The dividers belong to the opening

The error budget for a nautilus priced seven ways a growth factor read off a sawn shell can be wrong, and one of them dwarfed the rest: a pair of dividers walked along the spiral, which could read a nautilus 278 per cent too high. Three entries and one span then found that three of the other entries — the centre, the span and the oblique view — are one error priced at three spans, and that a real section has one span. Read together at that span, and with the dividers set aside, the budget refuses the golden spiral from about three quarters of a turn of shell upward.

The dividers were set aside because they are the historical method, and because the 278 per cent had not been examined. The essay asked the obvious question about them: a pair of dividers walks along the curve, how far it can walk is the span, and a short span gives fewer steps. Do the dividers belong to the span too, and priced at one span does the whole budget, dividers included, still refuse the golden spiral?

Where the inflation comes from #

A measurement in steps found the mechanism. Arc along a logarithmic spiral is proportional to radius, so equal steps of arc are equal steps of radius, and at the inner end one step can be most of the innermost whorl. If a single step carries the angle more than half a turn, the unwrapping that assigns each point its angle cannot tell which way the curve went, loses whole turns, and attributes the same rise in radius to less angle — so the factor comes back too high, and only too high. The floor on the number of steps is the arc divided by the square root of the growth factor less one, $(k^T - 1)/(\sqrt{k} - 1)$ in units of the innermost radius, where $k$ is the factor per turn and $T$ the turns walked.

For a nautilus growing 3.2 times a turn the floor is 12 steps over two turns, 41 over three, 132 over four and 425 over five, and every walk at the floor reads the factor to the last digit while every walk one step short reads it too high. Below the floor the error grows fast as the steps thin out: over five turns, 1.4 per cent at 400 steps, 21 at 60, 91.7 at 20 and 278 at nine, which is the fewest points the fit accepts. So the budget’s entry was a walk of nine steps along five turns, and it is a statement about that walk rather than about dividers.

Nobody walks a count #

The floor is written as a number of steps, and it is natural to read it the way the budget did — as though a person decided in advance how many times to walk the dividers and then did so over whatever span the section offered. Nobody measures that way. A person sets the dividers to an opening, puts one point at the inner end of the curve, and walks until the curve runs out. The count is whatever the arc allows.

That changes the arithmetic completely. The number of steps a walk takes is the arc over the opening: $(k^T - 1)/s$, with the opening $s$ in units of the innermost radius. The floor is the same arc over $\sqrt{k} - 1$. Their ratio is $(\sqrt{k} - 1)/s$, and the span has cancelled. A walk at a given opening is either above the floor at every span or below it at every span.

The criterion is the opening itself: less than $\sqrt{k} - 1$ of the innermost radius. For a nautilus that is 0.789. It has a plain geometric meaning: an opening that small, laid from the inner end, cannot reach past the half-turn point of the innermost whorl, so the first step — the only one that is ever at risk — never carries the angle more than half a turn.

Why the square root #

The threshold can be read off the curve. Stepping in equal arc is stepping in equal radius, so the first step from the inner end, at an opening $s$, reaches the radius $1 + s$ times the innermost one. A logarithmic spiral growing $k$ times a turn reaches $\sqrt{k}$ times any radius after exactly half a turn. So the first step carries the angle less than half a turn precisely when $1 + s < \sqrt{k}$, and the whole criterion is that the first step stop short of the radius the curve has half a turn in.

For a nautilus, $\sqrt{3.2} = 1.789$ and the opening must be under 0.789 of the innermost radius. For a slower shell growing twice a turn it is 0.414. For the golden spiral, whose factor per turn is $\varphi^4$, the square root is $\varphi^2$ and the safe opening is $\varphi^2 - 1$, which is $\varphi$ itself: dividers opened to 1.618 innermost radii are the widest that can walk a golden spiral exactly. A golden spiral tolerates a wider opening than a nautilus because it grows faster, and a faster spiral reaches its half-turn radius sooner.

