A measurement in steps

There are two ways to put points on a drawn spiral. Step round it in equal angles, which is what a protractor does and what every measurement in this collection has quietly assumed. Or walk along it in equal lengths, which is what a pair of dividers does and what anyone measuring a real shell with real instruments would do first.

The second is better, up to a count that can be written down in advance. Below that count it is not merely worse — it returns a specific wrong answer, always in the same direction, and the direction is towards a golden spiral.

Arc on this curve is radius #

The reason the two differ at all is a property of the logarithmic spiral rather than of either instrument. Arc length along the curve is proportional to the radius at the point reached, so the total arc from the innermost whorl outward, measured in innermost radii, is the growth factor raised to the power of the span, less one.

A step of equal arc is therefore a step of equal radius. Divide the curve into a fixed number of equal steps and the points fall where the radius falls, which on a curve that multiplies its radius each turn means they crowd into the outermost whorl.

Which is exactly what a fit wants #

That crowding is an advantage, and a large one. Over three and a half turns of a nautilus spiral an even-arc sample puts 69.9 per cent of its points in the outermost turn against 28.6 per cent for an even-angle sample; on a golden spiral it is 85.5 per cent.

Points in the outermost turn are the points least damaged by an error in the assumed centre, because they are furthest from it. Given enough of them the arc-walked measurement beats the angle-stepped one by 5.80 times against a quarter-radius centre error on a nautilus spiral over that span — 0.23 per cent of error against 1.34 per cent, with six hundred points either way.

That is a substantial improvement on a quantity this collection has had to measure rather than dismiss, and it is the reason to want the dividers rather than the protractor in the first place.

And what it costs at the inner end #

The same crowding starves the other end. If the points are spaced evenly in radius and the radius spans a factor of fifty-nine over three and a half turns at 3.2 — the innermost whorl is 1.706 per cent of the outer radius — then almost none of them land on the innermost whorl, and the innermost whorl is where the curve turns fastest.

So a single step of the dividers, out near the aperture, is a small fraction of a turn. The same step at the inner end is most of a whorl, and it can be more than a whorl.

The step that swallows a turn #

That is where the measurement breaks, and it breaks in the bookkeeping rather than in the geometry. Recovering a growth factor means fitting log r against an angle that has been unwrapped — made to increase along the curve rather than resetting every turn — and unwrapping works by adding a turn whenever the observed angle drops.

A step that carries the angle more than half a turn is a step the unwrapping cannot read. It cannot tell a step of two thirds of a turn forward from one of a third of a turn back, so it takes the shorter reading and loses the difference. Whole turns of angle go missing, and they go missing at the inner end where the steps are longest.

The law #

Which makes the condition exact rather than a rule of thumb. One step is the total arc divided by the number of steps, in innermost radii; a step subtends half a turn at the inner end when it reaches the square root of the growth factor, less one. Setting one against the other gives a floor on the number of steps:

$$N \ge \frac{k^{T} - 1}{\sqrt{k} - 1}$$

with k the growth factor per turn and T the span in turns. Nothing is fitted and nothing is measured; both sides are geometry.

Reading the law #

Both terms do something recognisable. The numerator is the total arc, so a longer span needs more steps in proportion to how much curve there is — and on this curve that is exponential in the span, not linear.

The denominator is how much angle one unit of arc costs at the inner end, which is set by the factor alone. A tight spiral turns a lot for a little arc and needs its steps close together in arc; a fast one turns very little for the same arc and would need few, except that the numerator has already exploded.

Seventy-four, and five hundred and twenty-one #

The two numbers worth carrying. A nautilus spiral at 3.2 per turn, measured over three and a half turns, needs 74 steps. A golden spiral at 6.854 over the same span needs 521.

Over two turns instead, the same two are 12 and 29. The floor rises with the span far faster than it rises with the factor, and it rises with the factor far faster than anybody would guess from looking at the two curves — which is the usual difficulty with this pair, since a short arc is compatible with almost any factor.

Exact in twelve of twelve #

The floor is a derivation, so it is checked against a measurement: for each factor and span, bisect on the step count to find the smallest one at which the recovered factor comes back exactly, and compare.

They agree at every one of the twelve combinations tried — 16 for a slow spiral over three and a half turns, 38, 74, 172, 521, 641, and the six two-turn cases below them. Three of the twelve have a floor beneath the recovery’s own nine-point minimum, where it cannot be observed at all, and are recorded at nine. Everywhere the floor is observable, the derived number is the observed one.

Below the floor the answer is too high #

One step below the floor is enough. A true 3.2 spiral over two turns comes back at 3.8948; over three and a half turns at 3.3709. A true golden spiral over two turns comes back at 8.6333, and one at 7.4 over two turns at 9.3069.

