Phyllotaxis
the spirals are counted, not admired
Generator

morphospace

The 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.
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

Drawn at its defaults, in shells and growth. It takes no options at all, so every essay calling it gets this exact drawing.

Called by 7 essays

the blast radius of changing it

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.

 Branching and transport

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.

 Branching and transport

Fitting the exponent

Assuming the exponent is three and reporting the error tells you how wrong the data is. Fitting the exponent and reporting what it comes out as tells you what the network is doing — and the machinery has to be shown returning something other than three, or it is not a fit.

 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.

 The claims, measured

The claim that survives

Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.

 Branching and transport

L-systems describe, they do not explain

Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing.

 Shells and 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.

All 105 generators