One number for a shell that changes

The fit that recovers a shell’s growth factor has been priced four times: what an unmarked centre costs it, how far a centre has to move to make a nautilus golden, whether its residual can tell a logarithmic spiral from something else, and what walking a pair of dividers along the curve does. Every one of those measurements drew a spiral with one growth factor and asked how well the fit got it back.

A shell does not have to have one. Raup’s space has no time in it, and an animal whose whorls expand faster as it matures is a trajectory through that space rather than a point in it. The question is what the fit’s single number means when the rate moves.

A shell whose rate changes steadily #

The simplest change there is makes the argument. A logarithmic spiral grows at a rate b, with r = a·e^(bθ) and an expansion per turn of e^(2πb). Let b itself run in a straight line from one value at the apex to another at the aperture, over an arc of L radians. The expansion per turn at each point of the arc then moves from W₀ at the start to W₁ at the end, and the logarithm of the radius is a quadratic in the angle rather than a line.

That is one change among many, and it is chosen because it is the gentlest: nothing happens abruptly, and nothing a reader would call a change in form happens at any one whorl. It is also the first term of any smooth change, so what it shows is what every smooth change shows to first order.

The fit reports the geometric midpoint #

Hand the fit such a shell, sampled evenly in angle as every spiral so far has been sampled, and it returns the geometric mean of the two ends. For 2.8 rising to 3.6 over four turns it returns 3.17490, and √(2.8 × 3.6) = √10.08 is 3.17490, to the last digit the fit carries.

The reason is short. A least-squares line through a quadratic sampled evenly and symmetrically has exactly the slope the quadratic has at the middle of the interval, because the curvature term is even about the middle and a line’s tilt is odd, so the curvature contributes nothing to the tilt. The slope at the middle is the rate at the middle, and the rate at the middle is the geometric mean of the two ends.

Geometric, not arithmetic #

The middle of 2.8 and 3.6 in the ordinary sense is 3.2, and the fit does not return it. It returns 3.1749, which is 0.79 per cent lower. The reason is that the rate b, and not the expansion W, is what changes in a straight line; W is the exponential of b, and the midpoint of a straight line in b is the geometric mean in W.

The difference is small here and it is a reminder of which quantity a fit to logarithms is linear in. A shell whose W changed in a straight line rather than its b would put the fit’s number slightly elsewhere, and the gap between the two readings grows with the change.

Where the midpoint is #

On a four-turn arc the fit’s number belongs to the point two turns along. Drawn as a curve along the shell, the expansion rises from 2.8 at the apex through 3.1749 at the middle to 3.6 at the aperture, and the fit’s horizontal line crosses the curve at exactly one place, which is the middle.

That is a correct statement about a part of the shell nobody measures anything at. The apex is where the shell began and the aperture is where it is now, and the fit’s number describes neither.

Every point weighted, the ends not at all #

There is an exact statement behind the midpoint, and it holds for any change, steady or not. The slope of a least-squares line through a curve sampled evenly on an interval is a weighted average of the curve’s own local slope, with weight 6x(1 − x) at the fraction x of the way along. The weight is 1.5 at the middle and falls to zero at both ends.

For a four-turn arc that puts 68.75 per cent of the weight on the middle two turns and 15.625 per cent on each of the first and last turns. So whatever a shell does in its last whorl, the fit hears less than a sixth of it, and what it does at the aperture itself it does not hear at all. For a steady change the weighted average happens to be the midpoint; for any other it is still an average in which the present counts for nothing.

Every case, a falling one included #

The result is not a feature of one pair of numbers. A shell rising from 3.06 to 3.35 fits at 3.20172; from 2.5 to 4.1 over three and a half turns at 3.20156; from 2.4 to 4.8 at 3.39411. Each is the geometric mean of its own two ends.

A shell whose expansion falls, from 3.6 to 2.8, fits at 3.17490 — the same number as the shell that rises from 2.8 to 3.6. The fit cannot tell a shell speeding up from one slowing down, because the geometric mean is symmetric in its two ends, and a direction of change is precisely what a mean discards.

Three shells, one number #

Put three of the six side by side. A shell that grows by 3.2 throughout fits at 3.2000; one rising from 3.06 to 3.35 fits at 3.2017; one rising from 2.5 to 4.1 fits at 3.2016. Their fitted factors agree to 0.06 per cent, and all three are accepted as spirals, with residuals of 0, 0.0301 and 0.1439.

