Model Description

A faithful, code-aligned description of the inhibition-field model behind the simulator.

TL;DR — how it works (one screen)

New primordia are born on a fixed ring at radius R.
Each step all primordia drift outward by v * dt.
The next angle is chosen by minimizing an inhibition field F(θ) = Σ K(d) across angleSamples candidates.
The discrete argmin is refined by a parabolic refinement to obtain θ*.
Optional Gaussian noise perturbs θ*; divergence mean/std are reported.

Overview (what we model)

We model a growing shoot apex as a 2D disk with a birth ring of radius $R$ . Primordia are point-like organs defined by polar coordinates $(r,\theta)$ . They are created on the ring ( $r=R$ ) and then drift outward over time. Each existing primordium contributes an inhibitory influence. A new primordium appears at the angle where the total inhibition is minimal.

This is an inhibition‑field model in the spirit of Douady–Couder, implemented as a fast discrete optimizer over candidate angles.

[!KEY IDEA] Spiral counts emerge from repeated argmin of an inhibition field on the birth ring while older primordia drift outward.

Diagram A — Choosing θ*: birth ring, active primordia, distances dᵢ

The field value at angle θ is computed by summing kernel contributions from active primordia using distances dᵢ(θ) from the ring point c(θ).

Objects & Parameters

Geometry and time

$R$ : birth ring radius (cfg.R)
$v$ : radial drift speed (cfg.v)
$dt$ : time step (cfg.dt)
$\texttt{maxR}$ : cutoff radius for "active" primordia (cfg.maxR)

Sampling / search

$\texttt{angleSamples}$ : number of discrete candidate angles on $[0,2\pi)$ (cfg.angleSamples)
Parabolic refinement: sub-sample minimizer around the best discrete sample

Kernel (inhibition strength vs distance)

The kernel configuration is a tagged union:

exp: $K(d) = A\,e^{-d/\lambda}$
gaussian: $K(d) = A\,e^{-d^2/(2\sigma^2)}$
softPower: $K(d) = A / (d^p + \varepsilon)$
hardCoreExp: exp kernel + hard exclusion for $d < d_0$
custom: user expression $K(d)=f(d,\text{params})$ via expr-eval

Noise

Optional angular noise applied after $\theta^*$ is computed:

enabled (cfg.noise.enabled)
$\sigma_\theta$ in degrees (cfg.noise.sigmaThetaDeg)
seed (cfg.noise.seed) for determinism

[!TRY THIS] Set noise.enabled=false to see a near‑deterministic convergence of divergence statistics; then enable noise with $\sigma_\theta \approx 0.1^\circ$ to observe a broader divergence distribution.

Step algorithm

Each simulation step repeats:

Drift — all primordia move outward: $r \leftarrow r + v \cdot dt$
Update active set — keep only primordia with $r \le \mathrm{maxR}$
Compute inhibition field — for each candidate angle, sum kernel contributions from active primordia
Find θ* — pick the angle that minimizes $F(\theta)$
Add primordium — place new point at $(R,\, \theta^*)$

Optional Gaussian noise can be added to θ* before placement.

Diagram C — Active set: primordia within maxR are included

In the code, the active set is implemented as a moving suffix index (firstActiveIndex) because radii increase monotonically with age under constant drift.

Diagram B — Field F(θ); minimum at θ*

The placement angle θ* is the argmin of F(θ) over the birth ring.

Inhibition field & argmin(θ)

For a candidate angle $\theta$ , define the candidate point on the ring:

$c(\theta) = (R\cos\theta,\ R\sin\theta).$

Each active primordium $j$ has Cartesian position:

$x_j = r_j\cos\theta_j,\quad y_j = r_j\sin\theta_j.$

Distance to the candidate:

$d_j(\theta) = \sqrt{(R\cos\theta - x_j)^2 + (R\sin\theta - y_j)^2}.$

Total inhibition field:

$F(\theta)=\sum_{j \in \text{active}} K(d_j(\theta)).$

Placement rule:

$\theta^* = \arg\min_{\theta \in [0,2\pi)} F(\theta).$

In code, the optimizer is "grid search + parabolic refinement":

[!IMPLEMENTATION MAPPING]

Discrete scan and argmin: src/sim/simulator.ts → findMinimumFieldAngle()

Refinement uses $F(\theta_{i-1}),F(\theta_i),F(\theta_{i+1})$ and clamps offset to $[-0.5,0.5]$

Kernels

exp

$K(d)=A\exp(-d/\lambda)$

gaussian

$K(d)=A\exp\left(-\frac{d^2}{2\sigma^2}\right)$

softPower

$K(d)=\frac{A}{d^p+\varepsilon}$

hardCoreExp

Same as exp, but invalidates a candidate angle if any active primordium is closer than $d_0$ :

$\exists j:\ d_j(\theta) < d_0 \Rightarrow F(\theta)=\text{HUGE\_PENALTY}.$

custom

Users can provide an expression string like:

A * exp(-d / lambda)

The simulator evaluates it with a stable parameter scope and safety guards (exceptions / NaN / Infinity → penalty).

[!IMPLEMENTATION MAPPING] src/sim/kernel.ts → compileKernel() (all kernel types) and validateExpression() (custom kernels).

