How far a centre must move

The objection to measuring a shell’s growth factor is always the same, and it is a good one: the centre is assumed, and a different assumption would give a different answer. The answer that matters is 6.854, because that is the number the shell is said to have and does not.

So the question has a shape a sweep can take. Is there a point in the plane — any point, at any distance, in any direction — from which a spiral drawn at 3.2 per turn reads as golden?

The question, stated so it can be answered #

The essay that prices what a centre costs answers a smaller question: how much a particular displacement moves the answer. That is a derivative at one radius in one hundred and eighty directions, and it is a sample of the plane rather than a statement about it.

The statement about it needs the plane. A gap of a factor of 2.14 is not going to be closed by a small error, so the interesting question is whether it can be closed by any error at all, and how bad that error would have to look.

That is a search rather than a sensitivity, and it returns a boundary rather than a number.

The plane, and how it was covered #

482,220 assumed centres: nineteen spans of arc, a hundred and forty-one displacements at each, a hundred and eighty directions at each of those.

The displacements are spaced logarithmically from 0.01 to 500 innermost radii, eight per cent apart, because the quantity runs over five decades and a linear grid would put every point in the last one. The directions are two degrees apart, so no direction is more than a degree from one that was tried.

Five hundred innermost radii is far outside anything a person could mistake for a shell’s centre. It is in the sweep because the question is whether the golden factor is reachable at all, and a search that stops at plausible errors cannot answer that.

The answer, in one line #

A spiral drawn at 3.2 comes back as the golden 6.854 at every span up to 1.15 turns, and at no span from 1.2 turns upward.

Fourteen of the nineteen spans refuse it entirely — not at the cheapest centre, but at every one of the 25,380 centres tried on each of those rows. The boundary lies between 1.15 and 1.20 turns of arc, and it is a boundary in span rather than in displacement.

Below the boundary #

Where it is reachable, it is not expensive. At 0.75 turns the cheapest centre that returns the golden factor sits 0.2957 innermost radii away, which is 12.4 per cent of the picture’s outer radius. At one turn it is 0.5463 radii, or 17.1 per cent.

Above that the price rises quickly: 0.6352 radii at 1.05 turns, 0.7459 at 1.10, 0.8802 at 1.15 — and then nothing.

What half a radius looks like in a picture #

A displacement of 0.5463 innermost radii at a span of one turn is 17.1 per cent of the outer radius of the drawing. That is a sixth of the way across the picture.

It is worth converting because “the centre was assumed” sounds like a subtle error and this one is not. A person placing the centre a sixth of the picture’s radius from where the spiral’s own centre is has put it visibly off the middle of the coil, on a drawing where the whole coil is one turn wide.

So even in the region where the golden reading is available, it is not available by carelessness. It is available by a mistake that would be obvious to anyone looking at where the point was placed.

The boundary is a span and not a displacement #

The natural expectation is that the answer would be a distance — the centre must be wrong by more than so much. It is not. There is no displacement that works at long spans and fails at short ones; there is a span past which nothing works.

That is the useful form of the result, because span is the quantity a measurement can report and often does. A growth factor quoted with the arc it was measured over is a claim that can be audited; one quoted without it cannot.

It also means the refutation is cheap. Anybody doubting a fitted factor need not argue about where the centre was put. They need only ask how much arc was read, and a section showing more than a turn and a fifth settles it.

Why arc closes the door #

The mechanism is the same one that makes the error fall as the inverse square of the span. A displaced centre bends the line that the fit puts a slope through, and the bend is confined to the small radii where the displacement is a large fraction of the radius.

Over three quarters of a turn almost the whole curve is at small radii, so the bend is most of the data and the slope can be made almost anything. Over two turns the outer whorl is nine times the radius of the inner one, dominates the baseline, and is untouched.

The picture is a lever arm. Adding arc lengthens it, and past a certain length no force applied at the short end moves it far enough.

The wall #

Drawing the cheapest golden-returning displacement against span shows the door closing rather than a curve running off the page. It rises from 0.2957 to 0.8802 radii across five spans and then stops, and the region past it is not a region where the displacement is large. It is a region where no displacement exists.

Five points is a short curve, and it is short because the thing being drawn stops existing. That is a different shape of result from a quantity that grows without bound, and it is worth seeing rather than reading.

The ceiling over the whole plane #

The other reading of the same sweep asks not can it reach golden but how high does it get. Over the whole plane, at each span, the highest factor any assumed centre returns:

160.0 at 0.75 turns, 17.47 at one, 12.57 at 1.05, 9.67 at 1.1, 8.00 at 1.15, 6.843 at 1.2 — just under the golden factor, and the first row that misses it — then 5.91 at 1.3, 5.26 at 1.5, 4.93 at 1.75 and 4.72 at two.

The fall through the short spans is steep and the crossing at 1.2 is narrow: 6.843 against 6.854 is a miss by two parts in a thousand. A sweep with a coarser displacement grid would have put the boundary in a different place, which is why the grid is eight per cent per step and the boundary is quoted as an interval between two swept spans.