The same walk over two spans #

Opened to one innermost radius, slightly over the threshold, the dividers take 17 steps along two and a half turns and 104 along four, and both walks read the factor too high — by 18.2 and 4.2 per cent. Closed to 0.78, just under it, they take 22 and 133 steps, and both walks are exact. Opened to two radii, 9 and 52 steps, 38.5 and 8.4 per cent too high. The first step decides every one of these, and whether it carries the angle past half a turn is set by the opening, not by how far the walk goes afterwards.

Exact at every span, or too high at every span #

Read at spans from half a turn to six, dividers opened to 0.3, 0.5, 0.7 or 0.78 of the innermost radius read the nautilus factor exactly every time — forty-four walks, every one to within rounding. Opened to 0.8, 0.9, one, one and a half or two radii, they read it too high at every span of three turns or more, and at two turns and a half for all but the narrowest. Below a turn and a half every opening is exact, because the fit’s own minimum of nine points is already above the floor there.

Over the threshold, the error falls as the span grows. Opened to one radius the dividers read 28.7 per cent too high over two turns, 10.9 over three, 4.2 over four, 1.6 over five and 0.6 over six. The damage is done in the first step, at the inner end, and a longer walk adds many good steps to one bad one, so its fit leans on the bad step less. Opened to 0.8 the error is 8.6 per cent over three turns and half a per cent over six.

So the dividers do depend on the span, but in the opposite direction from the other span entries. The centre and the oblique view punish a short span. A pair of dividers opened too wide punishes a short span too, for a different reason, and forgives a long one.

The walk the budget priced #

The 278 per cent is visible for what it is. Nine steps along five turns of a spiral growing 3.2 times a turn is an opening of 37 innermost radii; the first step spans most of the first three whorls, and the fit is handed nine points that say almost nothing about the inner half of the shell. The dividers were not the instrument that made the nautilus look golden. A pair of dividers opened to nearly the width of the whole shell was, and no one measuring a nautilus has ever walked one.

The earlier essay drew the same conclusion from the other side, that a golden spiral cannot be measured this way within its own floor. What the opening adds is that the floor is not a burden a longer span imposes. It is a property of the dividers’ setting, satisfied once and then satisfied at every span.

The budget at one span, dividers included #

Put the dividers back into the budget at one span. At each span the worst joint error of a centre a quarter of the innermost radius off and a camera ten degrees off the normal — 188 per cent at half a turn, 89 at three quarters, 39 at one turn, 7 at two, under 1 at six — is added to the dividers’ error at a stated opening and to the three entries that depend on neither, the uneven clock, a miscounted septum and a changed expansion, which come to 22.2 per cent together. The golden spiral is refused wherever the total falls below the 114 per cent that separates 3.2 a turn from the golden spiral’s 6.85.

With the dividers opened to half, one, two or five innermost radii, the budget refuses the golden spiral at every span from three quarters of a turn to six and does not refuse it at half a turn — the same window the joint reading found with the dividers set aside. Opened to one radius, the total is 112 per cent at three quarters of a turn, 61 at one turn, 58 at two, where the dividers contribute their largest share, and 24 at six. The dividers widen nothing and close nothing.

Only at openings of ten radii and more does the window break. Opened to ten, the budget fails to refuse at four turns alone, 169 per cent, because four turns is where that opening leaves a walk of ten steps. Opened to the budget’s own 37 radii it fails at four and five turns and refuses again at six, where the walk has lengthened to 29 steps. In every case the failure is a walk within a step or two of the fit’s minimum.

What the three quarters of a turn is made of #

The lower edge of the window does not move whatever the dividers do, and it is worth saying what it is. At half a turn the joint centre and tilt error is 188 per cent on its own, larger than the gap; at three quarters it is 89, and with the 22 per cent of span-free entries it comes to 112, just under 114. The window opens between those two spans because a quarter-radius centre and a ten-degree tilt, in their worst orientation, cost a short arc far more than a long one — how far a centre must move found the same kind of edge from the centre alone: centres that make a nautilus golden exist at spans up to 1.15 turns and at none from 1.2.