Their apertures expand by 3.2, 3.35 and 4.1, and their last whorls by 3.2000, 3.3123 and 3.8203. A table that lists each shell by its fitted growth factor lists these three as one shell. The collection’s own convention for what counts as a logarithmic spiral admits all three, and the number it records cannot separate them.

Sampled along the arc, it is a different number #

The collection samples spirals evenly in angle. Sampled evenly in arc length instead, the same shells return other numbers, because arc length grows with radius and an even-arc sample puts most of its points on the outer whorls.

The rising 2.8 to 3.6 shell fits at 3.2989 sampled along its arc, against 3.1749 sampled in angle and 3.4887 for its last whorl. The falling shell fits at 3.0559 along its arc. So the one number a changing shell reports depends on how its points were chosen, which is the same dependence the dividers measurement found in a milder form: an instrument that puts points where the curve is long reads the outer whorls.

The residual cannot see it #

The fit’s residual is the largest departure, in the logarithm of the radius, of the points from its line, and the collection refuses a spiral whose residual passes 0.15. On the rising 2.8 to 3.6 shell the residual is 0.0836, comfortably accepted.

On a shell rising from 2.5 to 4.1 over three and a half turns — a change of sixty-four per cent from apex to aperture — the residual is 0.1439. That is accepted too. A shell whose last whorl expands more than half again as fast as its first passes, by the residual threshold the fit applies, as one logarithmic spiral.

Why a twelfth #

The size of that residual is arithmetic. Over an interval of half-width h, a parabola k·x² has a least-squares line that is flat at k·h²/3, so it departs from the line by 2k·h²/3 at the two ends and by k·h²/3 at the middle. The ends are the worst.

For the changing spiral the curvature k is the change in rate over twice the arc, and the half-width is half the arc, so the worst departure is the change in rate times the arc over twelve — which is ln(W₁/W₀) × turns / 12. For 2.5 to 4.1 over three and a half turns that is ln 1.64 × 3.5/12 = 0.4947 × 0.2917 = 0.1443, against 0.1439 measured. The fit’s residual is the formula, to within the discreteness of the samples.

The shape of the residual does see it #

The size of the departure is below the threshold; its shape is not ambiguous. What is left after the line is k·(x² − h²/3): above the line at both ends of the arc and below it in the middle, for a shell whose rate rises, and the mirror image for one whose rate falls. A steady change leaves a U in the residuals, and a slowing one leaves an arch.

That pattern is visible in the first figure, and it is the one piece of evidence of a change that a single fit throws away when it reports a number and a largest departure. An arch or a U is a statement about direction, which the geometric mean cannot make. Whether that shape survives an assumed centre, which bends the residuals once a turn rather than once over the arc, is not measured here; on a spiral drawn from its true centre it is exact.

The largest change a fit hides #

Setting the formula equal to the threshold gives the largest change the residual accepts: exp(0.15 × 12 / turns). Over two turns that is a change of ×2.4596; over three, ×1.8221; over three and a half, ×1.6724; over four, ×1.5683; over six, ×1.3499. Bisecting on the fit itself confirms each to within 0.4 per cent.

So the check is more lenient the less arc it is given, which runs against intuition. A fit over the two turns the collection’s own span floor allows can hide a change of more than a factor of two. The rising 2.5 to 4.1 shell sits just under its span’s limit at ×1.64 against ×1.6724.

A doubling is refused #

The check does refuse something. A shell whose expansion doubles, from 2.4 to 4.8 over three and a half turns, leaves a residual of 0.2017 against a predicted 0.2022, and it crosses the threshold at both ends of its arc, as a residual set by the ends must.

That is the scale of change it takes. Everything between a steady shell and a doubling, over three and a half turns, reads as one spiral with one factor.

A tighter threshold buys little #

Lowering the threshold is the obvious response, and the formula prices it. To refuse a change of ten per cent over four turns the threshold would have to sit below ln 1.1 × 4/12 = 0.0318 in the logarithm, which is a departure of 3.2 per cent in radius at a single point. The shell rising from 3.06 to 3.35, a change of 9.48 per cent, leaves 0.0301 and would still pass.

A threshold that tight refuses any spiral with a single point 3.2 per cent out of place, whatever put it there. A residual answers a different question from whether the rate changed, and making it stricter makes it a stricter answer to that other question.