Every bar runs the same way. The error from too few steps is always upward, never downward, at every factor and every span tried.

Why it is always upward #

The sign follows from what was lost. The unwrapping has dropped whole turns of angle, so the measurement thinks the curve covered less angle than it did — but the radii are unaffected, because a radius is read directly and needs no unwrapping.

The same rise in log r is then attributed to less angle, and the slope comes out steeper. A steeper slope is a larger growth factor. There is no mechanism in this failure that could make the answer too small, which is why the direction is a property of the law rather than a pattern in the numbers.

The whole sweep, count by count #

Between nine steps and the floor the answer is not merely high, it wanders, because how many turns are lost depends on exactly where the steps land.

The worst of them on a nautilus spiral over this span is 6.296, at thirteen steps. That is 1.97 times the factor the curve was drawn at, and it is within nine per cent of a golden spiral’s 6.854. Past seventy-four the sweep is flat and exact, with nothing in between to warn that the flat part had not been reached.

The slider #

The crossing is worth watching rather than reading about, so it is the one drawing in this group that moves.

What changes across the stops is the innermost step, and only the innermost step. The outer part of the walk looks correct at every setting, which is the difficulty: nothing in the picture at sixty-one steps says the number underneath it is wrong.

The only route that pushes a nautilus towards golden #

Set against everything else measured on this instrument, that direction is unique. A misplaced centre cannot do it — no assumed centre in the plane returns a golden factor from a nautilus spiral past 1.2 turns of arc. Noise cannot do it: of the sixty angle-sampled readings with noise added, twenty-three move by more than a per cent and every one of the twenty-three moves downward.

Under-sampled dividers do it at every span past two turns, and at nine steps over three and a half turns they turn a golden spiral into 38.839 and a shell at 7.4 into 48.118.

And why it is refused anyway #

Because the arithmetic that inflates the factor also destroys the thing the recovery insists on. The lost turns are lost from the measured span: the reading that returns 6.296 says it measured 1.50 turns, not three and a half, and the recovery declines to report a growth factor from less than two turns of arc.

Every one of the six factors read at nine steps over three and a half turns collapses to that same 1.50 turns and is refused on span before anything else is consulted. The residual would have refused it too — 1.516 against a threshold of 0.15 — but the residual is not a test that can be relied on, and the span floor here is.

The refusal is not a coincidence #

It is the same fact stated twice. A step count below the floor loses turns; losing turns is what inflates the factor; and the turns that were lost are exactly the turns missing from the measured span. A reading cannot be inflated by this route without announcing by how much.

That is a better guard than the residual manages, and it is worth saying why: the span is measured in the same quantity that was corrupted, where the residual is a scalar summary of something else entirely.

Even respecting the span floor #

The honest version of the ceiling is the one that keeps only readings whose measured span survives. Those still inflate — 4.503 over three drawn turns, 4.859 over four, 6.136 over five — so the guard is not absolute and the route reaches within eleven per cent of a golden spiral before it runs out.

It does not reach it. On this evidence the under-sampled walk is a real hazard and not a rescue of the golden claim, and a reader who wanted it to be one would need a shell photographed over five turns and walked with nine divider steps.

A golden spiral cannot be measured this way #

The floor has a practical consequence that the numbers make immediate. Five hundred and twenty-one steps is not a pair of dividers; it is a digitised outline.

A nautilus over the same span needs seventy-four, which is an afternoon’s work with dividers and a photograph. The curve the claim is about is the one the method cannot reach, which inverts the usual complaint that the shell is hard to measure.

Nothing about that is a defect of the shell. It follows from the single parameter the curve has: a faster factor covers more arc in the same number of turns and turns through less angle per unit of arc, and both of those are the same one number doing its work in the numerator and the denominator of the floor.

The other route to the same wrong number #

There is a second way to make a golden spiral read as a nautilus, and it comes from the opposite direction. Take a genuine golden spiral over three and a half turns, sample it in equal angles, and locate every point to one part in a hundred of the outer radius.

It comes back at 3.181. That is a nautilus, from a golden shell, with no under-sampling anywhere in the measurement.

The noise it takes was a property of the fit, not of the shell #

This essay used to put that failure at one part in a thousand — one pixel in a good photograph — and report the reading as 3.744. It was wrong by a factor of ten in the noise, and the reason belonged to the same bookkeeping this essay opened with.

