Three points on a diameter

Every instrument priced so far needs a centre. The fit needs one, and a quarter of the innermost radius moves its answer by more than a per cent; two fits over two arcs need one more, and the same quarter invents a change of growth of either sign. No shell has its centre marked, so each of those floors is a floor under the instrument rather than under the shell.

Workers on coiled ammonoids have long measured Raup’s W another way, and the definition of that measure has no centre in it. The question is what it costs, and what it is better and worse at than the fit.

Two diameters on one line #

Draw a line from the outer wall at the aperture, past the coiling axis, to the outer wall on the far side, which is the wall half a volution back. Its length is the conch diameter, dm1. The same line, read from that far wall back past the axis, meets the outer wall again one whole volution behind the aperture, on the aperture’s own side. That second length, far wall to the wall a volution back, is the diameter half a volution earlier, dm2.

The difference between them is the apertural height, ah = dm1 − dm2. The whorl expansion rate is the ratio of the two diameters, squared to make half a volution a whole one: WER = (dm1/dm2)². Every quantity in it is a distance between two points on the wall, on one straight line.

Exact on a logarithmic spiral #

On a logarithmic spiral the measure is exact, and the reason fits on one line. Call the radii at the aperture, half a turn back and a whole turn back r₀, r₁ and r₂. Then dm1 = r₀ + r₁ and dm2 = r₁ + r₂, and a spiral growing by W a turn has r₀ = √W·r₁ and r₂ = r₁/√W. So dm1/dm2 = √W, and its square is W.

For a spiral growing by 3.2 with its aperture three and a half turns along, dm1 is 1.559017 of the outer radius, dm2 is 0.871517, and the apertural height between them 0.687500, which is 1 − 1/3.2. Squared, their ratio is 3.200000. Checked at five expansions and three positions of the aperture, it is W to the last digit a computer carries.

Where the centre enters #

The centre enters once, to aim the line. A worker holding the shell has to decide which line through the aperture passes through the coiling axis, and that decision is made by eye on a shell with no axis marked. An aimed line that misses the true centre cuts the far wall and the inner wall at slightly different points, and the two diameters are slightly wrong.

The instrument’s whole claim to indifference rests on how that error enters. If the diameters’ ratio moves in proportion to how far the aim is off, the calipers are one more instrument with a centre floor. If it moves in proportion to the square, a small aim error is a very small error.

The aim error enters as its square #

On the 3.2 spiral, over 180 directions of aim, the calipers are out by 0.0002 per cent at an aim error of a twentieth of the innermost radius, 0.0007 at a tenth, 0.0045 at a quarter and 0.0181 at a half. Each doubling of the aim error multiplies the calipers’ error by four. Fitted over those four aim errors, the slope on logarithmic axes is 2.00.

The least-squares fit over the same arc, from the same wrong centres, is out by 0.268, 0.535, 1.341 and 2.701 per cent, with slope 1.00. At a quarter of the innermost radius the calipers are three hundred times better.

Why a slope of two means stationary #

A slope of two over every direction of aim says something exact about the aimed line. If turning the line slightly one way changed the ratio in proportion to the turn, turning it the other way would change it by the same amount with the opposite sign, and the error would rise as the first power. It does not. So the line through the true centre is where the ratio is stationary: turning it either way changes the answer only at second order.

That is the whole mechanism of the instrument’s indifference. An instrument read at a stationary point takes its input error squared, and the fit, which reads the logarithms of radii measured from the centre itself, is not read at one.

A radius ratio from the same centre #

The contrast that isolates the mechanism is a ratio that uses the same three wall points but measures from the assumed centre. The aperture’s radius over the radius a volution back, r₀/r₂, read from the assumed centre, is also W on a perfect aim. From a centre a quarter of the innermost radius off it is out by 0.675 per cent, with slope 1.00.

The same three points, the same line, and a hundred and fifty times the calipers’ error. What the calipers avoid is not a bad aim; it is using the centre’s coordinates in the lengths.

Five expansions #

At a quarter of the innermost radius, the calipers are out by 0.1093 per cent at an expansion of 1.6, 0.0185 at 2.4, 0.0045 at 3.2, 0.0008 at 4.5 and 0.0001 at the golden 6.854: a 1,210-fold fall across the range. The radius ratio falls sixteen-fold, from 2.069 to 0.128 per cent.

The fit over the whole arc does not fall at all. It sits between 1.12 and 1.35 per cent at every expansion, because the arc always starts at an innermost radius of one, and it is the innermost whorl that a fixed centre error bends. The fit over only the outer two turns does improve with the expansion, from 2.877 to 0.231 per cent, by leaving that whorl out.

