previous essay’s own two-card recovery degrades when it is unknowingly pointed at an array rather than a true strip, and that composition is not measured here.

What a point in shadow can see of the sky and the lamp a low shadow cannot locate both continue past this one along the same floor, one replacing the card’s edge with a slab overhead and the whole hemisphere with the source, the other replacing the soft edge with a hard one thrown by several posts and asking how well the source’s position, rather than its shape, can be recovered. Between the three of them — a lamp’s size, a lamp counterfeited by several smaller ones, and a lamp’s own position — the questions a single soft shadow on a flat floor can be asked are close to exhausted, and each answer has turned out to need its own control: a curved floor for the first, a single lamp that is not a coarse array for this one, and a fully enclosed point for the next.