Every essay — page 24
What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it
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What light is
A spectrum is a function, not a colour. Illuminants, reflectance, blackbody radiation computed from Planck's law, and what has already been discarded before the eye is reached.
A reversed tube declares from its least reversed line
A spectral table declares its lamp's line width as the narrowest width any of its peaks admits, and that declaration survived every reversed line so far because the tube's other lines were left unreversed to set it. A real discharge reverses its strong lines together. With every mercury line behind its own column, in proportion to its strength, the prediction was that a one-nanometre table would refuse every strong line and declare from the weak violet line, and a two-nanometre table would declare an unsafe width from a green-line depth of two and a half. The violet line is never read as a peak at all. The yellow line, the least reversed of the three that are, sets the declaration at every depth and widens it as it reverses. Through one nanometre the table refuses the blue and green lines and goes on declaring from the yellow — unsafely, in the violet's window, the one line the table never checked. Through two nanometres behind a cooler layer it refuses nothing, and from two and a half its declaration fails in the blue line's window, down to seven tenths of the true peak.
A stricter fit declares a wider line
A two-nanometre table of a mercury lamp whose every line is self-reversed admits every reversed line as a Gaussian within half a per cent, and declares a width its own ceiling cannot stand on from a depth of two and a half. The obvious repair is to fit harder. The prediction was that at a tenth of a per cent the table refuses the reversed blue and green lines, admits the least-reversed yellow, declares unsafely from it between — and that a threshold that strict would refuse even unreversed lines on a real instrument's noisy table. The refusals come where predicted: the blue from a depth of three, the green from four. But the stricter threshold makes the declaration less safe, not more. It narrows the range of widths each line admits from below, so the yellow, never refused, declares a wider width at every depth, and the ceiling built on it falls below the true peak from a depth of two instead of two and a half. And a tenth of a per cent survives a quiet instrument: no unreversed line is refused until the table's noise reaches half a per cent — and then the first line refused is the yellow, the one that declares.
Two nanometres hide the yellow's notch from any fit
A plain Gaussian fit admits the reversed tube's 578 nm yellow line at every threshold on a two-nanometre table, and the ceiling built on its width fails from a column depth of two and a half. A fit with the notch in it — an emitted Gaussian with an absorbed one taken out of its centre — might see what the plain fit cannot. The prediction was that two nanometres leave too few entries to determine the notch, and one nanometre finds it from a depth of two. Two nanometres hide it completely: the best notch fit carries a negative notch, a spike, at every depth, and the readings a two-standard-error rule calls reversed are the same noise draws at every depth, each fitting a notch that erases the whole line. One nanometre finds it sooner than predicted — in every reading from a depth of one and half of them at a half — and its fitted depth climbs with the column. But its standard error cannot be trusted with seven entries and four numbers: the rule calls a third or more of unreversed readings reversed. The notch's fitted depth itself separates them cleanly.
The notch fit declares the true width, and the true width runs out
A lamp's table declares the narrowest line width its peaks admit, and builds on it a ceiling for how high any line can rise between the entries. A reversed tube breaks both: its lines are notched by cooler gas, a plain fit reads each notched line as a wider one, and on a one-nanometre table the ceiling falls below the true peak from a column depth of a half, until at two the table refuses every line and declares nothing. A fit with the notch in it reads the yellow's notch on that table, and the proposal was to declare the line it finds underneath — notch taken back out — with the prediction that the ceiling then holds to a depth of three. It holds to two. The notch fit returns the tube's own 1.2 nm core within two hundredths of a nanometre at every depth, clean and in noisy readings, and the ceiling built on it is safe to two and fails from two and a half, in the window beside the reversed blue line. So does the ceiling built on the true width itself, at the same depths to within a hundredth. The repair is perfect, and what fails past two is the ceiling's rule, which assumes the lines it bounds are not notched.
A ceiling that knows the notch holds to the deepest column
A lamp's table promises a ceiling for how high any line can rise between its entries, and for a reversed tube — every line notched by a cooler column — that ceiling fails on a one-nanometre table from a column depth of two and a half, because its rule assumes the line it bounds has no notch. The notch fit already reads the yellow line's notch, and the proposal was to build the ceiling for a notched line instead: in each window, the factor for a slit-blurred line carrying that window's notch. The prediction was that it holds to a depth of four. It holds to five, the deepest column the census runs, at every depth and in every one of 440 noisy readings. The fit's depth must be read as an optical depth, −ln(1 − d); taken as the first-order depth it fits as, the ceiling fails from four and a half. Part of the safety is the fit's own error: read that way, a deep notch comes out too deep — 4.25 for a true 2.29 at depth five — and an overstated notch only raises the ceiling.
