Phyllotaxis
the spirals are counted, not admired
Concept

Growth parameter — where it appears

The single quantity a placement rule is turned by, standing for how fast the pattern expands relative to how fast organs arrive. The golden angle appears as the settled output over a range of it, which is the result the bifurcation diagram is about.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The angle between the first two peaks, for six starting disorders. The same equations, the same ring, six different starting perturbations: 177°, 47°, 109°, 151°, 47°, 133°. A spiral needs the angle between successive elements to be the same one each time, and a stationary ring does not supply that.

A ring cannot make a spiral

The peaks on a Turing ring do not all appear at once — there is a first and a second. But which two lead is decided by the starting disorder, so the angle between them comes out at 177°, then 47°, then 109°, then 151°. A divergence angle is a relationship that repeats, and this one does not.

mechanism · Ring limit What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.

The bifurcation diagram

Sweep the one parameter of the rule and the settled angle traces a diagram — a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of it, which is a more useful thing to know than the constant.

emergence · Bifurcation

Named alongside it

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

BifurcationDivergence angleRateBranchCylinderDispersion relationDistichousEmergence orderFibonacciInitial conditionMeristemPrimordium

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