What sets the calipers’ error #

The fall with expansion is not a separate effect. The innermost point the calipers use is the wall a volution back, at a radius r₂ = R/W, and measured against that radius the aim error does the same thing at every expansion. At a hundredth of r₂ the calipers are out by 0.0018 to 0.0024 per cent; at a tenth, by 0.18 to 0.24; at three tenths, by 1.75 to 2.31.

Divided by the square of the share, every one of the twenty readings lies between 0.183 and 0.257. The calipers’ error is about a fifth to a quarter of the squared aim error, measured on the radius they reach, whatever the spiral.

The rule, worked on a nautilus #

On the 3.2 spiral with its aperture three and a half turns along, the outer radius is 3.2^3.5 = 58.62 innermost radii and the radius a volution back is 58.62/3.2 = 18.32. A quarter of the innermost radius is 0.25/18.32 = 1.36 per cent of it.

Squared, that is 0.000186, and a quarter of it is 0.0000465, or 0.0047 per cent. The measurement says 0.0045. A worker can estimate the instrument’s centre cost on the back of an envelope from the size of the shell and the aim, and it will come out right.

A slow spiral, aimed badly #

A slow spiral is the hard case, because its wall a volution back is close to the axis. At 1.6 a turn the outer radius is 5.18 innermost radii and r₂ is 3.24. An aim error of a whole innermost radius is 31 per cent of r₂, and on a drawing the line visibly misses the centre.

Aimed towards 52°, the diameters read 1.6230 against the spiral’s 1.6, out by 1.44 per cent. Over all 180 directions the calipers are out by 1.86 per cent RMS, and the fit from the same wrong centres by 4.74. The calipers still win, on a shell and an aim chosen to be unkind to them.

Where the calipers lose #

They do lose, and the place is worth stating exactly. At 1.6 a turn with the aim 2.61 innermost radii off — half the outer radius — the calipers are out by 20.09 per cent and the fit by 14.08. In 16 of the 180 directions the line misses a whorl entirely and there is no second diameter to measure.

That is an aim error of half the shell. Nobody holding a shell misplaces its axis by half its radius, and the measurement’s interest is in where between a good aim and that absurd one the calipers stop winning.

The crossover, counted two ways #

Counted in innermost radii, the crossover moves enormously with the expansion: 1.55 innermost radii at 1.6, 6.51 at 2.4, 14.9 at 3.2, 43.4 at 4.5 and 154 at 6.854. By that count the slow spiral loses ninety-nine times sooner than the golden one.

Counted as a share of the outer radius, which is how an aim error on a real shell is sized, the crossings are 29.97, 30.38, 25.45, 22.47 and 18.27 per cent. Every expansion keeps the calipers ahead of the fit until the aim is at least eighteen per cent of the shell’s radius off. On centre error alone, the calipers are the better instrument for every shell anyone would measure.

On a photograph #

Put the nautilus’s 3.2 spiral on a photograph whose outer radius is a thousand pixels. A quarter of the innermost radius is then 4.3 pixels, and the calipers’ 0.0045 per cent against the fit’s 1.341 is the price of an aim that careful.

A careless aim is ten times worse. At 2.61 innermost radii the aim is 4.45 per cent of the outer radius off, 44 pixels, a line that visibly misses the axis on any print. The calipers are out by 0.50 per cent and the fit by 10.60. Because the calipers’ error goes as the square, it falls to the 0.285 per cent that a reading error of one pixel costs them at √(0.285/0.50) × 2.61 = 1.98 innermost radii: an aim within 34 pixels, 3.4 per cent of the outer radius, costs the calipers less than misreading a wall by a single pixel does.

Reading error is a different error #

A worker with calipers also misreads where the wall is. That error is along the line, not in two dimensions, and three readings make two diameters that share the far point. So dm1 and dm2 are correlated through the middle reading, and the spread of the logarithm of WER is 2σ·√(1/dm1² + 1/dm2² + (1/dm2 − 1/dm1)²), with every length in units of the outer radius.

At σ of a thousandth of the outer radius on the 3.2 spiral that is 0.282 per cent, and twenty thousand simulated surveys give 0.285. Treating the two diameters as independent readings, each with two errors of its own, would say 0.375: it counts the shared point twice.

The spread, worked #

The formula is short enough to evaluate by hand, and doing it shows which reading matters. With dm1 = 1.559017 and dm2 = 0.871517, the three terms are 1/dm1² = 0.4114, 1/dm2² = 1.3166, and (1/dm2 − 1/dm1)² = (1.1474 − 0.6414)² = 0.2560. They sum to 1.9840, whose root is 1.4086, and twice that times a thousandth is 0.2817 per cent.