A coarse table can borrow a notch, but not a phantom
A one-nanometre table can bound a reversed lamp by building its ceiling for the notch its own fit reads. A two-nanometre table cannot read the notch, and its ceiling, built for an unnotched line, fails from a column depth of three and a half. The proposal was to borrow the notch from the finer table, and the prediction that the coarse ceiling then holds to five with a wider margin. It does: 8 to 42 per cent above the reversed lamp's true peak at every depth, against the fine table's 1 to 29. But borrowed from the fine table's noisy readings, one notch in 440 sinks the coarse ceiling below the true peak — a notch of 0.13 the fit read in noise where there was no column at all. Under a wide slit a shallow notch flattens a line's top, two flanking entries see more of it, and a ceiling that assumes the notch is set lower; the phantom set it too low. Borrowing only notches the depth rule calls — deeper than 0.2 — keeps all 440 safe; borrowing those more than twice their error does not.
A wide table needs no notch to borrow
A two-nanometre table of a reversed tube cannot read the notch a cooler column cuts in its lines, and its ceiling, built for an unnotched line, fails from a column of three and a half; borrowing the notch a one-nanometre table read rescues it at every depth. The prediction for coarser tables was that a shallow notch lifts their flanking-pair factor further — to a fifth at five nanometres — and that the five-nanometre ceiling on a borrowed notch still holds to five. Both hold: the rise is 15, 18 and 20 per cent at three, four and five nanometres, and every borrowed ceiling stays above the true peak. But a table three nanometres wide or wider never needed the notch. Its unnotched ceiling stands at least 4.7 per cent above the true peak at every column depth, because each of its entries carries so little of any line's peak that the ceiling built on them sits high above every line, reversed or not — and borrowing the notch takes up to twenty points of that margin away.
A broad line brings the notch back to a wide table
A table of a reversed tube read through a slit three nanometres wide or wider never needs the notch a cooler column cuts in its lines: its ceiling, built for an unnotched line 1.2 nanometres wide, stays above the true peak at every column depth. The reason offered was that a wide entry carries too small a share of a narrow line's peak to be fooled, and the prediction was that broader lines undo it — that at a core of 3.6 nanometres the five-nanometre table fails from a column of about three, and that a table needs the notch wherever its slit is less than about twice the line's core. The five-nanometre table fails from a column of three, exactly. Across twelve tables of three cores and four slits, the column at which the unnotched ceiling first fails rises with the slit-to-core ratio — from a half at 0.56 to three and a half or four near 1.7 — and every table at a ratio of 2.5 or more holds; but a ratio of 2.08 still fails, so the threshold sits a little above two. With the true notch, every table holds.
Wings make a broad line easier to bound from its core
A spectral table read through a wide slit can still bound a reversed mercury tube's true peaks with a ceiling built for an unnotched line, so long as the slit is more than about two and a half times the lines' Gaussian core; at a ratio of 2.08 a five-nanometre table fails from a column depth of 3.5. A real discharge line has Lorentzian wings, which widen the width a table declares, and the prediction was that wings move that threshold out in proportion to the declared width, while a ceiling built on the declared width keeps it where it was. Both halves fail, the first in direction. With wings a tenth of a 2.4 nm core's width the five-nanometre table holds to 4.5, and with a quarter it holds at every depth, by eight per cent: wings take power from a line's peak and give it to the table's entry. A ceiling built on the wider declared width fails three of six winged tables that the core-built ceiling serves.
Wings rescue a three-nanometre table only to its deepest column
Lorentzian wings on a reversed tube's lines make every four-, five- and six-nanometre table's ceiling, built for the lines' 2.4 nm core, safer by eight to twelve points of margin a step, so that wings a quarter of the core's width carry all three to every column depth. The three-nanometre table, at a slit-to-core ratio of 1.25, is the one a bare broad line failed earliest, from a depth of two. The prediction was that it still fails with a quarter's wings, from about three, and holds at every depth only with a half's, because a narrow slit averages less of a line's skirt. The first half holds, half a step early: a quarter's wings fail it from 2.5. The second fails by a hair: with wings half the core's width the three-nanometre table holds to a depth of 4.5 and falls 1.1 per cent short at five. Its margin climbs by 6.6, 9.0 and 13.2 points a step — the smallest lifts of any table, from the lowest start.
A ceiling built for the notch holds the three-nanometre table
A table of a reversed discharge lamp read through a three-nanometre slit cannot see the notch a cool column of vapour cuts into each line's core, and a ceiling built for an unnotched core — the bound on how high the lamp can peak inside each window — fails from a column depth of two, and even with broad wings at five. The prediction was that a ceiling built for each window's notched line holds at every depth with any wing, by more than two per cent. It holds at every depth with any wing. With bare lines it holds by three tenths of a per cent at its tightest, at a moderate column, beside mercury's violet lines. Knowing the notch costs the shallow columns and pays the deep ones, because a shallow notch flattens a line's top and a deep one splits it.
Where the model breaks
Seventeen observers in 1931, an error in the blue that was never fully repaired, colour vision deficiency, and the display this page is being read on.
