The bifurcation diagram

The rule has one parameter. It is the ratio of how fast the elements drift outward to how often a new one appears — how much room the meristem makes between primordia — and everything else about the model is fixed.

Sweeping it produces the diagram this site is arranged around.

What is on it #

The golden branch. Over a wide range of growth rates the model settles within a few degrees of 137.5°. This is the regime that produces sunflowers, pine cones, pineapples and most of what gets photographed, and its breadth is the reason the golden angle is common rather than rare.

The transition. Above the golden branch the settled angle climbs, passing through values that belong to no named sequence.

The two-whorl regime. At high growth rates the model settles on exactly 180°, and stays there. Each new element goes directly opposite the last. That is distichous phyllotaxis — the alternating leaves of a grass, a maple, an elm — and it is not a degenerate case or a failure of the model. It is the commonest leaf arrangement there is.

So the diagram contains, in one sweep of one parameter, both the pattern that gets called mathematically remarkable and the ordinary alternating arrangement of a garden shrub. They are the same rule at different growth rates.

Why a diagram beats a constant #

The popular telling gives a number and an air of mystery. The diagram gives a mechanism and a prediction.

It explains why the golden angle is common. Its branch is broad. A plant does not need to hit a precise growth rate to land on it; a wide range of rates gives the same answer, which is exactly what a robust developmental outcome should look like.

It explains why other angles occur. Lucas-number phyllotaxis is a real and repeatedly observed minority pattern, and a model that only produced the golden angle could not account for it. A model with branches can.

It predicts what changes the pattern. Anything that alters the ratio of primordium spacing to meristem size should move a plant along the diagram — and it does. Changing the growth conditions of a real plant can shift its phyllotaxis, and the transitions in a single plant’s life, as the meristem changes size, are a walk along this axis.

It says what to measure. A constant says what to look for; a diagram says what to vary.

Hysteresis, and the path taken #

One feature of the real model that the sweep here does not show, and which is worth naming because it matters biologically.

In Douady and Couder’s full treatment the system shows hysteresis: which branch it lands on can depend on the path taken to get there, not just on the current parameter value. A meristem that grows gradually from a high growth rate down through the transition stays on one branch; one that starts in the middle may land on another.

That is the mechanism behind the observed sequences in real plants, where a shoot’s phyllotaxis rises through the Fibonacci sequence as it develops rather than jumping to a final value. The pattern is not chosen once; it is inherited from the pattern that preceded it.

This site’s sweep restarts the model from scratch at every parameter value, so it shows the branches without the history. That is a limitation and it is worth stating: the diagram here is the set of destinations, not the map of which one a given plant reaches.

The Lucas branch #

The literature reports a Lucas-number branch — parastichy pairs of 4 and 7, 11 and 18, 29 and 47 — occurring in a small but consistent minority of real plants, and corresponding to a divergence angle near 99.5°.

The sweep here does not resolve it cleanly, and that is an honest limitation of a coarse implementation rather than evidence against it. A Lucas branch is narrower than the golden one, so a sweep at this resolution can step over it, and the discrete boundary sampling blurs exactly the distinctions that separate nearby branches.

What the site can show is that the Lucas angle produces Lucas counts — that a head built at 99.5° gives 47 and 76, neither of which is a Fibonacci number. That is enough to establish the point the diagram is making, which is that Fibonacci is a consequence of one branch and not a law of plants.

Where the diagram stops #

The sweep starts at a growth parameter of 0.18 and that is not an aesthetic choice.

Below it this implementation stops converging: the elements barely move between steps, several sit on the meristem boundary at once, and the settled angle wanders over a hundred degrees however long the run. The low-growth regime is precisely where the literature says the interesting high-order Fibonacci behaviour lives, and this model cannot reach it.

That is a real limit and it gets a figure of its own rather than a quietly chosen axis range.

Reading the branches #

The diagram has structure worth naming, because a bifurcation diagram is a picture of an answer set rather than an answer.

