Three entries and one span

The claim that a nautilus grows as a golden spiral asks for a growth factor of 6.854 per turn, and measured sections give about 3.2 — a gap of 114 per cent. The error budget for a nautilus set every way a growth factor read off a section can be wrong against that gap: an assumed centre, a short span of arc, a pair of dividers, an uneven clock, an oblique photograph, a miscounted septum, and a shell whose expansion changed. Added without cancellation the seven came to 332.6 per cent and did not refuse the claim, but one entry decided that — the dividers — and with it set aside the rest came to 54.4 per cent and refused it twice over.

That essay was explicit about its own weakness. Adding worst cases linearly assumes the errors conspire; adding them in quadrature assumes they are independent; and at least two of them were known not to be. A displaced centre and a short span are the same failure seen twice, and an oblique view costs a half-turn span twenty times what it costs three turns. It asked for the joint reading instead of the seven margins. This is that reading.

The budget as it was assembled #

Three of the seven bars are the ones this essay is about. The centre entry is a quarter of the innermost radius of offset, read over two turns of shell: 6.88 per cent. The span entry is the same quarter-radius offset read as a root mean square over spans of one turn and more: 22.4 per cent. The oblique entry is a camera ten degrees off the normal, read over half a turn: 2.95 per cent. Together, 32.2.

Each number is correctly measured. What they are not is three separate errors. The first two are one error, the displaced centre, priced at two different spans; and the third was priced at a third span. A photograph of a section shows one span of arc — however much shell survived and could be traced — and a budget for that photograph has to price everything at that span.

One error, priced at one span #

So the reading is redone as a section would present it. A nautilus spiral at 3.2 per turn is traced over a stated span, from its innermost point outward; the camera is tilted about an axis at a stated bearing; and the growth factor is fitted about a centre displaced by a stated share of the innermost radius in a stated direction. Twelve directions of the centre are read against twelve bearings of the tilt, at spans from half a turn to three.

At two turns a quarter-radius centre and a ten-degree tilt, in their worst orientation together, cost the reading 7.12 per cent. At one turn they cost 38.8. At half a turn they cost 188. The budget’s 32.2 per cent is not any of those. It overstates what the three errors do to a two-turn section by about four and a half times, and understates what they do to a half-turn section by nearly six.

Every orientation at half a turn #

The joint reading can also be looked at orientation by orientation, and at half a turn with a twenty-degree tilt it is worth looking at. Each circle is one combination of the direction the centre is displaced in and the bearing of the tilt axis; its area is the size of the error and its shade the sign.

The centre dominates. Its direction sets whether the reading is high or low and by roughly how much: from 176 per cent too high in the worst direction to 60 per cent too low in the opposite one, row by row. The tilt’s bearing then moves each row up or down by a few per cent in a pattern that changes with the row. The worst combination reaches 228 per cent where the centre alone reaches 176 and the tilt alone 12.5. Over every orientation the root mean square is 91.3 per cent.

Every orientation at two turns #

At two turns the same grid has the same structure at about a thirtieth of the size. The worst combination is 7.91 per cent, against 6.86 for the centre and 0.74 for the tilt; the root mean square is 4.74. Lengthening the span shrinks a centre error roughly as the square of the span, and a tilt error faster still, because a tilt’s effect has a period of half a turn and averages away over whole half turns.

The shape of both grids — rows set by the centre, columns nudging them — is the first answer to whether the errors are independent. Mostly, yes: each adds its own contribution. The departures are the interesting part.

In the worst case they add to more than their sum #

The budget’s pessimistic reading added worst cases, on the grounds that the worst a combination could be was the sum of the worsts. It is not. At every span and every size read, the worst joint error is at least the sum of the two worst single errors, and on a short span it is well above it: for a quarter-radius centre and a twenty-degree tilt, 21 per cent above at half a turn, 7 at a turn, 3 at three turns. For a tenth-radius centre and a ten-degree tilt the excess is 4 per cent at half a turn and under 2 from a turn up.

