Phyllotaxis: Living Lattice

1Overview

The simulator implements a geometric model of phyllotaxis based on Vogel's formula (1979). This model generates the characteristic spiral patterns observed in sunflower heads, pinecones, and succulent rosettes by positioning elements at regular angular intervals from a central point.

Each element's position is determined by two factors: its index in the sequence (which controls radial distance) and the divergence angle (which controls angular placement). By varying these parameters, we can reproduce the full range of natural phyllotactic patterns.

Core Principle

The model treats each primordium as a point in polar coordinates, where the angle increases by a constant divergence angle and the radius grows according to a scaling function.

2Mathematical Formulation

For each element n in the sequence, its position in polar coordinates (r, θ) is calculated as:

θ_n = n × α

r_n = c × √n

Where α is the divergence angle (typically the golden angle ≈ 137.508°), n is the element index, andc is a scaling constant that controls the overall size.

The square root relationship for the radius ensures that elements are packed with approximately equal density throughout the pattern, mimicking how plant organs maintain consistent spacing as they grow outward.

Key Equation

The golden angle α = 360° / φ² ≈ 137.508°, where φ (phi) is the golden ratio (1 + √5) / 2 ≈ 1.618.

3Simulation Parameters

The simulator exposes several parameters that allow exploration of different phyllotactic patterns:

Divergence Angle (α)

The angle between successive elements. Default is the golden angle (137.508°). Small deviations produce dramatically different patterns, demonstrating why the golden angle is evolutionarily optimal.

Number of Elements (N)

Total count of primordia to generate. Higher values reveal more complete spiral patterns and make Fibonacci numbers more apparent.

Scaling Factor (c)

Controls the spacing between elements. Larger values spread elements further apart; smaller values create tighter packing.

Element Size

The radius of each primordium. Can be constant or vary with distance from center to simulate growth patterns where outer elements are larger.

Try This

Set the divergence angle to exactly 137.5° and watch perfect Fibonacci spirals emerge. Then adjust by just 0.1° and observe how the pattern degrades into visible gaps or alignments.

4Spiral Counting Algorithm

The simulator includes an algorithm for counting visible spirals (parastichies). Two families of spirals exist in any phyllotactic pattern: one running clockwise and one counter-clockwise. The counts of each family are typically consecutive Fibonacci numbers.

The algorithm works by identifying nearest neighbors for each element and grouping elements that form continuous chains. For a pattern with divergence angle α, the expected spiral counts can be predicted from the continued fraction expansion of α/360°.

Example: Sunflower Head

A typical sunflower displays 34 clockwise spirals and 55 counter-clockwise spirals (or vice versa). These are consecutive Fibonacci numbers, a direct consequence of the golden angle divergence.

Mathematical Connection

The Fibonacci spiral counts arise because the golden angle divides 360° in the ratio 1:φ, and the convergents of the continued fraction for 1/φ are ratios of consecutive Fibonacci numbers.

5Rendering Implementation

The simulator renders patterns using SVG for precise geometric output. Each element is converted from polar to Cartesian coordinates:

x = r × cos(θ)

y = r × sin(θ)

Color mapping applies hue variation based on angular position, creating the characteristic spiral color bands. Optional spiral highlighting renders connecting lines between elements belonging to the same parastichy, making the Fibonacci structure visually explicit.

Visual Enhancement

The simulator uses color gradients that follow angular position, making spiral families visually distinct and easier to count.

6Limitations and Extensions

This geometric model captures the spatial arrangement of phyllotaxis but simplifies several biological realities. Real plants exhibit variation in element placement, three-dimensional curvature, and growth dynamics that this 2D static model does not capture.

Possible extensions include: time-based animation showing growth from center outward, 3D projections onto spherical or conical surfaces, and physically-based models incorporating auxin transport dynamics.

Future Directions

More sophisticated models simulate the actual biochemical processes (auxin transport, cell division) that generate phyllotactic patterns, rather than just reproducing the geometric outcome.

Model Description