A centre that invents a life history

A single fitted growth factor on a shell whose expansion changes reports the geometric midpoint of the arc and hides the change, with a residual that accepts a change of sixty-four per cent. The repair the essay on Raup’s three numbers proposed is to fit the shell twice, over an inner arc and an outer arc, and read a disagreement between the two as a finding about how the animal grew.

On a spiral drawn from its true centre that works: a steady change from 2.8 to 3.6 splits the halves by 13.39 per cent, exactly √(W₁/W₀) − 1. But no shell has its centre marked, and the fit needs one.

A spiral that does not change, split in two #

The object is the plainest there is: a logarithmic spiral growing by 3.2 a turn, over four turns, with nothing changing anywhere along it. Split at two turns, each half spans two turns, which is exactly the span floor below which the collection refuses a fit.

Any disagreement between the two halves’ fitted factors is therefore the instrument’s, and not the shell’s, because the shell has one factor. Fitted from the true centre the halves agree to the last digit, and the test reports no change, which is the right answer.

A quarter of the innermost radius #

Displace the assumed centre by a quarter of the innermost radius, towards 52°. The inner half then fits at 3.4193 a turn and the outer half at 3.2200. The shell appears to have slowed down, by a split of −5.83 per cent between its first two turns and its last two.

The displacement is not large. Across the whole spiral, whose outer radius is 3.2⁴ = 104.86 innermost radii, a quarter of the innermost radius is 0.238 per cent of the outer radius. On a drawing whose outer radius spans a thousand points, it is under two and a half points — about the width of the line the spiral is drawn with.

In every direction #

Turn that same displacement through 180 directions and the split swings through a full cycle. It is most negative, −5.83 per cent, towards 52°, and most positive, +5.75 per cent, towards 248°, and its root mean square over all directions is 4.09 per cent.

The outer half reads higher in 93 of the 180 directions and lower in 87. So the sign of the invented change is as good as a coin: whether the shell appears to have sped up or slowed down depends on nothing but which way the centre was wrong.

The genuine change it matches #

A steady change in expansion from W₀ to W₁ splits the halves by √(W₁/W₀) − 1, so a split s corresponds to a genuine change of (1 + s)² from apex to aperture. A root mean square split of 4.09 per cent is therefore the split a change of 1.0409² = 1.0835 produces — an animal whose last whorl expands 8.35 per cent faster than its first.

That is not a subtle change to report about an animal’s life. It is the size of change that would reasonably be reported as a finding about growth, and the spiral here has none.

Almost all of it is the inner half #

Split into its two halves, the error is lopsided. The inner half’s fitted factor is out by 4.55 per cent RMS over all directions; the outer half’s by 0.44 per cent, a tenth as much. The split between the halves is, to a good approximation, the inner half’s error on its own.

The reason is scale. The displacement is a fixed length, a quarter of the innermost radius, and the inner half’s points lie between one and 3.2² = 10.24 innermost radii from the centre while the outer half’s lie between 10.24 and 104.86. The same quarter is a far larger share of the inner radii, so it bends the inner half’s logarithms far more.

One length against three radii #

The arithmetic is short enough to carry. A point whose distance from the centre changes by a small fraction ε has a logarithmic radius that changes by about ε, at most. At the first point of the spiral a quarter of the innermost radius is a fraction of 0.25; at the join between the halves, 10.24 innermost radii out, it is 0.024; at the aperture, 0.0024.

So the inner half’s logarithms are disturbed by up to a quarter at one end and by two and a half hundredths at the other, and a line fitted through a set of logarithms disturbed that unevenly takes a false slope. The outer half’s run from 0.024 down to 0.0024, ten times smaller at each end, and its slope barely moves. A fit reads its rate from how the logarithms change along the arc, and a fixed length changes them most where the radii are smallest.

Either sign #

Turned towards 248°, the same quarter-radius displacement makes the inner half fit at 3.0073 and the outer half at 3.1803, a split of +5.75 per cent. The spiral now appears to have sped up, by very nearly the amount it appeared to slow down when the centre was wrong the other way.