So “adding without cancellation” was not the pessimistic bound it was taken for. The errors interact, and in the worst orientation the interaction makes things worse.

The interaction is the span #

The departure itself — each joint error minus the centre’s alone minus the tilt’s alone, at the worst orientation — is what the question of independence is really about, and it has one controlling variable. For a quarter-radius centre and a twenty-degree tilt it is 39.8 per cent at half a turn, 6.2 at three quarters, 3.4 at one turn, 0.36 at two and 0.13 at three. For a tenth-radius centre and a ten-degree tilt: 2.1, 0.37, 0.25, 0.03, 0.01. It falls with every lengthening of the span, by a factor of about a hundred from half a turn to two.

So the two errors are nearly independent on a long span and strongly coupled on a short one, and the coupling is the span itself. That is the precise form of the earlier observation that a displaced centre and a short span are the same failure seen twice: a short span is not a separate error but the condition under which every other error of reading is largest and most entangled.

Over random orientations, quadrature holds #

The quadrature reading fares better, because it was only ever claimed for independent errors in random orientation. Averaged over all 144 orientations, the root mean square of the joint error is within four per cent of the quadrature sum of the two single errors’ root mean squares at every span from a turn upward: 22.64 against 22.46 at one turn, 4.61 against 4.57 at two. At half a turn it is 86.65 against 85.60 for a ten-degree tilt, and 91.3 against 86.0 for a twenty-degree one.

That is the useful half. A worker who does not know which way the centre is off or which way the camera leans has no particular orientation, and for that worker quadrature is the right rule from a turn upward. A worker asking what the worst case could be should not add worst cases; the worst case is worse.

Whether the budget refuses the claim #

Put back into the budget, the joint reading changes the answer to the question the budget was built for. With the dividers set aside, the other three entries — an uneven clock, a miscounted septum and a changed expansion, none of them priced at a span — come to 22.2 per cent. Add the joint worst case of a quarter-radius centre and a ten-degree tilt at one span, and the budget is 25.1 per cent at three turns, 29.3 at two, 31.9 at one and a half, 61.0 at one, 111.5 at three quarters and 210.4 at half a turn.

The gap is 114.2 per cent. So the budget refuses the golden spiral at every span from three quarters of a turn upward — at three quarters by less than three per cent — and does not refuse it at half a turn, where the displaced centre alone can carry a section of a 3.2 shell past 6.854. The earlier essay’s conclusion, that the claim is refused twice over with the dividers set aside, holds for a section of two turns and more; it does not hold for a fragment.

What a fragment can and cannot say #

This is the same boundary how far a centre must move found by a different road: the centres that make a nautilus read as golden exist at every span up to 1.15 turns and at none from 1.2 upward. That reading asked how far the centre had to move, allowing any distance; this one prices a stated quarter of the innermost radius. Both put the line near a turn. The budget’s line falls a little lower, at three quarters, because at a turn the displacement a golden reading needs is larger than a quarter radius: the quarter-radius joint worst case there is 38.8 per cent, a third of the gap.

For a person with a real specimen the recommendation is plain. The span of arc the section shows is not one entry among seven; it is the number every other entry is priced at, and it should be stated with the growth factor, before any claim is argued from it. A count carries no error said the same about counts: a number without the condition it was read under cannot be compared with anything.

The centre alone, span by span #

Underneath the joint reading, the centre by itself already carries most of the span’s effect. A quarter-radius offset in its worst direction costs 176 per cent at half a turn, 87.6 at three quarters, 37.3 at one turn, 9.29 at one and a half, 6.86 at two and 2.82 at three. From half a turn to one the cost falls by 4.7; from one to two by 5.4; from two to three by 2.4, close to the square of the ratio of the spans, 2.25, that the centre’s own analysis found. The tilt adds a few per cent to that at every span, and the interaction a few more on the short ones.