The largest term is the shorter diameter’s. The wall a volution back is the reading the whole instrument is most sensitive to, because a fixed error along the line is the largest share of the shortest length — the same reason the fit is most sensitive to its innermost whorl, arriving by a different route.

Under noise, the fit wins where it matters #

Put the same size of error on the fit’s four hundred points, in both coordinates, and the comparison turns over. At 1.6 a turn the fit spreads by 0.016 per cent against the calipers’ 0.184; at 2.4 by 0.056 against 0.236; at the nautilus’s 3.2 by 0.143 against 0.285. The fit is the better instrument under noise at every expansion up to and including the nautilus’s.

That is not surprising once it is said. The calipers use three readings and the fit uses four hundred, and averaging is what a least-squares fit is for. The comparison is fair to the extent that a real section yields a traced outline of hundreds of points and three well-placed caliper readings, which is roughly the choice a worker has.

Where the fit collapses #

At 4.5 a turn the calipers take the lead, 0.358 against 0.443 per cent, and at the golden 6.854 the fit collapses to 6.95 per cent with a bias of −2.56. Its arc starts at an innermost radius of one on a spiral whose outer radius is 843, so noise of a thousandth of the outer radius is 0.84 of the innermost radius, as large as the first whorl itself.

The calipers never read that whorl. Their innermost point is a volution back from the aperture, 123 innermost radii out on the golden spiral, and the same noise is a small share of it. The conversion from two-dimensional scatter to an error along the line is under five per cent at every expansion here, so the comparison is not an artefact of how the error was modelled.

Which error the section has #

So each instrument wins against one error. On centre error the calipers win by orders of magnitude at every expansion. On reading noise the fit wins at every expansion up to 3.2. A real section has both, and which instrument is better depends on their relative sizes.

Take the nautilus’s 3.2 with a quarter-radius centre error and noise of a thousandth of the outer radius, and suppose the two errors independent. The calipers total √(0.0045² + 0.285²) = 0.285 per cent; the fit totals √(1.341² + 0.143²) = 1.349. For the fit to tie, its centre error would have to shrink until its own contribution was 0.247 per cent, which is a centre within about 0.046 innermost radii — a little under the twentieth at which two arcs stop inventing a change.

Reading the present #

There is a third difference, and it has nothing to do with error. On a shell whose expansion rises from 2.8 to 3.6 over four turns, the fit reports the geometric midpoint, 3.1749, and the last whorl expands by 3.4887. The calipers read 3.5053: the expansion 0.849 of a half-turn behind the aperture.

On the shell rising from 2.5 to 4.1 they read 3.8641 against the last whorl’s 3.8203 and the fit’s 3.2016; on the falling shell, 2.8776 against 2.8894 and 3.1749. Whichever way the expansion changes, two diameters at the aperture read the present, and the fit reads the middle of the animal’s life.

Why a little nearer than the last whorl #

The last whorl’s ratio reads the expansion a whole half-turn behind the aperture, and the calipers read it about 0.85 of one. The difference is in how dm1 is built. It is the aperture’s radius plus the far wall’s, and on a spiral growing by W the aperture’s part is √W/(1 + √W) of the whole — 0.641 at 3.2 — so the outer half-turn carries more than half the diameter’s weight.

That is the only sense in which the calipers are biased, and it is a bias towards the part of the shell that decides whether the last whorls touch.

What this does not establish #

It measures the outer wall only, so it says nothing about the axis distance D, and contact needs D as well as W. It assumes each wall point can be placed to the stated error, which a worn or broken venter may not allow. It uses one diameter; averaging several along the shell is a further instrument, likely to close some of the gap to the fit under noise, and is not measured.

It also says nothing about how any published expansion was measured. The nautilus’s 3.2 is quoted here from the literature, and whether that figure came from calipers, a fit or dividers walked along the curve decides which of the floors above lies under it.

What would withdraw it #

A logarithmic spiral on which two diameters half a volution apart, squared, do not return W from the true centre. An aim-error slope near one rather than two. A spread of the three-reading simulation that departs from the formula with the far point shared. A fit that loses to the calipers under noise at 1.6 a turn. Each is run every time the measurement runs, and each has held.

A centre that is fitted, not chosen #

The fit and the calipers have been compared here with the fit’s centre chosen first and held fixed, because that is how it has always been used here. The obvious third instrument lets the fit choose its own centre, minimising the same residual over the centre’s two coordinates as well as the growth factor. The next measurement is that instrument, and its test is plain: whether a fitted centre lands within the 0.046 innermost radii at which the fit would beat the calipers on a nautilus, and what reading noise does to a centre that is estimated from the same noisy points it is meant to correct. A chambered shell offers a different way round the centre altogether, and what the septa count measures it.