Error propagation — where it appears
Named by 16 essays across 2 fields — each of them below, with the objects they name alongside it.
Three points on a diameter
Ammonoid workers measure a shell's expansion without a centre: two diameters half a volution apart, squared. On a logarithmic spiral that is exact, and the centre is needed only to aim the line. A quarter-radius aim error costs the fit 1.341 per cent and the diameters 0.0045, because the aim error enters as its square. Reading noise is another matter: at a thousandth of the outer radius the fit's four hundred points beat the calipers' three readings at every expansion up to the nautilus's, and which instrument is better depends on which error the section actually has.
What the septa count
A nautilus's chambers are each a scaled copy of the last, and an earlier essay gave their ratio as about 1.3 — what a growth factor of 3.2 gives over a third of a turn. It does not: a third of a turn at 3.2 is 1.474 in length. A ratio of 1.3 is 4.43 septa a whorl as a length, 8.87 as an area and 13.30 as a volume, so the dimension decides the count threefold. And the count is an exponent in any reading of the growth factor taken from one chamber to the next: one septum miscounted at thirteen moves it by 9.14 per cent. A chamber and the one a whorl out give W³ with no count at all.
The error budget for a nautilus
Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.
The band nobody can be placed in
The boundary between shells whose whorls run into one another and shells whose whorls run free was located here to the last bit a double holds. A specimen is not a point on that line, it is a measurement with an error, and carrying the whole measured error budget onto the boundary turns the line into a band running from 1/(W(1+b)) to 1/(W(1−b)). At the budget with the dividers set aside that band covers 48.2 per cent of the box the morphospace figure here is drawn on, and its share runs from 6.7 to 59.5 per cent across the six boxes in use — the same box-dependence the contact region itself showed. The angle criterion carries the same error better above an expansion of 1 + √2 and worse below it, exactly.
What the axis distance costs
Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.
A floor no better fit can lift
The band of shells nobody can place was built from one of the two numbers the contact boundary relates, and the other has now been priced. Carried together they widen the band by a third and take 48.2 per cent of the morphospace box to 59.7. The number that matters is further down: with the whorl expansion measured perfectly, 12.6 per cent of the box is still undecidable, and at an expansion controlled to a hundredth 94 per cent of what remains belongs to the second number. And the two errors are not independent — they come out of one guessed centre, which traces a curve across the boundary rather than a rectangle around it.
Three entries and one span
The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.
The dividers belong to the opening
The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.
The rim sets the opening
A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.
The outline finds its own centre
A worker with a sawn shell has an outline and nothing else: the centre, the expansion W and the distance from the axis D all have to come off it at once, and three numbers fitted to one outline can trade against each other. Fitted together, they do not trade where it matters. Started from a centre guessed a quarter of the innermost radius off, the drawing locates its own centre to about a thousandth of that radius, and over a turn of section drawn to a thousandth of the rim it returns W to 0.12 per cent and D to 0.045 — where reading the same drawing at the guessed centre gives 16.8 and 0.98. W's error rides the centre's; D's does not, and W and D do not trade against each other at all. The trade the round trip was set to watch for appears only where the model is wrong: a view five degrees off the section's plane moves W by three times its own noise error while the fit's residual stays within the noise, because the free centre moves to absorb the squash.
The cut along the axis
A median section cannot see how far a shell travels along its axis, so the third of Raup's numbers seemed to need a second cut. It does not need the first. One cut through the axis shows every whorl twice a turn as a whole circle, and fitted with the axis free it returns W, D and T together — to 0.023, 0.085 and 0.031 per cent on a high spire drawn to a thousandth of its rim, from an axis laid a quarter of a radius off and two degrees out. On the shell the median section was read on, it reads the expansion twice as well over a turn and the distance from the axis as well. Laid by eye and left there, the axis costs T four per cent and a high spire's D thirteen, and the residual says so. What it does not always say is that the saw missed the axis: on a low spire a cut a fifth of the innermost radius off moves W by two and a half times its own noise error with the residual at the noise.
An opening that is not round
One cut through a shell's axis returns Raup's three numbers to a few hundredths of a per cent when every section is a circle. A snail's opening is closer to an ellipse. Read with circles, a cut of a shell whose opening is a quarter taller than it is wide gives W exactly and puts the whole misfit into the other two: D comes back 29 per cent low and T 4.9 per cent low, by a first-order form in which the circle takes the ellipse's mean half-axis. A nearly round opening is the dangerous one — an aspect of 0.99 moves D by fourteen of its own noise errors while the points miss their circles by only a quarter more than the noise. Fitted with ellipses, the cut reads all five shape numbers with W, D and T nearly as precise as before, and the opening's height does not trade against the spire's translation.
A displaced census leaves a hole, not a pile
A mislocated centre moves a shell's place in Raup's space rather than blurring it, and the prediction was that a whole census would be compressed against the contact line, in a pile that could pass for shells preferring the line. Twenty thousand shells read from guessed centres say otherwise. Nothing piles up. The density beside the line falls, by about a sixth in the nearest bin, because a shell near the line is as likely to be carried over it as along it. And that gives a census a test that needs no outside knowledge: a section seen to run free cannot be read below the line, and 2.9 per cent are, at a tenth of a radius of error. Seventy-nine specimens are enough for one to show up.
A hand that drifts takes back what the fit won
A tilt can be undone by searching for the view that makes a traced spiral straightest, and with an error independent from point to point the corrected fit beats the calipers twentyfold. A pen does not err that way. It wanders out for a few degrees and comes back. Give the same tracing error a length along the outline and the corrected fit's advantage is gone on one turn by a twentieth of a turn of drift, with the band as wide as the calipers', and on two turns falls from twenty-five times to about twice. The tilts it invents on a section with none peak at a twentieth to a tenth of a turn, not at the half-turn that looks like a tilt. The residual test still separates drift from a real tilt, on a turn or more.
Four walls closer than they looked
Two of the four falloff exponents had a wall with no upper end at all, and the other two were located to factors of two and a half and nearly four. Nine rises at eighty starting angles close every bracket — and the four walls turn out to sit inside a factor of 1.111 of one another, which is narrower than the narrowest bracket.
Two refinements that do not multiply
The design that located the wall did two things at once — doubled the starting angles and halved the rise spacing — and the arithmetic behind it assumed each would buy about a factor of two. The finer rises did ninety-nine per cent of the narrowing and the doubled angles added under one, because a bracket's ends are rises and no error bar can move them.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsMeasurement errorClaim testingModel scopeWhorlGrowth factorIdentifiabilityMorphospaceParameter spaceGolden spiralInvoluteLogarithmic spiral