So the budget has one span-dependence, and it is the centre and the view. The dividers, once opened sensibly, are exact; the three remaining entries do not care about the span. A section of three quarters of a turn or more refuses the golden spiral with every entry the budget has ever listed, priced honestly at the span it is.

The historical method was not the problem #

It is worth being fair to the dividers, since the budget’s original ranking made them look like the source of the golden-spiral claim. They are the oldest way of measuring a shell’s curve and, as the earlier walk found, the best one available once there are enough steps, because they put their points where the curve’s length is. The worst they can do, at a sensible opening, is nothing; at a careless one, a few per cent over a long span. A person using them to argue a nautilus was golden would have had to open them across most of the shell.

The direction of the error is what gave the dividers their place in the story. Every other entry in the budget can push the factor either way or pulls it down — an assumed centre and a section seen from the wrong angle both — and the dividers alone push it up, towards the golden spiral. So they are the one error that could turn a nautilus golden, and it is worth knowing what that would take. Over five turns a walk reads more than the 114 per cent needed with twelve steps or fewer, 120 per cent at twelve and 63 at sixteen; over four turns, with eleven or fewer, eleven clearing it by less than a point. Twelve steps along five turns is an opening of 28 innermost radii. Making a nautilus look golden with dividers takes dividers opened across most of the shell.

The nautilus question asked how a claim so easy to check survived so long. Whatever kept it alive, it was not a pair of dividers walked along the curve at an opening of an innermost radius or less, because that walk returns 3.2 at every span.

In millimetres #

The criterion is generous in units of the innermost radius and harsh in millimetres, because a nautilus’s innermost radius is small. A shell ten centimetres in radius read over five whorls at 3.2 a turn has an innermost radius of 0.3 millimetres, and the safe opening is a quarter of a millimetre: no hand walks dividers that fine along a sawn section.

The way out is the one a person takes anyway, which is to start where the whorl can be read. Started at a radius of 3 millimetres the walk covers three turns of the same shell and the safe opening is 2.4 millimetres; started at 10 millimetres it covers two turns and the opening can be 7.9. A wider start buys a practical opening at the price of a shorter span, and the budget above says what the shorter span costs: at two turns the joint centre and tilt error is 7 per cent, and the budget refuses the golden spiral with room to spare. So the dividers and the span meet after all — not through the step count, but through where the walk begins.

What the pricing leaves out #

It walks from the inner end. A walk that starts further out has a larger innermost radius and a correspondingly larger safe opening, since the criterion is relative to wherever the first step is laid, so the numbers here are for the most demanding start.

It prices the dividers on a perfect curve. A drawn curve’s thickness and a section’s broken inner whorls both limit how small an opening can be used, and the septa interrupt a real nautilus’s line. Those are practical costs of a small opening, not errors of a large one, and they are not measured here.

And it adds worst cases. Three of the entries at one span were found to add in quadrature over random orientations, so the totals above are an upper bound on a typical section’s error, and the window for a typical section is wider.

Findings that would overturn it #

A span at which dividers opened to less than 0.789 of the innermost radius read a nautilus spiral too high. An opening over it that reads the factor exactly at some span of three turns or more. A span from three quarters of a turn upward at which the whole budget, with the dividers opened to five innermost radii or less, does not refuse the golden spiral.

Still open: the opening on a real section #

The criterion is in units of the innermost radius, which a sawn section rarely shows cleanly: the first whorls of a nautilus are often broken or filled. The measurement is the same walk started at the innermost whorl a section actually preserves — a radius several times the true innermost one — asking how the safe opening scales with where the walk starts, and whether a practical pair of dividers, set to the width a hand can read on a real section, is inside the criterion for the whorl it starts on. If the safe opening grows with the starting radius as the geometry says, the dividers are safe everywhere a person can put them.