A test that reports a change of either sign from one error, depending only on its direction, cannot be read as evidence of a change in either direction until the direction of the error is known — and the direction of a centre error is exactly what nobody choosing a centre knows.

A difference is more fragile than a fit #

The same quarter-radius displacement moves the single fit over all four turns by 1.06 per cent RMS. Split into halves, it moves their difference by 4.09 per cent, very nearly four times as much. At a twentieth of the innermost radius the two are 0.21 and 0.82 per cent, the same ratio.

Both steps cost. A half has less arc than the whole, and a fit’s tolerance to its centre falls with its arc. And the difference is taken between the two halves the centre treats least alike, the one it disturbs most and the one it disturbs least. A test built from a subtraction between them inherits the worse of the two almost whole.

In proportion to the displacement #

The split rises in proportion to the displacement. Over four turns it is 0.82 per cent RMS at a twentieth of the innermost radius, 1.64 at a tenth, 4.09 at a quarter and 8.22 at a half, the last matching a genuine change of ×1.1711. Each doubling of the displacement doubles the split.

Over three and a half turns every figure is larger — 1.15, 2.31, 5.78 and 11.60 per cent — because each half then spans only one and three quarter turns. Less arc per half means the same centre error bends a larger share of each fit.

A twentieth of the innermost radius #

At a twentieth of the innermost radius the split still swings from −1.16 per cent towards 58° to +1.15 per cent towards 242°, with a root mean square of 0.82 per cent. That matches a genuine change of ×1.0164, and it still splits in either sign, the outer half reading higher in 91 directions of 180.

A twentieth of the innermost radius is under half a thousandth of the outer radius on this spiral — about half a point on a drawing whose outer radius spans a thousand. Because the split is proportional, there is no displacement small enough to be free; there is only a displacement small enough to be smaller than something else.

The floor is the centre, not the noise #

The obvious objection is that measured points are noisy anyway, so the centre is one error among others. Measured, it is the larger. Point noise of a thousandth of the outer radius, scattered in both directions, splits the halves by 0.74 per cent RMS over two hundred realisations; three thousandths splits them by 2.28 per cent, and a ten-thousandth by 0.07.

A centre error of a thousandth of the outer radius is about a tenth of the innermost radius on this spiral, and it splits the halves by about 1.6 per cent — more than twice what noise of the same size does. Put the other way, a centre displaced by only 0.0455 innermost radii already splits the halves as much as a thousandth of noise does, and that is 0.434 of a point on a drawing whose outer radius spans a thousand.

Noise has no direction #

The two errors differ in a second way, and it matters more than their sizes. At a thousandth of noise the mean split over two hundred realisations is 0.026 per cent against an RMS of 0.74, and the outer half reads higher in 105 of 200: the scatter points no way in particular, and averaging many traces of one section shrinks it.

A centre error does not scatter. Held in one direction it splits the halves by the same amount, in the same sign, every time the section is traced. Tracing the section again from the same chosen centre, or averaging ten tracings, leaves the split exactly where it was. The one error a careful worker can reduce by repetition is the smaller one.

What half a point means #

That crossing is the practical content of the measurement. For the centre’s contribution to be no larger than that of scatter of one point on a radius of a thousand, the centre has to be placed to within 0.43 of a point: more than twice as precisely as the points of the curve are themselves placed.

Nothing a person or a routine does by eye places the centre of an unmarked shell that well. How far a centre must move to make a nautilus golden measured the tolerance of a single fit to its centre; this is the tolerance of a difference between two fits, and that is the more fragile quantity.

What a genuine change has to clear #

Genuine changes split the halves exactly as the formula says. A change of ×1.02 from apex to aperture splits them by 0.995 per cent, 0.24 times the quarter-radius centre floor; ×1.05 by 2.47 per cent, 0.60 times; ×1.0835 by 4.09 per cent, the floor itself; ×1.1 by 4.88 per cent, 1.19 times; ×1.2 by 9.54 per cent, 2.33 times; and ×1.5 by 22.47 per cent, 5.49 times.

