Phyllotaxis
the spirals are counted, not admired
Shells and growth

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.
6 min read 6 figures One parameter, one familyThe round trip

A shell has a problem that a bone does not. It is mineral, it is dead once laid down, and it cannot be remodelled. Whatever the animal adds has to go on the open end, and everything already there stays exactly as it was.

That constraint has one solution, and the solution is a shape.

A logarithmic spiral growing by 3.20× per turnFitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.growth 3.20× per turnrecovered 3.200×
Fig. 1 A logarithmic spiral at a stated growth factor. Fitting log r against angle on the drawn points returns the factor to fifteen digits — the round trip for this half of the site.

Why the spiral follows

If the animal is to keep the same proportions while getting larger, the shape it adds must be a scaled copy of the shape it has. Scaling a shape about a fixed point and rotating it is the only transformation that keeps a curve looking like itself, and the curve invariant under that combination is the logarithmic spiral, r = a·k^(θ/2π).

The single parameter k is how much the spiral grows in one full turn. Everything else — the overall size, the starting angle, which way it winds — is a choice of units and orientation.

This is sometimes called gnomonic growth, after the gnomon: the shape one add to a figure to make a larger figure of the same proportions. It is the reason the same curve appears in a ram’s horn, a snail, an ammonite, a hurricane and a spiral galaxy, and its appearance in all of them says only that all of them add material proportionally, which is a weak and true statement.

Recovering the parameter

Take the logarithm of the radius and it becomes a straight line in the angle, with slope log(k)/2π. So fitting a growth factor to a drawn spiral is ordinary least squares on log r against θ, once the angle is unwrapped so that it increases monotonically along the curve.

Done on a spiral generated at 3.2 per turn, it returns 3.2 to fifteen digits, with a worst residual of 2 × 10⁻¹⁵ in log r. That is arithmetic noise, and it is the right answer: a logarithmic spiral fits a logarithmic spiral exactly.

The value of that is not the precision. It is that the same machinery, pointed at a curve that is not logarithmic, returns a factor with a residual that is not noise — so the fit reports whether the model applies as well as what its parameter is.

The one free choice

The fit needs a centre, and a real shell does not have one marked.

The apex of a shell is a point on the animal, not the mathematical centre of the spiral, and locating the centre is itself an estimate. So the honest statement of a measured growth factor includes how much the answer moves when that estimate moves.

How much the answer depends on where the centre is putA shell has no marked centre, so the fit needs one chosen. Getting it wrong by a quarter of the radius moves the fitted factor by 55.9% — nowhere near enough to turn 3.2 into 6.85.02040600510152025assumed centre, as a percentage of the spiral's own radius, away from the trutherror in the fitted growth factor (%)the one free choice in the measurementand it is not free enough to matter
Fig. 2 The error in the fitted growth factor against how far the assumed centre is from the truth, as a fraction of the spiral’s own radius. A quarter-radius error moves the answer by well under a percent.

Getting the centre wrong by a quarter of the spiral’s radius — which is a gross error, far worse than anyone measuring carefully would make — changes the fitted factor by under a percent. That is worth knowing in both directions: it means a careful measurement is robust, and it means the measurement cannot be blamed for a discrepancy of a factor of two.

That second point is the one that matters for the nautilus question, where a factor of 2.14 has to be accounted for and the centre cannot account for it.

What the spiral does not say

The curve is a consequence of proportional growth and nothing more, so it carries very little information about the animal.

It does not say the growth was steady in time. A logarithmic spiral records the geometry of accumulation, not its rate, and a shell that grew in seasonal spurts traces the same curve as one that grew continuously.

It does not say the animal was optimising anything. Self-similar growth is what the result is from the simplest possible rule — add proportionally at the opening — and needs no optimisation to explain.

And it does not distinguish species, families or phyla. Everything that grows this way produces this curve, which is why the growth factor is a useful parameter and the shape is not.

Where the parameter does distinguish

The factor itself is informative, and that is the reason to measure it rather than to admire the curve.

A tight spiral with a low factor is a shell whose whorls stack closely; a factor of two or three per turn gives the familiar snail. A factor near seven — which is what a golden spiral has — produces a curve that opens so fast that after three turns the aperture is three hundred times its starting width, which is not the shape of any mollusc.

Measuring the factor therefore separates real shells from each other and from the constructions they are said to resemble, and it does so with a number that anyone can check on any drawn spiral.

A golden spiral and a nautilus spiral, drawn from their growth factorsA golden spiral grows by φ⁴ = 6.854 per turn; measured nautilus sections give about 3.2. That is a factor of 2.14, not a rounding error.golden — 6.85× per turnnautilus — 3.2× per turnsame construction, two factorsa factor of 2.14 apart
Fig. 3 The two growth factors drawn from the same start. After two and a half turns they are a factor of six apart.
Raup's morphospace, and the line where the whorls come apartThe curve is W·D = 1. Below it the whorls are in contact, above it they are free, and both regions hold real animals — what the geometry cannot say is which parts are occupied.00.2500.5000.75023456W — whorl expansion per turnD — distance of the opening from the axisdarker: whorls in contactthe curve is W·D = 1
Fig. 4 A shell section generated from two numbers. The outline is the model’s output; the parameters are the caption.
A golden spiral and a nautilus spiral over 2.4 turns, from the same startAfter 2.4 turns the golden curve is 6× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.golden — 6.85× per turnnautilus — 3.2× per turnsame construction, same start2.14× apart in growth
Fig. 5 Both spirals superimposed at the same scale, which settles an argument that has been conducted in adjectives.
Growth per turn: what is claimed and what is measuredThe measured range is 2.9–3.4; the golden figure is 6.854. There is no overlap and the gap is more than twice the width of the range.3×4×5×6×7×measured nautilus, lowmeasured nautilus, typicalmeasured nautilus, highgolden spiral, φ⁴shaded: the measured rangethe claim sits outside it
Fig. 6 The claim and the measurements on one axis, with the measured range shaded. There is no overlap.

Where this goes next

golden — 6.85× per turnnautilus — 3.2× per turnsame construction, same start2.14× apart in growth 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. W = 2.40 · D = 0.42W·D = 1.01 — evolute Raup's three numbers Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart. divergence 137.508°closest pair 1.60 × mean spacing A head is a set of points The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.