The golden branch occupies most of the parameter range and is the wide plateau near 137.5°. Starting the model from an empty meristem at almost any G in that region and letting it settle produces this branch, which is what makes 137.5° the common case rather than a special one.

The Lucas branch appears as a second, thinner curve near 99.5°. It is reached from different initial conditions at the same G — the branches coexist rather than succeeding each other — and it is a real attractor rather than a transient. Its spiral counts are Lucas numbers, which is how one recognises it in a plant rather than in a plot.

The whorled regime sits at high G, where the settled angle is exactly 180° and elements appear in opposite pairs. This is distichous phyllotaxis — the alternating leaves of a grass — and it is not an approximation to anything. The model lands on half a turn precisely, because with that much room per element the previous one is the only one that matters and the best place is directly opposite it.

The transition between the spiral regimes and the whorled one is where the diagram earns its keep. It is not gradual: the settled angle jumps.

Why a diagram beats a constant #

The usual telling of this subject offers a number. The model offers a map, and the difference matters in three ways.

A constant cannot be falsified usefully. A plant with a divergence angle of 99.5° either refutes “the angle is 137.5°” or is dismissed as an exception, and neither response is informative. The same plant is a prediction of the diagram — a specific branch at a specific parameter range.

A constant cannot explain variation. Real plants show a spread of angles, and the spread has structure: certain values are common, the ones between them are rare. That is exactly what a branch diagram looks like when sampled, and it is not what noise around a constant looks like.

A constant hides the parameter. G is the ratio of radial drift to plastochron — how fast the meristem makes room relative to how fast it makes primordia — and it is a physiological quantity that differs between species and changes during a single plant’s development. The counts changing along a radius is that change, seen from the outside.

What the parameter is, physically #

G is not an abstraction, and it is worth pinning down.

The meristem produces primordia at intervals — the plastochron — and each primordium, once formed, drifts outward as the tip grows. G measures how far a primordium gets before the next one appears, relative to the size of the meristem.

Small G: primordia appear faster than they can get out of the way, so the ring is crowded, each new one interacts with many predecessors, and the settled angle comes from a compromise among them. This is the high-count end.

Large G: each primordium is well clear before the next appears, so the new one sees essentially only its immediate predecessor and goes opposite it. This is the whorled end.

The whole diagram is a statement about how many neighbours matter, and the branches are the different ways a system settles when the answer to that question changes.

The low-growth edge #

The left end of the sweep is where this implementation stops, and it is plotted rather than trimmed.

Below about G = 0.18 the discrete model does not settle within the iteration budget. That is a property of this implementation — a sampled circle, a finite interaction window, a fixed number of steps — rather than of the model, and the continuous formulation is well behaved there.

It is also, awkwardly, where the literature’s Fibonacci ladder lives: the classic result is that as G decreases the system climbs through successive Fibonacci pairs, 21/34 giving way to 34/55 and so on, and that ladder is in the region this sweep cannot resolve.

Showing the boundary rather than cropping to the region that works is the honest option and is what the essay on where the model stops is about. A diagram that ended at 0.18 with no explanation would read as a complete picture of a model that has more in it than this.

What a bifurcation diagram is, generally #

Worth a paragraph for readers meeting the object rather than the subject, because the diagram’s structure carries meaning independent of phyllotaxis.

A dynamical system with a parameter has, for each parameter value, some set of states it settles into. Plotting those settled states against the parameter gives the diagram. A branch is a curve of settled states; a bifurcation is a parameter value where the number or stability of branches changes.

The famous example is the logistic map, whose diagram period-doubles into chaos. This one is tamer: branches that coexist over wide ranges, with transitions where one loses stability.

The general lesson from either is the same and it is why the object is worth knowing. A system with a parameter does not have an answer; it has an answer set. Reporting one value of one branch as “the” behaviour is a category error, and it is what quoting 137.5° without the diagram does.

Coexistence, and what it needs from the model #

The two spiral branches overlapping in parameter is the structurally interesting part, and it is worth being clear about what produces it.

If the branches merely succeeded one another — golden below some G, Lucas above — the diagram would be a description of the parameter and nothing more, and a plant’s branch would be determined by its growth rate.

