What the centre costs

A growth factor is recovered from a drawn spiral by least squares on the logarithm of the radius against the unwrapped angle. The radius is measured from somewhere, and where that somewhere is has to be assumed.

This collection has published a figure for what a wrong assumption costs, twice, and the figure is too small by about a factor of five at the spans a shell section actually offers.

The one free choice #

Everything else the fit needs is given. The coordinates are read off the section, the angle is unwrapped by walking the curve, and the model has one parameter. The centre is the single quantity that comes from a judgement rather than from the shell.

That is why it is the objection anybody raises first, and it is why the essay deriving the spiral from growth named it as the one thing to measure sensitivity against. A shell’s apex is a point on the animal, not the mathematical centre of the curve, and the two are not the same point.

So the honest statement of a fitted growth factor includes how far the answer moves when the assumption moves. The question is what that quantity is.

What the collection says it costs #

The sentence, in the form it has been carrying: displacing the assumed centre by a quarter of the spiral’s own radius — a gross error, far worse than careful work would produce — moves the fitted factor by well under one per cent.

Nothing about that sentence names a span. It is a statement about a displacement and a result, with no mention of how much arc the fit was given, and the amount of arc turns out to be the only thing that matters.

A figure quoted with no span attached is a figure that is true at some spans and false at others. This one is true above four and a quarter turns, and false at every span a shell section offers.

What it costs #

Displace the assumed centre by a quarter of the innermost whorl’s radius, in each of a hundred and eighty directions two degrees apart, and take the root mean square of the error in the recovered factor. On a spiral drawn at 3.2 per turn the answer is:

4.562 per cent at two turns. 2.520 at two and a half. 1.910 at three. 1.341 at three and a half. 1.056 at four. It first falls under one per cent at 4.25 turns.

Two turns is what a well-preserved section shows without ambiguity. Three and a half is generous. At neither is the published figure within a factor of two of the measurement, and at two turns it is out by more than four.

Six factors, one answer #

The same displacement was applied to all six growth factors this collection draws — 1.6, 2.4, 3.2, 4.5, the golden 6.854 and 7.4 — and the span at which the error crosses one per cent is 4.25 turns for every one of them.

That is the part worth noticing. The six factors span a range of 4.6 in the quantity being measured, and they agree to the resolution of the sweep on the span at which the measurement becomes accurate. Whatever governs the cost of a bad centre, it is not the shell.

The law under the numbers #

The error falls as the inverse square of the span. Multiplying it by the square of the number of turns gives a constant between 0.1643 and 0.1719 from three turns to eight — a spread of 4.6 per cent across a range in which the error itself falls by a factor of seven.

An inverse square is a strong statement about what to do. Doubling the arc divides the error by four, and no other choice available to a person measuring a shell does anything comparable. Better coordinates, more points, a more careful eye at the apex: none of them appears in the law.

The law is fitted to nothing. It is read off a sweep that computed the error at each span independently, and the constant is what those independent numbers turn out to share.

Why the factor barely enters #

At two turns the six errors run from 4.166 per cent at a factor of 7.4 to 5.832 per cent at 1.6. That is a spread of 1.40 across the whole ladder of factors, against 5.5 between two turns and four on any single one of them.

So the growth factor does move the answer, slightly, and in the direction that is easy to get backwards: the slowest spiral is the one a bad centre hurts most. A tight coil puts its innermost whorl closer to the assumed centre in units of the outer radius, and the innermost whorl is where the damage is.

The size of the effect is what matters here. Anyone reporting a growth factor with a sensitivity attached can quote the span and ignore the factor, and be wrong by at most 40 per cent of a quantity that is itself small.

The correction, and its size #

The published sentence is wrong by about a factor of five at two turns, taking “well under one per cent” at its most generous reading, and by rather more taking it literally.

That is not a rounding. It is the difference between a measurement that can be quoted without a caveat and one that cannot: 4.56 per cent on a factor of 3.2 is a range of roughly a seventh, which is wide enough that two workers displacing their centres in different directions would disagree in the second digit.

The correction changes what a careful measurement should report. It does not change what the measurement says.

Where the sentence appears #

In three places, any of which a reader may have met before this one.

The essay that prices the nautilus gap uses it to dismiss the obvious objection to its own argument, in the section listing what the measurement cannot be blamed on. The essay that derives the spiral from growth uses it twice, once when introducing the centre as the one free choice and once in the summary of what the fit needs. And the essay asking what a mechanism would have to show quotes it in passing, as the one case on this site where a measurement of form settled something.

All three are linked here rather than described, because a reader who met the figure there should be able to reach the measurement that replaces it. None of the three has had its argument changed by this, and all three have had a number in them changed.

What survives it #

Everything the nautilus argument rests on.