What the number is asked for #

A growth factor is quoted for a reason, and the reason here is usually contact: whether a whorl runs into the one before it. For a shell coiled in a plane the whorls touch when W·D is below one, which is the line located exactly for a circle and decided by one angle for a spire.

That W is the expansion between the whorls that meet. At the aperture those are the last whorl and the one before it, and their expansion is not the midpoint’s.

The last whorl’s number, worked #

The ratio of the last whorl’s radius to the one a turn earlier is the local expansion half a turn back from the aperture. On a four-turn arc that is seven eighths of the way along, so it is W₀^(1/8) × W₁^(7/8). For 2.8 to 3.6 its logarithm is 0.125 × 1.02962 + 0.875 × 1.28093 = 0.12870 + 1.12081 = 1.24951, and the last whorl expands by 3.4887.

The fit says 3.1749. The two differ by nearly ten per cent, on a shell whose residual was less than three fifths of the threshold.

Three numbers for the present #

The shell now carries three candidates for its current expansion. The aperture’s own rate is 3.6, the last whorl’s ratio is 3.4887 and the fit reports 3.1749. The first describes a whorl not yet built; the second is what two whorls that exist, and can touch, actually do; the third belongs to the middle of the arc.

Only the second decides anything about the last whorls, because contact is a relation between two whorls a turn apart. It is also the one a fit cannot return, since the fit gives the last whorl less than a sixth of its weight.

Opposite verdicts at the aperture #

At an axis distance of 0.3, the fit’s number puts the last whorls of the 2.8 to 3.6 shell in contact, since 3.1749 × 0.3 is 0.9525, and the last whorl’s own number frees them, since 3.4887 × 0.3 is 1.0466. Between D = 0.2866 and 0.3150 the two verdicts differ, a band 0.0283 wide.

The band grows with the change. It is 0.0104 wide for a shell rising from 3.06 to 3.35, 0.0506 for 2.5 to 4.1 and 0.0646 for a doubling. For a steady shell it has no width at all, which is the only case in which the fit’s number and the aperture’s are the same thing.

A shell that slows down #

The falling shell gives the same fit, 3.1749, and a last whorl of 2.8894. At D = 0.3 both numbers put its last whorls in contact, so here the fit happens to agree with the aperture — and the band of disagreement has moved to D between 0.3150 and 0.3461, where a shell slowing down is called free by its fit and in contact by its last whorl.

The error runs the other way and has the same kind of size. A reading that did not know which way the shell had changed could not say which side of the band it was on.

What two arcs could tell #

The obvious repair is to fit the shell twice, over its inner half and its outer half, and compare. On the 2.8 to 3.6 shell the inner half fits at 2.9816 and the outer at 3.3808, a split of 13.39 per cent, which is exactly √(W₁/W₀) − 1 = √1.2857 − 1 = 0.1339. A steady change leaves a split that says how large the change was, and its sign says which way.

That is the test the essay on Raup’s three numbers recommended, and it works on a spiral drawn from its true centre. Whether it works from a centre somebody had to choose is a separate question, and the next measurement answers it.

What this does not establish #

It says nothing about whether real shells change their expansion, or how. The steady change in rate is one model of a change among many; a change concentrated in the last whorl would put the fit’s number further from the aperture still, and a change in D alongside a change in W would move the contact verdict in ways not measured here.

It also does not say that the nautilus’s measured 3.2 is a midpoint of anything. It says that a single fitted number on a shell that changes is a midpoint, and that nothing in the fit’s own output reveals whether the shell changed.

What would withdraw it #

A changing spiral, sampled evenly in angle, whose fitted factor is not the geometric mean of its two ends. A residual that grows faster with the change than ln(W₁/W₀) × turns/12. A change between a steady shell and a doubling, over three and a half turns, that the threshold refuses. Each of those is one spiral to build, and each is built every time the measurement runs.

The two-arc test, and the centre it needs #

The next measurement takes the repair seriously. A split between an inner half and an outer half reports a steady change exactly — from the true centre. The test is what the split reports on a spiral that does not change at all, fitted about a centre displaced by a quarter of the innermost radius: if that split is as large as the split a genuine change leaves, then the two-arc test measures the centre, and a disagreement between two fitted arcs is not evidence that an animal changed how it grew.