The recovery’s unwrapping added a turn whenever the observed angle dropped by more than half a turn and never subtracted one when it rose. On a golden spiral over this span the innermost whorl is a thousandth of the outer radius, so noise at that scale reverses the angular order of neighbouring points there — and every reversed step was promoted to a forward step of nearly a whole turn. The angle climbed where the curve had not, and the same rise in log r divided by a manufactured span read low. The routine was destroying the spiral, and it was doing it in exactly the way an under-sampled walk does: by getting the number of turns wrong at the inner end.

The search that found it had fitted every point a second way as a formality. With the unwrapping repaired, the same golden spiral at one pixel in a thousand comes back at 6.766 — a bias of 1.3 per cent, not 45.

The one that does happen announces itself, and so does everything else #

At one part in a hundred the reading is wrong and it is not disguised. Its residual is 3.453, twenty-three times the threshold the recovery uses.

What it does not do is stand out from the rest. Measured identically, a genuine nautilus spiral comes back at 3.193 — accurate to 0.22 per cent — and carries a residual of 1.470, itself ten times the threshold. At the noise that fakes the reading, nothing measured that way is believed at all, which is a blunter answer than this essay used to give and a better one: there is no noise anywhere in the ladder at which a golden spiral reads as a nautilus while a genuine nautilus measured the same way still passes.

Trimming turns it round #

The repair is one line and it says where the damage was. Discard the inner turns, keep the outer two, and the same golden spiral with the same noise comes back at 6.778 — a bias of 1.11 per cent — with a residual of 0.969.

A 54 per cent error becomes one per cent by throwing away the innermost whorl. The same trim on a nautilus reading gives 3.1957, and on a 7.4 spiral it turns a 58 per cent error into two.

The trimmed reading is still refused: 0.969 is six times the threshold, and a fit accurate to one per cent is declined on its scatter. That is the same casualty the residual essay prices — the arc that carries the noise is the arc that carries the shape — arriving here as the price of the trim rather than as a defect of it.

The two failures are the same whorl #

Which is the finding that ties the essay together. Under-sampled dividers break at the inner end because the steps are longest there. Noise breaks the fit at the inner end because a fixed error in position is a fractional error in radius that is largest where the radius is smallest, and on a golden spiral over this span the innermost whorl is 0.119 per cent of the outer radius.

One instrument puts too few points on the innermost whorl and the other measures them too badly. Both consequences arrive as a growth factor with nothing visibly wrong with it.

So the honest statement of a measured factor has to name which part of the curve produced it, in the same way that a spiral count is a statement about an annulus rather than about a flower. A factor read from three and a half turns and a factor read from the outer two are two measurements, and on a golden spiral at one part in a hundred they differ by a factor of 2.13.

Eight rows that moved upward #

The two halves meet in a place that was almost missed. Across the whole noise table, all but eight of the readings that moved by more than a per cent moved downward — and those eight run from 2.14 to 7.19 per cent upward.

Every one of the eight is an arc-sampled golden or 7.4 spiral over three and a half turns at two hundred and forty points, against floors of 521 and 641. They are not noise results at all; they are this essay’s failure appearing inside somebody else’s table, in the same shape as three angles that turned out to be three samples of a grid and a quantisation larger than the disturbance it was measuring.

What the crossover says about which method to use #

Arc beats angle above the floor and loses catastrophically below it, so the choice is not between two instruments but between a count and a curve. The smallest count at which arc sampling beats angle sampling is 96 for a nautilus spiral over three and a half turns and 600 for a golden one; for a spiral at 7.4 over that span it does not happen within six hundred points at all.

So the rule for a shell is: count the turns, compute the floor, and if the outline cannot carry that many points, step in angle instead and accept the larger centre sensitivity. The worse method with enough points beats the better one without them, which is the same lesson as a sample that returns a narrow wrong answer rather than a wide one.

What this does not show #

It does not show any published shell measurement was made below the floor. Nothing here reads a real specimen; the curves are drawn at stated factors and the dividers are simulated, and whether anybody has ever walked a shell in thirteen steps is not a question this can answer.

It does not show the law holds outside the range tried — six factors from 1.6 to 7.4, two spans, and a recovery with a nine-point minimum of its own. And it says nothing about a shell that is not a logarithmic spiral, which is the case the residual was supposed to catch and does not.

What it leaves #

A measurement with a computable minimum, which is a rarer thing than it sounds. Most of the requirements in this collection are empirical — a run length, a window, a grid step — and are found by sweeping until the answer stops moving. This one is derived first and confirmed after, at every combination where it can be observed.

The general form is worth keeping. An instrument that samples a curve has a resolution below which it does not degrade gracefully but aliases, returning a confident wrong answer with no wide interval attached, and the count at which that begins is usually a property of the geometry rather than of the instrument. Working it out costs an afternoon and is the only thing that distinguishes a measurement from a number, which is what a protractor’s requirement turned out to be.