Phyllotaxis

Spiral leaf arrangement as a propagator network where the golden angle emerges as Nash equilibrium.

Trit: +1 (PLUS - Generator) Color: #3DD98F (green — growth)

Core Insight

The Douady-Couder model of phyllotaxis IS a propagator network (SDF Ch7):

• Cells = primordia (partial information: position, auxin, inhibition)
• Propagators = Gaussian inhibition kernels (constraint transformers)
• Scheduler = plastochron clock (SICP Ch5 explicit control)
• Quiescence = golden angle spiral (Nash equilibrium)

The golden angle (137.508°) is not designed — it EMERGES from the constraint network, just as Nash equilibria emerge from propagator-based game solving.

Architecture

                    ┌─────────────────────────────────────┐
                    │        Phyllotaxis Propagator        │
                    │         (SDF Ch7 × Nash)             │
                    └──────────────┬──────────────────────┘
                                   │
          ┌────────────────────────┼────────────────────────┐
          │                        │                        │
    ┌─────▼──────┐          ┌──────▼──────┐          ┌──────▼──────┐
    │   Julia     │          │  TypeScript  │          │   Scheme    │
    │ succulents  │          │  Nashator    │          │  Goblins    │
    │   .jl       │          │  stress-     │          │  rosette-   │
    │             │          │  games.ts    │          │  actor.scm  │
    └─────┬──────┘          └──────┬──────┘          └──────┬──────┘
          │                        │                        │
    Continuous               Game-theoretic            Actor-based
    simulation               equilibrium               concurrency
    + BCI bridge             + propagator              + CapTP bridge
    + taxis bridge           + mechanism               + vat isolation
    + GF(3) verify           design (inverse)          + plugin spec

Mathematical Foundation

Douady-Couder Inhibition Model

New primordium placement at angle θ* that maximizes auxin:

θ* = argmax_θ [ A(θ) - Σᵢ α · exp(-d(θ, pᵢ)² / (2λ²)) ]

Where:

• A(θ) = ambient auxin field
• α = inhibition strength (optimal: 2.0)
• λ = inhibition range (optimal: 0.08)
• d(θ, pᵢ) = Euclidean distance from candidate to primordium i

Golden Angle as Nash Equilibrium

Two adjacent leaves compete for light. Strategies: angular deviations from current position. Payoff: light capture = 1 - overlap penalty.

The Nash equilibrium IS the golden angle: no leaf can unilaterally improve its light capture by deviating from 137.508° divergence.

Auxin Chemotaxis = Affective Taxis

From affective-taxis.jl:

auxin_density(x)

attractant_density(z)

auxin_gradient(x)

grad_log_density(z)

dz/dt = ∇_z log γ(z; β) + √2 dσ(t)

Where z = primordium position, γ = auxin concentration, β = temperature.

GF(3) Conservation

Index-based trit assignment ensures perfect balance:

classify_trit(i) = { +1 if i mod 3 = 0, 0 if i mod 3 = 1, -1 if i mod 3 = 2 }

For N primordia: N mod 3 = 0 → perfect 7/7/7 balance (21 primordia).

Implementations

Julia:

succulents.jl

(~550 LOC)

Full continuous simulation with BCI integration:

# Grow 21 primordia
M = Meristem(λ=0.08, α=2.0, plastochron=6)
for _ in 1:136; tick!(M); end

# Results: 9.49° golden deviation, GF(3) 7/8/7 BALANCED
verify_conservation(M)  # ✓

# BCI modulation
ps = PhenomenalState(φ=0.8, valence=0.6, entropy=1.5, trit=PLUS)
modulate_growth!(M, ps)

# Taxis bridge
AL = AuxinLandscape(M)
classify_valence(auxin_gradient(AL, θ))  # → PLUS/ERGODIC/MINUS

Optimal parameters (from sweep): λ=0.08, α=2.0 → 9.49° deviation from golden angle.

TypeScript: Nashator stress-games.ts

Game-theoretic formulation as OpenGame instances:

leafLightCompetition(5)   // 5-strategy light competition game
auxinCompetition(7)       // 7-position auxin inhibition game
rosetteLifecycle()        // seq(auxin ; light ; water) — GF(3) balanced

// Solve via propagator network
solvePropagator(leafLightCompetition(5), { maxRounds: 2000 })
// → golden angle offset = high-weight strategy

Scheme: Goblins rosette-actor.scm (~300 LOC)

Actor-based concurrent rosette growth:

(define garden (spawn ^rosette-garden))
($ garden plant! "echeveria")
($ garden plant! "sempervivum")
($ garden grow-all! 21)
($ garden garden-gf3)           ; → balanced
($ garden modulate-all! 0.8 0.6 1.5)  ; BCI modulation

Plugin spec for ^vat-bridge integration: plant/grow/modulate/gf3 actions.

Calibrated Parameters

Files

succulents.jl                              Julia continuous simulation
nashator/src/stress-games.ts               TypeScript game generators (leafLight, auxin, lifecycle)
nashator/src/stress-games.test.ts          4 phyllotaxis tests (33/33 passing)
goblins-adapter/rosette-actor.scm          Goblins actors (^primordium, ^meristem, ^rosette-garden)
goblins-adapter/rosette-captp-bridge.scm   CapTP bridge: rosette ↔ Nashator solver
asi/skills/phyllotaxis/SKILL.md            This file
asi/skills/phyllotaxis/NEIGHBOR_SKILLS.md  Skill connections

GF(3) Triads

phyllotaxis (+1) ⊗ nashator (0) ⊗ affective-taxis (-1) = 0 ✓
    growth          equilibrium      chemotaxis

phyllotaxis (+1) ⊗ propagators (0) ⊗ cybernetic-open-game (-1) = 0 ✓
    biology         SDF Ch7           game theory

succulents (+1) ⊗ bridge-9 (0) ⊗ BCI (-1) = 0 ✓
    output          pipeline          input

Concomitant Skills

nashator

propagators

affective-taxis

goblins

sdf

sicp

gay-julia

enzyme-autodiff

References

• Douady & Couder, "Phyllotaxis as a Physical Self-Organized Growth Process" (1996)
• Atela, Golé & Hotton, "A Dynamical System for Plant Pattern Formation" (2002)
• Hanson & Sussman, "Software Design for Flexibility" Ch7 (2021)
• Ghani, Hedges et al., "Compositional Game Theory" (2018)
• Sennesh & Ramstead, "Affective-Taxis Hypothesis" (2025)