The ceiling a shell section is allowed #

None of that is a measurement anybody would report, because a fit over three quarters of a turn is refused before its answer is read. Imposing the two-turn floor leaves seven spans, and their ceilings are 4.463 at two turns, 3.899 at two and a half, 3.934 at three, 4.010 at three and a half, 4.259 at four, 4.373 at five and 4.467 at six.

The highest anywhere is 4.467, and it is the answer to the question this essay exists for.

An inflation of 1.40 where the claim needs 2.14 #

A spiral drawn at 3.2 read from the worst centre in the plane, at any span a measurement is allowed to use, comes back at 4.467. That is an inflation of 1.40.

The claim being tested needs 2.14. So the most a misplaced centre can buy is under two thirds of the way there in ratio, using a centre chosen with knowledge of the answer, at the span that happens to be most generous, in the direction that happens to be worst.

The gap is not close, and it is not close in the direction that would matter. The whole category of explanation — the measurement is sensitive to a choice nobody can make correctly — is exhausted by 4.467, and the shortfall is a factor of one and a half after every concession has been made.

The first refusal: two turns #

The two-turn floor is a rule about what the fit is allowed to answer, not a discovery. It is there because a shorter arc does not pin a growth factor: over half a turn a slow spiral and a fast one differ by less than the width of a drawn line, which is the whole reason the nautilus claim survived as long as it did.

The sweep shows what the floor is worth. It removes every span at which the golden factor is reachable, with four spans to spare — the boundary is at 1.2 turns and the floor is at two.

That is a floor doing its job for a reason that was not the reason it was set, which is the kind of coincidence worth checking rather than enjoying. The check is the second refusal.

The second refusal: the residual #

Every fit reports the worst departure of a point from the fitted curve, in the logarithm of the radius, and a fit is refused when that exceeds 0.15 — a worst point 16.2 per cent off the curve it is supposed to lie on.

The cheapest golden fit in the whole sweep, at 0.75 turns, carries a residual of 0.163. Every golden fit from a full turn upward is above 0.41: 0.413 at one turn, 0.482 at 1.05, 0.650 at 1.1 and 1.076 at 1.15.

So every centre that returns the golden factor is over the threshold already, and the cheapest one is over it before the span floor is consulted at all.

What a residual of 0.163 means #

The number is a departure in the logarithm of the radius, so it converts to a fractional error in radius directly. A residual of 0.163 is a worst point about 18 per cent away from the radius the fitted spiral puts it at.

That is a fit anybody would look at and reject. The curve and the points visibly part company at one end, and the parting is at the inner end, where a displaced centre does its damage.

The ceilings are worse still. The fits that produce 4.467 and its neighbours carry residuals between 3.39 and 4.77 in log r — worst points 29 to 118 times the fitted radius away from the curve, which is 23 to 32 times the threshold. Those are not fits with a large error; they are arrangements of points that have nothing to do with the curve drawn through them.

The inner decade, where a real error lives #

The sweep runs to five hundred innermost radii because the question demanded it, but nobody misplaces a centre by five hundred radii. The region a real error occupies is the first two decades.

Inside it the picture is almost entirely pale, which is the shading for a cell where every direction returns within five per cent of the factor the spiral was drawn at. A quarter-radius error sits in that region at every span above two turns, which is the measurement of what it costs read off a picture instead of a table.

Where the ceiling fits actually sit #

The centre that produces the ceiling is not near the shell. At three and a half turns it sits 3.05 innermost radii from the true centre — outside the third whorl of the drawn spiral — and at six turns it sits 57.4.

That is the honest way to report a ceiling: not as a number the measurement might return but as a description of the arrangement that returns it. A point outside the third whorl is not an estimate of where the coil began; it is a point somewhere else in the picture.

Between the residual and the position, the ceiling fits fail every test a reader could apply by looking, which is what makes 4.467 an upper bound rather than a possibility.

The tail that goes back up #

One feature of the ceiling curve is not explained here and is recorded rather than smoothed. Over the whole plane the ceiling falls from 160.0 at three quarters of a turn to 4.72 at two turns, and then rises again — 4.75 at two and a half, 5.16 at three, 5.23 at three and a half, 5.38 at four, 5.60 at five, 5.75 at six.

With the span floor imposed the same shape appears one span later: 4.463 at two turns, down to 3.899 at two and a half, and back up through 3.934, 4.010 and 4.259 to 4.467 at six.

The minimum is real and the rise past it is real, and no mechanism for either is offered. The rising limb is where the extreme displacements live — 57.4 innermost radii at six turns — so the natural guess is that a very distant centre reads the outer whorls as a different curve rather than reading the spiral badly. That is a guess, it is not tested here, and the result does not depend on it: every point on the curve is below the golden factor by a wide margin.

The two unwrappings, and the defect the search found in its own instrument #

The fit has to unwrap the observed angle so that it increases along the curve, and there is more than one way to do that. The route this collection used adds a turn whenever the angle drops by more than half a turn, and never subtracts one when it rises. Fitting every centre a second way, by accumulating each step with its sign, was meant to be a formality.

