What the septa count

The nautilus question settled that a nautilus does not grow as a golden spiral: about 3.2 a turn against the golden 6.854. It also said that a nautilus’s chambers are spaced so that each is a scaled copy of the last, and it quoted their ratio as about 1.3, what a growth factor of 3.2 gives over the third of a turn between septa.

That sentence carries two claims, and only one of them survives arithmetic. The copying is exact, and it is worth more than it was used for. The number does not close, and what it would have taken to close is a statement about how many septa a whorl holds.

A partition that copies itself #

Lay septa down at equal angles, n of them to a whorl, in Raup’s tube, which grows by W a turn. Turning the whole shell about its apex by a whorl’s n-th part and scaling it by W^(1/n) carries every point of the tube to the point one septum on, because that is what a logarithmic tube is. So it carries each chamber exactly onto the next.

Nothing else is needed for the result. Every part of a partition that copies itself scales by the same linear factor, and a length scales by that factor, an area by its square and a volume by its cube.

Length, area and volume #

From one chamber to the next, a length therefore grows by W^(1/n), an area by W^(2/n) and a volume by W^(3/n). At 3.2 a turn and thirteen septa a whorl those are 1.0936, 1.1960 and 1.307896.

The three differ by a factor of three in the exponent, which is the entire source of the trouble below. A reader told only that “successive chambers differ by 1.3” has not been told which of the three it was, and at 3.2 a turn the three readings describe three different shells.

Checked the long way #

The copying argument is short enough to distrust, so it is checked on the drawn tube. In cylindrical coordinates a solid whose section at each angle is some region has volume equal to the integral, over angle, of that region’s area times the distance of its centroid from the axis, exactly, however the region moves along the axis.

Integrated that way over the rings of a 3.2 tube at axis distance 0.1, successive chamber volumes at thirteen septa a whorl grow by 1.307896 at every chamber, and at four septa by 2.392558. The quadrature agrees with the closed form to four parts in a hundred million. Translating the tube along its axis, as a spire does, changes no ratio.

The closed form #

For a tube whose section is a disc of radius ρ·s centred at distance κ·s from the axis, with s = W^(θ/2π), the integral can be done exactly. Chamber c, running from c/n to (c + 1)/n of a whorl, has volume π·ρ²·κ·(W^(3(c+1)/n) − W^(3c/n))·2π/(3 ln W).

At 3.2 a turn and axis distance 0.1, ρ is 0.45 and κ is 0.55, so the first chamber at thirteen septa a whorl is 0.34989 × 0.30790 × 1.80062 = 0.194 in cubic units of the radius the outer wall starts at. The only part of the formula that depends on which chamber it is, W^(3c/n), is a power of one number, and the ratio between neighbours is that number, W^(3/n), whatever ρ and κ are.

What “about 1.3” could have been #

Read the other way, a ratio q between successive chambers fixes the septa per whorl: n = dim × ln W / ln q, where dim is one, two or three. At 3.2 a turn a ratio of 1.3 is 4.43 septa a whorl as a length, 81.2° apart; 8.87 as an area, 40.6° apart; and 13.30 as a volume, 27.1° apart.

So the dimension decides the count threefold. A shell with four or five septa to a whorl and a shell with thirteen can both be described as having “chambers about 1.3 times the last”, and nothing in the phrase says which.

The sentence that does not close #

The collection’s sentence took the ratio to be what 3.2 gives over a third of a turn. As a length that is 3.2^(1/3) = 1.474, not 1.3, and as a volume it is 3.2 itself. Earlier in the same essay a quarter turn was used instead, 3.2^(1/4) = 1.337, which is close to 1.3 and describes a shell with four septa to a whorl.

The two cannot both be the spacing, and neither yields 1.3 over a third of a turn. The repair states what each reading requires rather than choosing one, because which reading the “about 1.3” was is not recoverable from the sentence.

Four septa to a whorl #

The quarter-turn reading, drawn, is a shell with four septa to a whorl, 90° apart. Its chambers grow by 1.3375 in length, 1.7889 in area and 2.392558 in volume from each to the next. A worker quoting the length ratio of that shell and a worker quoting the volume ratio of the thirteen-septum shell would both be quoting a number near 1.3.

The figure makes a second point without trying. Whatever the count, the chamber a whorl out is the same size relative to this one, because a whorl out is a whole turn and a whole turn scales everything by W.