That makes the old span entry’s meaning clear. It was the centre’s error averaged over spans of a turn and more, which is a number about a set of sections of different lengths, not about any one of them. The residual a fit leaves cannot say which span a section has either; only the section can, by how far round its whorls can be traced.

The tilt alone does not fall steadily #

The tilt’s own contribution has a shape the centre’s does not. A ten-degree tilt in its worst bearing costs 2.95 per cent at half a turn, then 0.28 at three quarters, 0.73 at one turn, 0.32 at one and a half, 0.18 at two and 0.08 at three; a twenty-degree tilt costs about four times as much at each, with the same dip at three quarters. The cost is not a steady decline with span.

The reason was found when the tilt was first priced: a tilt adds a term of period half a turn to what the fit sums, and there is a discrete set of spans at which that term cancels exactly, the first at 0.7151 turns and the next at 1.2294. Three quarters of a turn sits next to the first, and one and a half turns is closer to the second than one turn is. So the one span-dependent entry that is not the centre has spans at which it is nearly free, and the span at which the budget first refuses the golden spiral, three quarters of a turn, happens to be one of them.

What a careful worker buys at half a turn #

The quarter-radius offset and the ten-degree tilt are the budget’s own conditions, chosen as errors a worker would not think to control. A worker who does control them — a centre found to a tenth of the innermost radius, a camera within ten degrees of square — changes the picture on a short span completely.

With a tenth-radius centre and a ten-degree tilt, the joint worst case is 52.3 per cent at half a turn, 14.3 at one and 2.85 at two, and the matched budget without the dividers is 74.5, 36.4 and 25.0 per cent: under the 114 per cent gap at every span, half a turn included. With a twentieth of the radius and five degrees it is 44.2 per cent at half a turn. So a fragment of half a turn can refuse the golden spiral after all, if its centre is found carefully; what it cannot do is refuse it with the budget’s stated carelessness. The span and the care are the two numbers that decide it, and they trade against each other.

Why adding worst cases is worse than it looks #

The superadditivity has a plain geometric reading. A displaced centre makes the fitted angle-to-radius relation curve, with a curvature that depends on where along the spiral the section starts and which way the centre is off; a tilt adds a ripple with a period of half a turn. On a long span both average towards a straight line and their small leftovers add. On a short span neither averages, and a tilt that happens to squash the section along the direction the centre is displaced in makes the displaced centre look further off than it is. The worst orientation is the one where the two effects line up, and there they reinforce rather than merely add.

That is also why quadrature survives: lining up is one orientation out of many, and a random orientation mostly does not line up.

What this does not establish #

The other four entries’ dependence on these three or on each other. The dividers, an uneven clock, a miscounted septum and a changed expansion were not read jointly with anything, and the dividers in particular — the entry that decided the original budget — is a function of the span as well, since a divider walk needs arc to walk along. That coupling is not measured here.

Nor does it price a real specimen. The offsets and tilts are the budget’s own stated conditions, a quarter of the innermost radius and ten or twenty degrees; a careful worker does better than either, and the joint reading at smaller errors is in the grids above.

What would withdraw it #

A span, offset and tilt at which the worst joint error is less than the sum of the worst single errors. A span from a turn upward at which the root mean square joint error departs from the quadrature sum by more than four per cent. A matched-span budget that refuses the golden spiral at half a turn, or fails to at two.

Still open: whether the dividers belong to the span too #

The dividers decided the original budget, at 278 per cent, and they were set aside as the historical method rather than measured against the span. But a pair of dividers walks along the curve, and how far it can walk is the span; a short span gives fewer steps and a coarser reading. The measurement is the dividers read at the same spans as the centre and the tilt, and read jointly with them: whether the entry that dominates the budget is itself a function of the span, and whether, priced at one span, the whole of the budget — dividers included — refuses the golden spiral from some span upward.