Phyllotaxis
the spirals are counted, not admired
Concept

Logarithmic spiral — where it appears

A curve whose radius grows by a fixed factor each turn, which is the shape a shell's outline follows. Its growth factor can be recovered from a drawing to many digits, which is how the claim about the nautilus is settled.

Named by 15 essays across one field — each of them below, with the objects they name alongside it.

A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.

Growth as a rule

A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.

shells · Spiral A golden spiral and a nautilus spiral over 2.5 turns, from the same start. After 2.5 turns the golden curve is 7× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.

The nautilus question

A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.

shells · Nautilus What a quarter-radius centre error costs a 3.2× spiral, against what the collection publishes. Root-mean-square error in the recovered growth factor when the assumed centre is displaced by a quarter of the innermost whorl's radius, against how much arc is measured. It is 4.56 per cent at two turns, 2.52 per cent at two and a half, 1.91 per cent at three and 1.34 per cent at three and a half. It first falls under one per cent at 4.25 turns — and at 4.25 turns at all six of the growth factors surveyed, so the span rather than the factor is what decides it.

What the centre costs

The fit that recovers a shell's growth factor needs a centre, and no shell has one marked. Displacing it by a quarter of the innermost whorl's radius moves the answer by 4.56 per cent at two turns, which is about five times the figure published earlier.

shells · Spiral fit Every assumed centre from 0.01 to 500 innermost radii, at 19 spans, against a spiral drawn at 3.2. One row per span of arc, one cell per assumed displacement on a logarithmic grid from 0.01 to 500 innermost radii, shaded by the highest growth factor any of 180 directions returns there. A displaced centre reaches 6.854 at every span up to 1.15 turns and at no span from 1.2 upward, so the boundary is a span rather than a displacement. The dashed rule is the two-turn span floor, and the cheapest golden fit anywhere leaves a residual of 0.163 against a threshold of 0.15 — so a golden reading is refused twice over.

How far a centre must move

Four hundred and eighty-two thousand assumed centres, at nineteen spans and a hundred and eighty directions each, asked whether a spiral drawn at 3.2 can be made to read as the golden 6.854. It can, at every span up to 1.15 turns and at none from 1.2 upward, and every centre that manages it is refused twice over.

shells · Spiral fit The fit pointed at twelve curves, and the residual each one leaves. Every curve is handed to the same recovery from its own true centre with no noise anywhere, and every one of them returns a growth factor. A circle comes back at exactly 1.0000 with a residual of zero, and so does every ellipse tried, whatever its aspect. An Archimedean spiral read from its second turn comes back at 1.306 with a residual of 0.090 and is accepted; it is refused only when the arc includes its own first turn, at 0.1976. Across every growth factor and span the fit was tested at, the band inverts — a genuine spiral reaches 0.670 while archimedes-4 sits at 0.035 — so no threshold separates them.

The residual is not the test

The fit that recovers a growth factor also hands back a residual, and that residual has been read as what separates a genuine logarithmic spiral from something that merely looks like one. Pointed at twelve curves it fits a circle exactly, accepts an Archimedean spiral, and refuses a golden one that is right to three decimal places.

shells · Spiral fit The step floor derived, against the step floor measured — exact in 12 of 12, with 3.2 over 3.5 turns marked. One step of the dividers subtends half a turn at the inner end when it reaches the square root of the growth factor less one, which puts the floor at the factor to the power of the span, less one, over that. The smallest count at which the factor actually comes back exactly is then found by bisection, and the two agree in 12 of 12 cases: 74 steps for a 3.2 spiral over three and a half turns and 521 for a golden one. The three that appear not to agree are the ones whose floor falls below the fit's own nine-point minimum, where it cannot be observed.

A measurement in steps

Walking a pair of dividers along a shell's spiral is the oldest way to measure it and the best one available once there are enough steps, because it puts the points where the curve is. Under a count that follows exactly from the geometry it inflates the answer instead, and it is the only route measured here that pushes a nautilus towards a golden spiral.

shells · Spiral fit How far a shell growing from 2.8 to 3.6 a turn departs from the one spiral a fit gives it. The shell's logarithmic radius along its 4-turn arc, less the straight line a single growth factor fits. The fit returns 3.17490 a turn, which is the geometric mean of the two ends, 3.17490. The largest departure is 0.0836 in the logarithm against the 0.15 the collection refuses a spiral past, so the fit accepts this shell; a change that is steady in its rate leaves ln(W₁/W₀) × turns/12 = 0.0838.

One number for a shell that changes

An animal is under no obligation to grow at one rate from hatching to maturity, and the fit that recovers a shell's growth factor returns one number whatever it is given. Handed a shell whose expansion rises steadily from 2.8 to 3.6 a turn, it returns 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepts. A change of sixty-four per cent over three and a half turns passes as one logarithmic spiral, and at the aperture, where contact is decided, the one number and the last whorl give opposite verdicts.

shells · Spiral fit The two halves of a 3.2 spiral fitted about a centre 0.25 innermost radii off, towards 52°. A logarithmic spiral growing by 3.2 a turn over 4 turns, which does not change, split into an inner half and an outer half, each fitted about a centre displaced by 0.25 of the innermost radius towards 52°. The inner half returns 3.4193 and the outer half 3.2200, a split of −5.83 per cent. The panel on the right enlarges the first whorl, where the true centre and the assumed one can be told apart; across the whole spiral the displacement is 0.238 per cent of the outer radius.