Against noise alone the same changes look robust: a change of ×1.05 is 3.32 times the thousandth-of-noise floor, and ×1.02 is 1.34 times. The centre floor is what takes them away.

The worst direction sets the bound #

A root mean square is the right summary of a centre error of unknown direction, and the wrong bound for a claim. A reading of a change has to survive the direction the centre was actually wrong in, and the worst of the 180 splits a quarter-radius error produces is 5.83 per cent — the split of a genuine change of 1.0583² = 1.120, twelve per cent from apex to aperture.

So a two-arc test from a centre that could be a quarter of the innermost radius off cannot distinguish a change of less than about twelve per cent from no change at all, and for a change it does see, it cannot say which way it went without the direction of the centre error.

Shorter halves are worse #

Over three and a half turns the quarter-radius cycle runs from −8.13 per cent towards 52° to +8.22 per cent towards 244°, with a root mean square of 5.78 per cent, which is the split a genuine change of ×1.1189 produces. The inner half’s error is 5.75 per cent and the outer half’s 0.75.

Shells are rarely measured over four clean turns; the inner whorls are the ones hidden or broken. Every turn a real section loses from the arc raises the floor.

On a photograph #

Put the spiral on a photograph whose outer radius is a thousand pixels. The innermost radius is then 1000/104.86 = 9.54 pixels, and a quarter of it is 2.4 pixels. A centre placed by eye within two or three pixels of the true one is a careful placement, and it leaves a split floor of 4.09 per cent RMS and 5.83 in the worst direction.

To bring the floor down to one per cent, the split being proportional to the displacement, the centre has to be within 0.05 × 1/0.818 = 0.061 innermost radii, which is 0.58 of a pixel. That is a centre found by computation from the curve, not placed, and even then the point scatter of a traced outline, at a pixel, sets a floor of its own of three quarters of a per cent.

What the test can still do #

The two-arc test is not useless, and saying what survives is part of the result. A split larger than the 5.83 per cent a quarter-radius error reaches in its worst direction cannot be manufactured by a centre error of that size in any direction, and a change of a fifth from apex to aperture, which splits the halves by 9.54 per cent, clears it by more than half again.

And a centre known to a twentieth of the innermost radius lowers the floor to 0.82 per cent, where a change of five per cent clears it three times over. What the test cannot do is detect a modest change from a centre chosen by eye.

What this corrects #

The advice to fit a shell twice over different arcs and read a disagreement as growth has been repaired where it was given, with the centre caveat attached. The advice was not wrong about what a genuine change does. It left out what the centre does, and the centre does more.

It also sharpens what a residual can and cannot test. A residual could not see a steady change; two residuals, or two factors, can see one from the true centre, and see an invented one from any other.

What this does not establish #

The centre error is modelled as a fixed displacement in a stated direction, and the noise as independent scatter in both coordinates. Neither is a measured property of any real section, and how well real centres are located is not known here. A centre fitted along with the growth factor, rather than chosen first, is a different instrument and is not measured.

The genuine changes are the steady change in rate the essay before this one modelled; a change concentrated in one whorl splits the halves differently. And the spiral is the nautilus’s factor, 3.2; a spiral that expands more slowly has more of its radius in its inner whorls, and where the floor sits for it is not measured here.

What would withdraw it #

A centre displacement whose split between halves is no larger than point noise of the same size. A split whose sign does not reverse as the displacement is turned. An outer half whose error is as large as the inner half’s. A centre error that averages away over repeated tracings. Each is one spiral to fit, and the measurement fits all of them every time it runs.

An instrument that needs the centre only to aim #

The next measure to test is the one ammonoid workers already use, which is built so that the centre barely matters. Two diameters of the shell half a volution apart, squared, give the expansion with no centre coordinate in them; the centre is needed only to aim the line the diameters lie on. The test is how that aim error enters — if it enters as its square rather than its first power, a quarter-radius error that moves a three-and-a-half-turn fit by more than a per cent should move the diameters by under a hundredth of one, and three points on a diameter measures whether it does, and where on a slowly expanding shell it stops being true.