The gap to be accounted for there is a factor of 2.14 — 3.2 against 6.854 — which is an inflation of 114 per cent. The largest error a quarter-radius displacement produces at any usable span is 4.56 per cent. The correction moves the objection from fifty times too small to twenty-five times too small, and the conclusion is identical on either arithmetic.

It is worth being exact about what has been repaired and what has not. The claim the centre cannot account for the nautilus gap is true and is now supported by a measurement rather than by a figure that was not right. The claim a quarter-radius error costs well under one per cent is false and has been withdrawn.

The stronger version of the first claim needs a sweep rather than a single displacement, and that is what the whole plane of assumed centres answers.

A quarter of what #

The unit matters and is easy to lose. Every displacement here is in units of the innermost whorl’s radius — the radius at the start of the measured arc, which the spiral is drawn to have equal to one.

That is the small end of the picture, not the large one. A quarter of the innermost radius on a spiral read over three and a half turns at 3.2 per turn is 0.4 per cent of the outer radius, because the innermost whorl is itself 1.706 per cent of the outer radius at that span.

So “a quarter of the spiral’s radius” is a much smaller error than it sounds, and calling it gross was already generous. It is a displacement invisible in a photograph of the whole shell, and it moves the answer by more than four per cent over two turns.

How far a centre has to move for a stated error #

Read the same sweep the other way. On a 3.2 spiral over two turns, a 1 per cent error needs a displacement of 0.0592 innermost radii, 5 per cent needs 0.2775, 10 per cent needs 0.5563, and 25 per cent needs 1.6415.

Over three and a half turns every one of those grows: 0.1886, 0.9556, 2.2361 and 8.3191. The same displacement that costs ten per cent at two turns costs under five at three and a half.

Neither column reaches the 114 per cent the nautilus claim needs, at any displacement listed, which is the arithmetic behind the previous section stated in the units a person measuring would use.

Two spans a section actually offers #

Two turns and three and a half are the working numbers, and they bracket what a shell gives. Under two turns the fit is refused outright, for reasons a separate measurement makes the case for. Much above three and a half and the innermost whorl is too small to place a point in.

Between them the error at a quarter radius falls from 4.562 per cent to 1.341, which is the factor of 3.4 an inverse square predicts for a factor of 1.75 in span. Everything a careful worker can do about a bad centre is contained in that ratio.

Why arc is the only thing that buys accuracy #

The fit is a straight line through the logarithm of the radius against the angle, and its slope is the growth factor. Displacing the centre bends that line, most at the small radii where the displacement is a large fraction of the radius, and least at the large ones where it is a small fraction.

Adding arc adds points at large radii, which are the points the displacement barely touches, and it extends the baseline the slope is measured over. Both effects push the same way.

This is why nothing else in the measurement helps. More points at the same span sample the same bent line more densely; a sharper eye at the apex moves the displacement rather than the span. The one lever is how much of the shell survived.

What a photograph has room for #

A picture holds a fixed ratio of largest measurable radius to smallest. At a thousand to one — generous for a photograph of a sectioned shell — the number of turns available is set entirely by the growth factor: 14.70 at 1.6, 7.89 at 2.4, 5.94 at 3.2, 4.59 at 4.5, 3.59 at the golden 6.854 and 3.45 at 7.4.

That is a hard constraint and it is not about equipment. A spiral spends its dynamic range at its own rate, and a fast one spends it quickly.

Set the per-cent standard beside it. The measurement needs 4.25 turns to reach one per cent against a quarter-radius centre error, and only the three slowest factors have that much room in a thousand-to-one frame.

The golden spiral is the one that cannot be measured #

Reading those two numbers together gives the result of this essay that was not expected.

A nautilus at 3.2 per turn has 5.94 turns available and needs 4.25, so it can be measured to a per cent in an ordinary frame with arc to spare. A golden spiral at 6.854 has 3.59 available and needs the same 4.25, so it cannot.

The curve that is hardest to measure accurately, against exactly the objection the nautilus argument has to answer, is the golden one. That is a fact about the arithmetic of dynamic range rather than about shells, and it applies to any claim that a fast-opening curve has a particular factor.

It also sharpens the reading of published measurements: a reported factor near 6.854 carries more centre-sensitivity than a reported factor near 3, for the same care and the same camera.

The innermost whorl is the whole problem #

At three and a half turns the innermost whorl is 19.30 per cent of the outer radius at a factor of 1.6, 1.706 per cent at 3.2, 0.119 per cent at 6.854 and 0.091 at 7.4.

Those are the radii the displacement is measured against, and they are why the fast spirals are fragile. A fixed displacement in the frame is a large fraction of a small innermost radius and a negligible fraction of a large one.

