Phyllotaxis
the spirals are counted, not admired
Shells and growth

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.
5 min read 6 figures One parameter, one familyState the claim, state the test

David Raup published a model in 1966 that generates coiled shells from three numbers.

W — how much the whorl expands in one turn. D — how far the generating opening sits from the coiling axis. T — how far it travels along the axis per turn.

Gastropods, ammonites, bivalves, brachiopods, nautiloids: essentially every coiled shell in the fossil record and every one alive is a point in that three-dimensional space.

A shell section at W = 2.40, D = 0.42W·D = 1.01, so the whorls are free of each other — an evolute shell, like a ram's horn or a planispiral ammonite. Both are things animals grow.W = 2.40 · D = 0.42W·D = 1.01 — evolute
Fig. 1 A section at stated W and D. Nothing about the shape is drawn; the outline is the model’s output, and the two parameters are the caption.

What the compression means

Three numbers is very few, and the immediate question is whether the model is doing work or whether it is flexible enough to fit anything.

It is not flexible. The model has no freedom to bend a whorl, thicken a wall, or adjust a profile locally. Given W, D and T the shape is fixed, and a shell that does not lie in the space cannot be produced by adjusting the parameters — it is simply outside the model.

So the fact that real shells lie in it is a genuine constraint being satisfied, and it says something specific: shells grow by accretion at an aperture, with the aperture scaled, rotated and translated at fixed rates. That is a claim about the growth process, and the model’s success is evidence for it.

The most useful line in the space

A whorl at angle θ occupies a radial interval, and the next turn’s inner edge clears this turn’s outer edge exactly when W·D ≥ 1. One product, one line, and it separates two shapes of shell that anyone can tell apart.

Above the line the whorls are free of each other — an evolute shell, like a planispiral ammonite or a ram’s horn, where each turn is visible and separate. Below it the whorls are in contact, each wrapping over the one before — an involute shell, which is what most gastropods are.

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. 2 The slice at T = 0, with the boundary drawn. Both regions hold real animals; the line is where the geometry changes character.

The label that was wrong

The first version of this called the region below the line impossible, and drew it in the colour reserved for failed claims.

That is simply false. Whorls in contact are not a contradiction — they are the ordinary condition of most snails, where each whorl wraps over its predecessor and the shell is a compact spiral rather than an open coil. Nothing about it is geometrically inconsistent.

The error was a specific and instructive kind: taking a boundary that the geometry genuinely computes and attaching to it a meaning the geometry does not supply. The line is real; “impossible” was an interpretation, and it was wrong.

What the model cannot say

Raup’s actual finding, the one the model is remembered for, is not in the geometry at all.

He plotted where real shells fall in the space and found that large regions of it are empty. Not geometrically excluded — perfectly constructible, and unoccupied. Some of the emptiness is functional, some is developmental, and some may be historical accident.

That is an empirical result obtained by measuring shells, and this site does not have it. Drawing the boundary is geometry and can be computed; drawing the occupancy is a census and cannot. The figures here show where the geometry changes and stop, and saying so is the difference between reporting Raup’s model and claiming his result.

The same distinction applies throughout: the packing measurements compute what a criterion says and cannot say what a plant does, and the dynamical model computes what a rule produces and cannot say what a meristem uses.

The third parameter

T is what turns a flat coil into a tower.

At T = 0 the whorls stay in one plane and the shell is planispiral — an ammonite, a nautilus, a ram’s horn. Increase T and each whorl steps along the axis, producing the conical spire of an ordinary garden snail or a whelk. Very large T gives a shape like an ice-cream cone with a thread wrapped round it.

The reason the figures here mostly show T = 0 is that a section through a planispiral shell is a plane curve, and a plane curve can be drawn honestly on a page. A high-T shell needs a projection, which introduces choices — and this fleet has a whole site about the choices a projection introduces.

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. 3 A logarithmic spiral at a stated growth factor, with the factor recovered from the drawn points to fifteen digits.
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. 4 The two growth factors drawn from the same start. After two and a half turns they are a factor of six apart.
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.
The two costs, and where their sum is leastPumping cost falls as r⁻⁴ and upkeep rises as r². Minimising the sum gives flow proportional to r³ — checked here across a fourfold range of flow, where Q/r³ holds to 3.9%.010200.50011.502vessel radiuscost per unit lengthleast total costpumping falls, upkeep risesthe sum has one minimum
Fig. 6 A tree whose every junction obeys the cube law by construction, checked at all sixty-three of them.

Where this goes next

growth 3.20× per turnrecovered 3.200× 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. 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. branch angle 30° · daughter ratio 1.0063 junctions checked The cube law A branching network built to move fluid for the least work obeys one relation at every junction — the cube of the parent radius equals the sum of the cubes of the daughters. It is a minimisation result, it is checkable on a real tree, and it is the rare biological rule with a derivation.