Plant Morphology: Lindenmayer’s Original Application

L-systems were not invented for computer graphics. They were invented to solve a problem in developmental biology: how to describe the sequence of cell divisions and differentiations that produce the characteristic structures of growing organisms. The fact that L-systems later became the standard tool for procedural vegetation is a consequence of their biological accuracy, not a coincidence.

Lindenmayer’s 1968 Model

Aristid Lindenmayer was studying Anabaena catenula, a filamentous cyanobacterium that grows as a chain of cells. The cells divide and differentiate in predictable sequences: a cell of type A divides into two cells of types A and B, while a cell of type B divides into two cells of types B and A. The result is a repeating pattern of cell types along the filament.

Lindenmayer wanted a mathematical notation that captured two features of this process. First, all cells in the filament divide simultaneously --- not one at a time. Second, the type of daughter cells depends on the type (and possibly the neighbors) of the mother cell.

The L-system he proposed in the Journal of Theoretical Biology had exactly these properties. The alphabet was the set of cell types. The axiom was the initial filament. The production rules specified how each cell type divides. Parallel application of the rules at each step models the simultaneous division of all cells.

For Anabaena, the context-sensitive L-system a < b -> a, b < a -> b (where the production depends on the left neighbor) accurately predicts the spatial pattern of cell types observed in real filaments under the microscope. This was not a visual approximation --- it was a formal model of the developmental program, validated against observed cell-type sequences.

From Cell Sequences to Branching Structures

Lindenmayer’s original model was one-dimensional: a string of cell types representing a linear filament. The extension to branching structures came through two innovations.

First, the bracket notation [ and ] introduced by Prusinkiewicz allowed the string to encode branching topology. The symbol [ marks the beginning of a branch; ] marks its end and returns to the parent stem. A string like F[+F][-F]F encodes a main stem (the outer F’s) with two lateral branches.

Second, the turtle graphics interpretation (described in the grammar page) converted strings into geometric structures in two or three dimensions. This allowed the same formal apparatus --- alphabets, axioms, productions --- to generate visual output that could be compared directly with photographs of real plants.

The combination was developed primarily by Prusinkiewicz and his collaborators at the University of Calgary throughout the 1980s and 1990s. Their 1990 book The Algorithmic Beauty of Plants demonstrated L-system derivations for dozens of specific plant species, establishing L-systems as a standard tool in both computational biology and computer graphics.

Fitting Models to Real Plants

The methodology for connecting L-system models to real plants involves three steps.

Measurement. Collect morphological data from real plants: internode lengths (the distance between branching points), branching angles, the number and arrangement of branches at each node, the diameter of stems at different levels, and the phyllotactic pattern (the angular arrangement of leaves or branches around the stem).

Rule inference. Identify the production rules that generate a branching structure consistent with the measurements. This is inverse modeling: given the output geometry, find the grammar. In practice, it is done by trial and refinement rather than algorithmic inversion, because the mapping from grammar to geometry is many-to-one (multiple grammars can produce similar geometry).

Statistical validation. Generate a population of virtual plants from the fitted grammar (using stochastic productions to introduce variation) and compare the statistical distributions of morphological features (internode length distribution, branching angle distribution, total height distribution) to the corresponding distributions measured in a real plant population.

Prusinkiewicz, Hammel, Hanan, and Mech (1997) described this approach as the “virtual laboratory” method. Their results showed that stochastic parametric L-systems could reproduce the visual appearance and statistical distributions of specific species --- Lychnis coronaria, Mycelis muralis, and several tree species --- to a level of accuracy that botanical experts found convincing.

What fits well: branching topology (the pattern of which branches go where), allometric relationships (the proportional changes in segment length and width with branching order), and overall silhouette shape.

What requires additional mechanisms: tropisms (growth toward light or gravity), competitive shading (branches in the shade grow differently from branches in sunlight), and resource transport (a branch’s growth depends on nutrient supply from the root system). Pure L-systems model geometry, not physics. Functional-structural plant models (FSPM) couple L-systems with models of carbon allocation, water transport, and light interception. These are no longer simple L-systems, but they retain the L-system as the structural backbone.

Phyllotaxis: Fibonacci and Spirals

The most visually striking pattern in plant morphology is phyllotaxis: the arrangement of leaves, florets, and scales in spiral patterns related to the Fibonacci sequence. Sunflower heads have 34 spirals in one direction and 55 in the other. Pine cones have 8 and 13. The ratio between successive Fibonacci numbers converges to the golden ratio phi = (1 + sqrt(5))/2 approximately equal to 1.618, and the angular separation between successive leaves on a stem converges to the golden angle 360/phi^2 approximately equal to 137.5 degrees.

The mechanism was clarified by Douady and Couder (1992) in a physical experiment: drops of ferrofluid placed sequentially on a dish, with mutual repulsion, spontaneously arrange in Fibonacci spirals. The pattern arises because each new element (leaf primordium, floret, ferrofluid drop) forms in the position farthest from all existing elements. The golden angle is the angular interval that maximizes the minimum distance between successive elements --- it is the most “spread out” arrangement possible.

L-system models incorporate phyllotaxis by setting the angular divergence between successive lateral branches to the golden angle. Parametric L-systems modeling the apical meristem (the growing tip of a plant shoot) with an inhibitory field around each new primordium reproduce the Fibonacci spiral pattern. The connection between L-systems and phyllotaxis is not ad hoc; it follows from the same developmental logic: local rules (place the next primordium where inhibition is lowest) applied iteratively at the growing tip produce a global pattern (Fibonacci spirals) that no individual step planned.

Prusinkiewicz and colleagues at Calgary have connected L-system phyllotaxis models to molecular data on auxin transport in the meristem. The hormone auxin accumulates at sites of low auxin concentration in neighboring cells, effectively implementing a local-maximum rule that matches the Douady-Couder mechanism. This connection between L-system geometry, physics experiments, and molecular biology is one of the strongest validations that L-systems capture genuine developmental mechanisms rather than merely producing visual approximations.

The caveat: the Fibonacci pattern in plants is an observation, not a theorem. Not all plants show Fibonacci phyllotaxis --- some have opposite leaves (180 degrees), whorled arrangements (120 degrees or 90 degrees), or distichous patterns (alternating sides). The L-system framework can model all of these patterns with appropriate parameter choices, but the prevalence of Fibonacci arrangements specifically requires an explanation rooted in developmental constraint, not just formal capacity.

Further Reading