A centre that invents a life history

The collection's advice for a shell that might have changed how it grew was to fit it twice, over different arcs, and compare. On a spiral that does not change at all, a centre displaced by a quarter of the innermost radius splits the two halves by 4.09 per cent — the split a genuine 8.35 per cent change from apex to aperture produces — in either sign, depending only on which way the centre is wrong. Point noise of the same size splits them by less than half as much, and averages away where the centre does not. The floor under the test is the centre, not the noise.

shells · Spiral fit Two diameters half a volution apart on a 3.2 spiral, aimed through the true centre. A logarithmic spiral growing by 3.2 a turn, its aperture 3.5 turns along. A line from the aperture through the true centre meets the outer wall half a volution back and a volution back. The conch diameter dm1 is 1.559017 of the outer radius, the diameter half a volution back, dm2, is 0.871517, and the apertural height between them 0.687500. Squared, dm1/dm2 is 3.200000 against the spiral's own 3.2, exact: both lengths are distances between wall points on one line, so the centre only aims it.

Three points on a diameter

Ammonoid workers measure a shell's expansion without a centre: two diameters half a volution apart, squared. On a logarithmic spiral that is exact, and the centre is needed only to aim the line. A quarter-radius aim error costs the fit 1.341 per cent and the diameters 0.0045, because the aim error enters as its square. Reading noise is another matter: at a thousandth of the outer radius the fit's four hundred points beat the calipers' three readings at every expansion up to the nautilus's, and which instrument is better depends on which error the section actually has.

shells · Spiral fit One spiral at 3.20× per turn, marked at equal intervals of time under four rate laws. The curve is identical in all four panels — every mark lies on r = W^(θ/2π) exactly, whichever clock put it there — and the marks are not. Under a constant angular rate the three whorls hold 14, 13, 14 marks; under a constant length added the three whorls hold 3, 9, 29 marks; under a constant area added the three whorls hold 1, 3, 37 marks; under a constant volume added the three whorls hold 1, 1, 39 marks. The growth factor is a rate per turn of the shell's own coiling, and a turn is not a unit of time; the whole of what an animal's growth rate means is in the spacing of these marks and none of it is in the curve.

A spiral with no clock

The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.

shells · Spiral The ratio of two whorls' line counts is the growth factor raised to the clock's own power. Each curve is W^p for one law: flat for a clock that advances the angle at a constant rate, W for one that adds a constant length at the opening, W² for a constant area and W³ for a constant volume. At 3.20× per turn those are 1.00, 3.20, 10.24, 32.77. So a count of lines in two successive whorls, divided, and read back through a growth factor the curve already gives, names the law — and the four are further apart the faster the shell expands.

What the growth lines carry

A shell's curve says nothing about how fast the animal grew, and its growth lines say all of it. Under a law that holds the pth power of the radius constant per unit time, the time spent crossing one whorl is proportional to the change in that power across it — so the lines in successive whorls stand in the ratio of the growth factor raised to p, and each whorl holds the count the closed form predicts to within the one line rounding can move. Dividing two counts and taking the logarithm against a growth factor the curve already gives returns p: 17 of 20 readings name their own law, the furthest 0.012 from a whole number. The other three are not wrong, they are uncountable — at 4.5 per turn under a volume clock the inner whorl of the pair holds one line.

shells · Spiral A shell section square on and seen 30 degrees off. The same three turns of a spiral built at 3.2 per turn, drawn as a camera normal to the coiling plane sees it and as one 30 degrees away from normal does. The tilted view is the plane compressed by 0.8660 along one direction. A fit to the first returns 3.200000 with a residual of 1.8e-15; a fit to the second returns 3.19527 with a residual of 0.07763. Nothing in the second picture says it is not a shell.

A section seen from the wrong angle

A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.

shells · Spiral fit The nautilus error budget with its three span-dependent entries priced at one span, against the gap the golden claim asks it to clear. The budget with the dividers set aside, but with its centre, span and oblique entries replaced by the worst joint error of a quarter-radius centre offset and a ten-degree tilt at one span of arc. 0.5 of a turn: 210.4%; 0.75 of a turn: 111.5%; one turn: 61.0%; 1.5 turns: 31.9%; 2 turns: 29.3%; 3 turns: 25.1%. The upper level line is the gap, 114.2%; the lower is the budget as it was priced, 54.4%, with its three coupled entries at 32.2% together. The budget clears the gap from 0.75 of a turn and not below it.

Three entries and one span

The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.

shells · Nautilus A pair of dividers opened to 1 of the innermost radius, walked along two and a half turns and along four. A nautilus spiral growing 3.2 times a turn, stepped at a fixed opening of 1 of its innermost radius from the inner end outward. Over 2.5 turns the walk takes 17 steps against a floor of 22 and reads the factor +18.2% too high; Over 4 turns the walk takes 104 steps against a floor of 132 and reads the factor +4.2% too high. The opening at which every span reads exactly is the square root of 3.2 less one, 0.789 of the innermost radius: under it the first step never carries the angle past half a turn, whatever the span.

The dividers belong to the opening

The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.

shells · Nautilus A pair of dividers opened to 1/50 of a shell's outer radius, walked over the outer 3 of its six turns. A nautilus spiral of six turns growing 3.2 times a turn. The section is taken to preserve only its outer 3 turns; the lost inner whorls are drawn faint. Dividers opened to 1/50 of the outer radius walk from the innermost preserved whorl to the rim in 48 steps; the right panel magnifies the first steps and the whorls inside them. In radii of the whorl the walk starts on the opening is 0.66, against the safe 0.789, so the walk reads the factor exactly. A walk at this opening is exact over any span shorter than 3.16 turns.

The rim sets the opening

A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.

shells · Nautilus

Named alongside it

The objects these essays reach for when they reach for this one.

Growth factorClaim testingHonest limitsWhorlMeasurement errorModel scopeResidualSpan of arcNegative resultAssumed centreError propagationGolden spiral

All concepts