A whorl apart #

A chamber and the chamber one whorl out differ in volume by exactly W³, which at 3.2 is 32.768. Integrated on the tube, the ratio is 32.768000 at four septa a whorl and 32.768000 at thirteen.

That pair needs no count, because the count cancels: n chambers of W^(3/n) each make W³. It needs no centre, because volumes are not measured from a centre. The expansion read from it is the cube root of a ratio of two volumes, and it is the reading the rest of this essay keeps returning to.

The whorl before takes a share #

A real nautilus is involute: each whorl wraps the one before it, so part of what the tube would occupy is already occupied, and a chamber is the tube less the earlier whorl. That changes every volume. The question is whether it changes any ratio.

And the ratios do not move #

At 3.2 a turn and axis distance 0.1 the earlier whorl takes 2.766 per cent of the first chamber; at axis distance zero, 3.051. At 2.4 a turn it takes 6.921 per cent, at 1.6 a quarter, 24.125, and at 1.3 nearly half, 45.294. At 3.2 with axis distance 0.4 it takes nothing, because W·D = 1.28 and the whorls do not touch, the line a circle’s contact is decided by; a shell with a spire has its own contact condition, and the share would vanish on that line instead.

With that share excluded, the ratios stay within 0.0006 per cent of W^(3/n) from chamber to chamber at 3.2, and within 0.0089 per cent at 1.3, where nearly half the chamber is gone. A whorl apart they stay within 0.0125 per cent of W³ in the worst case.

Why the exclusion is a measurement #

The reason is the same copying: the part the earlier whorl takes is itself carried onto the next chamber’s part by the same turn and scale. But an argument of that kind is exactly what a quadrature can accidentally reproduce, if its grid is scaled along with the shell.

So the excluded share is counted on a square grid whose cell is a fixed length for every chamber. Later chambers are resolved more finely than earlier ones, and nothing about the numerical method copies itself. The ratios agree anyway, to under a hundredth of what was taken out, which makes the agreement a measurement of the shell rather than a property of the arithmetic.

A slow shell #

At 1.6 a turn and thirteen septa a whorl, each chamber is 1.114563 times the volume of the last and a chamber a whorl out is 1.6³ = 4.096 times the size. The whorls wrap each other heavily; a quarter of each chamber lies inside the whorl before.

Drawn, the shell looks nothing like a nautilus, and its chamber ratios obey exactly the same law. What makes the law useful is that it does not care about the shape of the shell, only about the partition being even.

The count is an exponent #

Turn the law round to read the growth factor from chambers, which is the reason to measure chambers at all. From one chamber to the next, W = q^(n/3) for a volume ratio q, so the septa per whorl enter as an exponent.

A volume ratio of 1.3 read at eleven septa a whorl gives W = 2.617; at twelve, 2.856; at thirteen, 3.117; at fourteen, 3.402; at fifteen, 3.713. One septum miscounted between thirteen and fourteen moves the growth factor by 9.14 per cent.

As a length, worse #

Read as a length ratio the exponent is the count itself, W = q^n, and a ratio of 1.3 gives 2.197 at three septa a whorl, 2.856 at four, 3.713 at five and 4.827 at six. One septum at four moves W by thirty per cent.

This is the practical content of the ambiguity in “about 1.3”. A reader who takes it as a length and counts four septa infers 2.856; a reader who takes it as a volume and counts thirteen infers 3.117. Neither is 3.2, and the gap between them is set by the reading, not by the shell.

Errors, amplified and not #

The exponent amplifies measurement error too. A one per cent error in a volume ratio becomes (1.01)^(13/3) − 1 = 4.41 per cent in W at thirteen septa a whorl, and 5.10 at fifteen. A chamber-to-chamber reading multiplies every error in the volumes by about a third of the count.

Compared with the chamber a whorl out, the same one per cent becomes (1.01)^(1/3) − 1 = 0.33 per cent. The whorl-apart pair divides the error by three where the chamber pair multiplies it by four, and needs no count to do it.

Counting the septa from the volumes #

The two readings together give something neither gives alone. The whorl-apart pair supplies W with no count, and a chamber-to-chamber ratio q supplies W^(3/n). Dividing their logarithms gives the count itself, n = 3 ln W / ln q, so the septa a whorl can be measured from volumes rather than supplied.