The same arithmetic in units of a frame: with the two-turn floor imposed and a thousand points across the picture, the innermost whorl is 390.6 points across at 1.6, 97.7 at 3.2 and 21.3 at 6.854. Twenty-one points is a region a person can place a centre in to about a point, which is a per cent of that radius, which is a displacement of 0.01 in the units used here.

The picture of a displaced read #

A drawing makes the mechanism visible in a way the sweep does not. Take a spiral drawn at 3.2 over three and a half turns, assume a centre a quarter of the innermost radius away from the true one, and draw the circles one turn apart at whatever factor the fit returns.

The outer whorls are almost unaffected, which is the point. All the damage sits where the displacement is comparable to the radius, and there is only one whorl in that condition.

What the displacement does to the residual #

The fit reports a worst residual as well as a factor, and the residual is the part that says whether the model applies. A displaced centre raises it, because a spiral read from the wrong point is not a logarithmic spiral about that point.

That gives a second reading of the same damage, and it is why the residual gets an essay of its own. What it does not give is a test: a residual can be raised by a bad centre, by noise, or by the curve not being a logarithmic spiral at all, and it does not separate them.

For the purposes here the residual is a symptom rather than a diagnostic, and the factor is what is being measured.

The same curve from its own centre #

The comparison the displaced drawing needs is the undisplaced one, and it is the round trip this half of the collection is built on: a spiral drawn at a stated factor, read from its own centre, returns that factor.

Having the exact case is what makes the inexact ones interpretable. A fit that returns the input to the last digit when the input is right establishes that a wrong answer elsewhere came from the thing that was changed, which is the same discipline the exact recovery of a lattice’s two numbers is run under.

A tenth of a per cent is not available #

Push the standard one decimal further and the sweep runs out of arc. At every one of the six factors, the span at which a quarter-radius centre error falls under 0.1 per cent is beyond twelve turns, which is beyond what the sweep covers and far beyond what any shell holds.

An inverse square says why: a factor of ten in accuracy costs a factor of 3.16 in span, so 4.25 turns becomes about thirteen. Nothing in the way the measurement is made changes that exponent.

So a growth factor is a two-digit quantity when the centre is assumed and a three-digit one only when the centre is known — which for a drawn curve it is, and for a shell it never is.

What this does not show #

The sweep displaces the centre of a spiral that is a logarithmic spiral, and reports what the fit returns. Three things it therefore says nothing about.

It says nothing about a shell that is not a logarithmic spiral. Every curve here was drawn from the model being fitted, so the only error present is the centre.

It says nothing about noise in the coordinates, which is a separate perturbation with its own sweep and its own direction of bias.

And it says nothing about whether 3.2 is the right number for a nautilus. That figure comes from elsewhere and is the thing being perturbed here; the sweep would produce the same law about a spiral drawn at any factor, which is exactly what the six-factor agreement shows.

What it does not settle about the nautilus #

One reading of this essay would be that the nautilus argument has been weakened, and it has not, but it has been made to rest on something else.

It used to rest on a single displacement and a figure that was wrong. It now rests on a measured error of 4.56 per cent at two turns, which is twenty-five times too small to close a gap of 114 per cent — and, more decisively, on a sweep of the whole plane of assumed centres that finds no displacement anywhere returning the golden factor once the arc exceeds 1.2 turns.

The second of those is the argument the objection actually deserved, because a single displacement in a hundred and eighty directions is a sample of the plane and not a statement about it.

A figure with no span attached #

The general form of the defect is worth naming, because it is not a slip in arithmetic.

A sensitivity is a derivative, and a derivative has to be quoted with the point it was taken at. Four and a half per cent at two turns and one per cent at four and a quarter are the same measurement; well under one per cent is that measurement with the axis thrown away.

This collection has hit the same shape before — counts quoted without the radius they were taken at are the worked example, and there the missing field changed the answer by ten per cent. Here it changed it by a factor of five.

The habit that catches it is the one this site runs on: state the number, state what it is a number of, and give it a test it could fail.

The number to quote, and what would refute it #

For a logarithmic spiral read over a span of T turns from a centre displaced by a quarter of the innermost whorl’s radius, the error in the recovered growth factor is about 0.168 divided by T squared, in units of the factor itself, at every growth factor between 1.6 and 7.4.

At two turns that is 4.2 per cent, at three and a half 1.4, and it reaches one per cent at 4.25 turns. Those are the numbers to carry, and the law is more useful than any of them because it answers the question at a span nobody has swept.

What would refute it: a span at which the product of the error and the square of the turns falls outside the range 0.1643 to 0.1719, or a growth factor outside the six at which the one-per-cent crossing is not 4.25 turns. Both are cheap to test and neither has failed here. And the law is stated for one displacement — a different displacement is a different constant, which is the measurement the next rung makes over the whole plane rather than at one radius.