Growth factor — where it appears
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.
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 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Logarithmic spiralWhorlBranchCensusClaim testingEvoluteFibonacciGnomonic growthφ, the golden ratioGolden spiralInvoluteLeast squares fit