It was not. The two agreed to the last bit at all 223,624 assumed centres at which the angle never runs backwards, and parted company at 27,541 of the 258,596 at which it does — 10.7 per cent of them — by up to 2,906 per cent. A disagreement that is exact on one side of a line and enormous on the other is not two conventions differing; it is one of them being wrong.

The one-sided rule was the wrong one, and the mechanism is legible. A centre the curve winds around unevenly makes the observed angle genuinely go backwards for a stretch. The one-sided rule cannot represent that, so it promotes each backward step to a forward step of nearly a whole turn — and the same rise in log r is then divided by an angular span the points never covered. The recovered factor reads low.

Which of the two the field records #

Both, now. The routine was repaired to fold each step into a half turn either way and accumulate it with its sign, and the two routes agree to the last bit at all 482,220 centres — including every one of the 258,596 at which the angle runs backwards, which is where they used to part. The second route is kept rather than retired, because two separately written unwrappings agreeing over half a million centres is evidence and one agreeing with itself is not.

Almost nothing in the sections above moved, and what did moved upward: 5.08 to 5.23 at three and a half turns over the whole plane, 3.853 to 4.010 with the span floor imposed. That is the direction to expect, since the route being replaced read low wherever the two differed. The golden-returning centres did not move at all, because the nearest disagreement had sat at 3.046 innermost radii, outside the second whorl, and every one of them lies below a single radius — thirty times closer.

Why the repair does not reach the answer either #

A search that finds a defect in its own instrument owes the reader the arithmetic on whether the answer survives it, and here the answer is that the answer never depended on it.

The bound is 4.467, at six turns, exactly what it was before the repair: the row that carries it did not move at all, and the golden factor it has to reach is 6.854. The three refusals that produce it are untouched — the golden-returning centres are refused by the two-turn span floor and by residuals the unwrapping never enters, and the ceiling fits are refused by residuals of 3.39 and above, which no unwrapping repairs. The largest movement anywhere in the essay is a fortieth, in a quantity that would have to move by half to change what it says.

What the repair buys is not a different answer. It is that these are now measurements of the spiral rather than of the routine that read it, which is the whole reason a second route was written and is worth more than the fortieth. A search that cannot catch its own instrument is a search reporting the instrument.

What a plane buys over a displacement #

The difference between this essay and the one that measures what a centre costs is the difference between a sample and a search, and it is worth stating because both are called the same thing in ordinary usage.

A sensitivity is a statement about the neighbourhood of the true answer: move a little, see how much it moves. It can never say that something is unreachable, only that it is not reached nearby.

A search over the whole parameter can say unreachable, and it is the only thing that can. It is the same move as sweeping the angles a pair of counts allows rather than reporting the one the count suggests, and it returns the same kind of object: a region rather than a value. That is why the objection deserved 482,220 centres rather than one radius, and why the result is a boundary — a span at which an entire category of explanation runs out — rather than another per cent.

What this does not show #

Three limits, and the first is the one that would be easiest to overread.

It does not show that the nautilus figure of 3.2 is right. That number comes from published measurements of real shells and is the thing being perturbed here; a sweep that starts from it cannot test it.

It does not test a shell that is not a logarithmic spiral. Every curve in the sweep was drawn from the model being fitted, so the only defect present is the assumed centre. A shell whose coiling departs from the model is a separate question, and the residual is a poor instrument for it: it accepts curves that are not logarithmic spirals at all.

And it does not test noise, a section cut off the coiling plane, or a specimen chosen because it looked right. Each is its own perturbation with its own sweep.

What it does not settle about real shells #

The result is about an instrument pointed at a drawn curve, and a real measurement has more in it than that.

A real section is read by hand, at a resolution the person chooses, from a shell that may be worn at the apex — which is precisely where the arc that pins the factor is not. A shell read from its second whorl outward is a shorter span than a shell read from its first, and span is the quantity everything here turns on.

So the practical advice the sweep supports is narrow and worth having: report the arc. A growth factor with a span attached is a claim about which this measurement has something to say, and a growth factor without one is not — the same missing field that stops a shell becoming a point in a morphospace from being comparable with another.

The fit itself is not in doubt. Read from its own centre it returns the factor the curve was drawn at, to the last digit, and everything measured here is the distance between that and an assumption.

What would refute it #

A single assumed centre, at a span of 1.2 turns or more, from which a spiral drawn at 3.2 returns 6.854 with a residual under 0.15. That is one point, and it would overturn both the boundary and the ceiling at once.

Nothing in 482,220 tries produced one, and the sweep was built so that a failure would be visible rather than absorbed: the assertions behind these drawings check that every refusing span refuses at every cell, and that every reaching span’s cheapest fit is over the threshold. An assertion that has never rejected anything proves nothing, and both of those would fail loudly on a single counterexample.

The cheaper thing to look for is a different route to the same inflation, and there is one: a measurement made in steps rather than in angle does push a nautilus upward. It fails for a different reason, and that reason is a span as well.