They overlap. At a single G both are stable, and which one a run reaches depends on the initial condition: the angle between the first two primordia, before any pattern exists. That makes the branch a matter of history rather than of physiology, and it is why the Lucas branch is rare rather than confined to unusual plants — its basin is smaller, not its parameter range.

This is checkable in the model: fix G, sweep the seed angle, and record which branch each run lands on. The basins are visible as intervals, and the golden one is wider.

The sweep, and what it costs #

An implementation note, since a reader reproducing this will hit it.

Each point on the diagram is a full run of the model — a hundred or so placements, each requiring the repulsion field to be evaluated at 720 boundary samples against a window of previous elements. A sweep of a few dozen parameter values takes about a second, which is fine once and not fine when the same figure appears on several pages and is regenerated for every drag frame.

So the results are memoised by parameter, and the memo is shared across figures. The settling figure and the diagram use the same cached runs, which also means they cannot silently disagree — a pleasant side effect of caching for speed.

The general form: on a site whose figures are computed rather than drawn, an expensive computation appearing in several figures should be computed once and keyed, both because it is faster and because it makes the figures consistent by construction.

What to do with the diagram #

Three uses, in decreasing order of how often they get made.

Locate a plant on it. Recover the divergence angle from counts and a radius and see which branch it is on. That turns “this specimen is anomalous” into “this specimen is on the second branch”, which is a statement with consequences.

Predict what should be rare. The branches’ basin widths order the frequencies, and the ordering is checkable against a survey of measured angles — the only prediction on this site that would take a season of fieldwork to test.

Read off what the parameter would have to do. A plant whose counts rise through development is a plant whose G is falling, and the diagram says by how much.

None of those is available from a constant, which is the argument for the diagram restated as a list of things one can do with it.

The whorled end, and why it is exact #

The high-growth regime deserves more than a mention, because it is the one place on the diagram where the model produces an exact value and the exactness is informative.

At large G the previous element has moved well clear before the next appears, so the repulsion the new element feels is dominated by exactly one predecessor. The minimum of a single repulsive source on a circle is directly opposite it, and the angle is 180° with nothing to approximate.

That is distichous phyllotaxis — the alternating leaves of a grass, an iris, a maple — and it is the commonest arrangement in plants after spiral. It is usually treated as a separate phenomenon from spiral phyllotaxis and given its own explanation. On the diagram it is the same rule at a different setting.

Reaching two apparently unrelated botanical arrangements from one parameter is the strongest thing the diagram does, and it is invisible if the model is quoted only for the value it produces in the middle of its range.

What the diagram does not contain #

Three absences, so the object is not over-read.

It contains no biology. G is a ratio of rates and the rule is a repulsion; nothing in it is a cell, a hormone or a gene, which is precisely what the ferrofluid experiment established was sufficient.

It contains no arithmetic. The branches land where they land, and the fact that the main one sits at a maximally irrational value is a separate measurement that the sweep knows nothing about.

And it contains no counts. The spiral numbers everyone quotes are downstream — they follow from the angle and the radius one counts at, and the diagram would look identical if Fibonacci numbers had never been named.

Three things kept apart that popular accounts merge into a single story about plants using the golden ratio. Separated, each one can be checked.

One knob, and the other one nobody turned #

This diagram is a sweep of the growth parameter, and until the earlier work it was the only parameter of the rule that had ever been swept. The repulsion’s falloff exponent — fixed at an inverse cube, because that is the interaction between two magnetised ferrofluid droplets — was not reachable: the function that runs the model took no exponent and passed none down.

Threaded through and swept, it moves the settled angle by three and a half degrees across a sixteenfold range, from 139.50° at an exponent of a half to 136.00° at eight, without ever leaving the golden branch.

That is a licence for this diagram and a warning about generalising from it. The figure is robust to the one arbitrary constant in the rule, which had been assumed rather than known. And it is robust because of the disc’s geometry — on a stem, where the neighbours accumulate linearly with distance instead of logarithmically, the same exponent decides whether there is a pattern at all.