phyllotaxis

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) maps to attractant_density(z), and auxin_gradient(x) maps to grad_log_density(z). The Langevin dynamics are identical:

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

Parameter	Symbol	Optimal	Range Tested	Unit
Inhibition range	λ	0.08	0.05-0.30	radius
Inhibition strength	α	2.0	2-8	dimensionless
Growth rate	g	0.01	0.005-0.02	radius/tick
Plastochron	P	6	4-10	ticks
Taxis threshold	ε	0.001	0.001-0.01	gradient
Golden deviation	Δ	9.49°	—	degrees

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

Skill	Trit	Interface
nashator	0	Golden angle = Nash equilibrium; leafLightCompetition, auxinCompetition
propagators	0	SDF Ch7 cells = primordia; inhibition = constraint propagation
affective-taxis	-1	Auxin chemotaxis ≡ interoceptive taxis; Langevin dynamics
goblins	0	^rosette-garden actor; vat isolation; CapTP bridge
sdf	-1	Ch7 propagators, Ch8 degeneracy (parameter sweep fallback)
sicp	+1	Ch3 mutable state, Ch5 explicit control (scheduler)
gay-julia	0	GF(3) coloring, golden spiral, color conservation
enzyme-autodiff	-1	Gradient of inhibition kernel via Enzyme.jl

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)

Para(Optic) atlas

Part of: para-mensch-commons.

ALife atlas

Part of: alife-commons. Family: morphogenesis-and-growth. Canonical: lindenmayer-systems.