Metrics (divergence mean/std)

The simulator reports divergence angle statistics over the most recent $N$ steps (default $N=200$ ):

$\Delta_n = \mathrm{wrap}_{[0,2\pi)}(\theta_n - \theta_{n-1}),\quad \text{mean} = \frac{1}{N}\sum \Delta_n,\quad \text{std} = \sqrt{\frac{1}{N}\sum (\Delta_n-\text{mean})^2}.$

[!IMPLEMENTATION MAPPING] src/sim/simulator.ts → computeDivergenceMetrics(N=200) returns {mean, stdDev, count} in degrees.

Why you sometimes see 19 spirals

Spiral counts (parastichies) are tied to how the emergent divergence angle $\alpha$ is approximated by rationals.

If $\alpha/(2\pi)$ is close to a rational $p/q$ , you get near‑alignments every $q$ steps, which can make a $q$ -family of spirals visually prominent.
The golden angle corresponds to the "most irrational" number (continued fraction $[1;1,1,1,\dots]$ ), whose convergents are Fibonacci ratios — hence Fibonacci spiral pairs dominate.
With different kernel reach, maxR, sampling resolution, or noise, the effective $\alpha$ can drift toward other good rational approximations, temporarily highlighting non‑Fibonacci counts (like 19).

[!KEY IDEA] The simulator doesn't "pick Fibonacci numbers". It repeatedly minimizes $F(\theta)$ ; visible spiral counts are an emergent geometric consequence of the resulting divergence statistics.

Mapping to code

Core algorithm and data structures:

Simulation step: phyllotaxis-modelling/src/sim/simulator.ts → step()
Drift: driftAllPrimordia()
Active set as suffix: firstActiveIndex + advanceFirstActiveIndex()
Active arrays: buildActiveArrays() stores dense activeXs/activeYs for speed
Argmin field + refinement: findMinimumFieldAngle()
Noise: GaussianRNG in src/sim/prng.ts + wrapAngle()
Kernel compilation: src/sim/kernel.ts
Config & presets: src/sim/config.ts (defaultConfig(), PRESETS)
Validation: src/sim/validation.ts

UI hooks (for context):

src/ui/ui.ts wires controls, import/export, and field plot toggles.

Performance notes

This implementation is optimized for a tight inner loop:

Candidate angle $\cos/\sin$ arrays are precomputed once at init (candidateCos, candidateSin).
Active set is a moving suffix (older primordia drift outward monotonically).
Active coordinates are stored in dense Float64Arrays each step for cache-friendly iteration.
Hard-core kernel can early-exit with a large penalty.

FAQ

What does the kernel actually do?

The kernel $K(d)$ sets how strongly a primordium inhibits candidate positions at distance $d$ . Changing the kernel shape (and its length scale like $\lambda$ or $\sigma$ ) changes how many primordia matter at once and how sharp the minimum of $F(θ)$ is.

Why do we need angle sampling + refinement?

We approximate $\arg\min F(θ)$ by scanning a discrete grid of angles for speed, then recover sub-grid precision by fitting a parabola to the minimum sample and its two neighbors.

References

How these papers relate to the model

Model feature	Key references
Argmin on inhibition field ("least crowded spot")	Hofmeister → Douady & Couder
Kernel $K(d)$ (local repulsion)	Douady & Couder, Levitov, Adler
Golden angle / Fibonacci emergence	Douady & Couder, Adler, Levitov
Mathematical lattice theory	van Iterson
L-systems / computational approach	Prusinkiewicz & Lindenmayer

Core bibliography

Hofmeister, W. (1868). Allgemeine Morphologie der Gewächse. Leipzig: Engelmann. — First statement of the rule: each primordium forms in the "least crowded" spot on the meristem.
van Iterson, G. (1907). Mathematische und mikroskopisch-anatomische Studien über Blattstellungen. Jena: Fischer. — First purely mathematical treatment; introduced cylindrical lattice packing that underlies spiral counts.
Adler, I. (1974). A model of contact pressure in phyllotaxis. Journal of Theoretical Biology, 45, 1–79. — Shows that contact pressure leads to Fibonacci progressions and convergence to the golden angle.
Levitov, L. S. (1991). Energetic approach to phyllotaxis. Europhysics Letters, 14(6), 533–539. — Variational/energy-minimization derivation of why noble numbers (including golden ratio) are selected.
Douady, S. & Couder, Y. (1992). Phyllotaxis as a physical self-organized growth process. Physical Review Letters, 68(13), 2098–2101. — Landmark experiment: ferrofluid droplets on a ring spontaneously produce Fibonacci spirals via local inhibition.
Douady, S. & Couder, Y. (1996). Phyllotaxis as a dynamical self-organizing process (Parts I–III). Journal of Theoretical Biology, 178, 255–312. — Full theoretical model: iterative placement, bifurcation diagrams, and the "Hofmeister rule" formalized.
Prusinkiewicz, P. & Lindenmayer, A. (1990). The Algorithmic Beauty of Plants. New York: Springer-Verlag. — Computational models (L-systems) for plant morphogenesis including phyllotaxis.