Phyllotaxis
the spirals are counted, not admired
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.
5 min read 5 figures Form is not mechanismState the claim, state the test

Start with the string X. Replace every X with F[+X]F[-X]+X and every F with FF. Do it five times. Hand the result to a turtle that reads F as “go forward”, + and − as “turn”, and brackets as “remember and restore where one are”.

Four hundred and twenty-two line segments come out, and they look like a plant.

An L-system after 4 rewrites of two rulesX → F[+X]F[-X]+X and F → FF, walked by a turtle turning 22.5°. 130 segments, and not one of them knows anything about light, water or auxin.491 symbols · 130 segmentsa description, not a mechanism
Fig. 1 Five rewrites of two rules, walked by a turtle. Nothing in the rules refers to light, gravity, water or hormones.

What is genuinely good about them

Lindenmayer introduced these in 1968 as a model of how filamentous organisms develop, and they have real virtues.

They capture self-similarity honestly. A plant’s branching repeats at several scales, and a rewriting rule is exactly a statement about a structure that contains copies of itself.

They are compact. Two rules and an iteration count produce arbitrarily complex structures, which matches the observation that a genome does not store a tree’s shape.

They are parametric. Adding parameters to the symbols — a length, an angle, an age — turns a shape grammar into something that can respond to conditions, and open L-systems can take input from a simulated environment.

They are used. Plant modelling for graphics, for agronomy, and for architecture runs on them, and the results are good.

What they do not do

They contain no mechanism whatsoever.

The turn angle is a parameter, not a consequence of anything. The branching ratio is chosen. Nothing competes for light, nothing is limited by water, nothing has to support its own weight, and no signal passes between parts. A tree that grew by these rules would be indifferent to being planted upside down.

That is not a criticism of L-systems, which were never claimed to be mechanistic. It is a criticism of what happens next: a picture this convincing invites the conclusion that the rules capture something about how the plant works, and it does not follow.

The general problem

Patterns are cheap, and visual conviction is a poor guide to explanatory power.

A great many rules produce branching trees. A great many produce spirals. A great many produce hexagonal tilings. Matching an output constrains the space of possible mechanisms far less than it feels like it does, and the more striking the match, the stronger the false impression.

This is the same caution that applies to the ferrofluid experiment, which produces spiral phyllotaxis with no biology in it, and to the dynamical model, which produces the golden angle without being told it. In every case the honest reading is the same: the model shows what kind of process could produce the pattern, and narrows the field of candidate mechanisms without selecting one.

What distinguishes a description from a mechanism

A useful test, since the distinction is doing so much work.

A description reproduces the output. Change the conditions and it has nothing to say unless one change the description too.

A mechanism predicts the response. Change the light, the water, the gravity, the hormone, and it says what should happen — and can be wrong.

By that test, Murray’s law is a mechanism: it derives a relation from stated costs and predicts what happens to a junction when the flow changes. An L-system is a description: it reproduces a shape and says nothing about what would happen to the plant in shade.

Both are worth having. The error is treating the second as though it were the first because its output is prettier.

Where the line actually falls

It is worth being fair to the modern versions, which are less naive than the classic ones.

Environmentally-sensitive L-systems let the rewriting depend on simulated light or space, so a branch that finds itself shaded develops differently. That is a genuine step toward mechanism, because the response to a condition is now in the model rather than in the parameters.

The line is not between L-systems and other formalisms; it is between models whose parameters encode the answer and models that derive it. A rewriting rule with a fixed branch angle encodes it. A model in which the angle emerges from competition for light derives it — and could be wrong about a plant, which is what makes it worth testing.

The two costs, and where their sum is leastPumping cost falls as r⁻⁴ and upkeep rises as r². Minimising the sum gives flow proportional to r³ — checked here across a fourfold range of flow, where Q/r³ holds to 3.9%.010200.50011.502vessel radiuscost per unit lengthleast total costpumping falls, upkeep risesthe sum has one minimum
Fig. 2 A tree whose every junction obeys the cube law by construction, checked at all sixty-three of them.
The exponent fitted from the junctions, rather than assumedSweeping k and asking where r₀ᵏ = Σ rᵢᵏ holds best gives 3.000 — Murray's 3, recovered rather than imposed.02468234exponent khow badly r₀ᵏ = Σ rᵢᵏ fails at that kk = 3127 junctionsfitted k = 3.000
Fig. 3 The exponent fitted from the junctions rather than assumed, which returns three on a tree built to obey it.
Two runs of the same rule from unrelated starting anglesBoth settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°
Fig. 4 The rule itself, part way through a run. The next element goes to the minimum of the repulsion curve, and nothing in the rule names an angle.
How nearly each angle is a simple fraction of a turnA dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches 0e+0.00.2000.400204060denominator qhow nearly q × angle is a whole turn (0 = exactly)golden 137.508°137.3°135° = 3/8distance to the nearest whole turnno dips means no rows
Fig. 5 The pattern itself, generated from a stated angle. Nothing is placed by hand, so every claim about it is a claim about the rule.

Where this goes next

1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00° 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. branch angle 30° · daughter ratio 1.0063 junctions checked 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. 05e+51e+62e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.40 · 14 elements in playthe minimum is where the next one goes 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.