How well is set by the same exponent. At thirteen septa a whorl on a 3.2 shell, ln q is 3 × 1.16315/13 = 0.2684, and a one per cent error in q is an error of 0.00995 in its logarithm. The count then comes out uncertain by 13 × 0.00995/0.2684 = 0.48 septa: a single chamber ratio measured to a per cent cannot tell thirteen from fourteen. A regression over many chambers, which recovered W to 0.41 per cent at five per cent scatter, can.

When the count is not whole #

The whorl-apart pair has a price, and it is identifying “a whorl out”. That is exact only when the septa divide the whorl evenly. At 13.3 septa a whorl the chamber nearest one whorl on is thirteen chambers on, which is short of a whorl, and it reads W as 3.2^(13/13.3) = 3.117, 2.59 per cent low.

At 12.5 septa a whorl the nearest partner is thirteen on, past a whorl, and reads 4.76 per cent high; at 13.5 it is fourteen on and reads 4.40 high; at 14.2 it is fourteen on and reads 1.62 low. The error is never more than half a septum’s share of W, which at thirteen a whorl is 3.2^(0.5/13) − 1, about 4.6 per cent.

The price of needing no count #

So the pair needs no count only when it can find the right partner, and finding it is a matter of looking along a radius rather than counting. A worker who can see which chamber lies directly outside a given one has done what the pair requires; one who cannot is back to counting.

On a section the radial partner is usually visible, which is why the pair is the stronger reading in practice. But the error is not zero when the partition is uneven, and it is not a small error.

Scattered volumes #

Real chambers are not a perfect partition, so scatter each volume of a 32-chamber, thirteen-septum shell by a normal error in its logarithm and recover W three ways. At five per cent scatter a regression of log volume on chamber number, given the right count, recovers W to 0.41 per cent RMS; whorl-apart pairs, with no count, to 0.44.

Given one septum too many, the regression is out by 9.37 per cent with a bias of 9.36: the error is almost entirely bias. At ten per cent scatter the regression with the count is at 0.82 per cent and the miscounted one at 9.42. Scatter averages away over thirty-two chambers; a miscount does not average at all.

The same move as the calipers #

Two instruments now read a growth factor by comparing like with like at a fixed separation. Two diameters half a volution apart need a centre only to aim a line, and a chamber and its partner a whorl out need no centre at all. Both avoid the quantity that makes a single fitted number on a real section fragile, and a comparison of two fitted arcs more fragile still.

Both also read locally. Neither was designed to detect a shell whose expansion changes, but a pair a whorl apart reads the expansion over that whorl, and a run of pairs along the shell would read how it changes. That is not measured here.

What this corrects #

The sentence in the nautilus essay that gave the chamber ratio as about 1.3 over a third of a turn now states what a third of a turn gives at 3.2, what a ratio of 1.3 needs as a length and as a volume, and points here. Its published range for the growth factor, given as 2.9 to 3.4 in one place and 3.0 to 3.4 in another, is made consistent at 2.9.

The copying the sentence described was right, and it is the stronger half of the claim: it makes every chamber a carrier of W, and it makes a pair a whorl apart carry W without the count.

What this does not establish #

It does not say how many septa a nautilus lays down in a whorl, or that they are equally spaced; the count is an input throughout. It does not model the siphuncle, the curvature of the septa, or the body chamber, whose volume does not follow the partition. It takes the tube to be Raup’s, with a circular section, although the copying argument holds for any section that is carried onto itself.

What would withdraw it #

A drawn tube whose successive chamber volumes differ from W^(3/n) by more than its quadrature error. An involute tube whose ratios move when the earlier whorl is taken out on a grid that does not scale. A regression that stays unbiased under a miscount. Each is checked every time the measurement runs.

Whether the chambers can carry a golden ratio #

The golden claim about the nautilus was refused on its whorls. It survives, in some retellings, on its chambers, and the law here says exactly what that would require. A chamber ratio of φ = 1.618 at 3.2 a turn needs n = 3 ln 3.2 / ln φ = 7.25 septa a whorl as a volume, 4.83 as an area and 2.42 as a length. The next measurement is that test: whether any count a chambered shell could have puts φ in one of its chamber ratios, and whether a reading that finds φ there is a finding about the shell or a choice of dimension.