Repulsion — where it appears
The angle is an output
137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.
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.
Droplets with no biology in them
Douady and Couder dripped magnetised ferrofluid into a dish of silicone oil, and got spiral phyllotaxis with Fibonacci parastichy numbers out of a system containing no cells, no genes and no plant. That is the strongest evidence the pattern is physics — and the clearest warning about what a model can claim.
What a mechanism would have to show
This site says of every model it draws that reproducing a pattern is not explaining it. That is easy to repeat and hard to make precise. Here it is made precise — a list of what an account of phyllotaxis would have to establish, with each item marked according to whether the models on this site establish it.
Where the model stops
Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.
The tree and the attractor
The dynamical model settles on the golden angle over a range of one parameter and on the Lucas angle outside it, and it cannot reach the low-growth end at all. The lattice tree reaches everywhere, has no dynamics in it, and produces the same two angles as limits of two paths. Two routes, one pair of numbers.
A pattern with a rate
Every lattice on this site is a static object indexed by a parameter, and a plant is not. Put the rise on a clock, place each node where the repulsion from the ones below it is least, and the object that comes out has a history — which is the first thing here that could disagree with the ladder.
The lag that is not there
A pattern built out of its own history should hold its old parastichy pair past the point where a fresh lattice would have changed, and the gap should grow as the shoot is hurried. Over a fifteenfold range of rate it does not — every transition lands within a tenth of a rung of where the static ladder puts it.
The rate decides the branch
Forty nodes of Lucas lattice, carried down to the same fine rise twice. Hurried, the pattern holds the Lucas ladder through three more forks at 99.5°. Given room, it abandons it at the first fork it reaches and walks to 8/13 at 137.5°. Same seed, same rise, two ladders — with a threshold between them at about ninety nodes per rung.
Continuity from a coarse start
At a fine rise, one divergence in seven gives a Fibonacci pair. Grow a stem from a coarse start at a divergence nobody chose, down to that same rise, and sixteen runs out of sixteen end on 8/13. The expansion phase's interpretation was that continuity does the work; this is the measurement it never had.
How far a primordium reaches
The placement rule's repulsion falls as an inverse cube because that is what two magnetised droplets do, and nothing about a plant supplies the exponent. Asking what it controls produced one tidy wrong answer and one measured right one — and the difference between them is the difference between a total and a variation.
The exponent that barely matters
A code comment on this site claimed since its foundation phase that the repulsion's falloff exponent hardly changes the answer. It could not be tested, because the function that would have taken it never passed one down. Tested at last, it is true on a disc — by three and a half degrees across a sixteenfold range — and on a stem it decides whether there is a pattern at all.
A window that makes a pattern
A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.
Named alongside it
The objects these essays reach for when they reach for this one.
BranchRiseLadderFibonacciMeristemBifurcationDivergence angleAttractorConvergenceNoiseThe placement ruleTransitions