Phyllotaxis Seed Engine

UUON Foundation Inc. — Phillip Aguilar Ruiz III @uuon-foundation/phyllotaxis-seed-engine · USAL-1.0 · v1.0.0

Reproductive System / Seed Propagation — clouud ecosystem Layer 10

Live: https://uuon-foundation.github.io/phyllotaxis-seed-engine/

What This Engine Does

Models recursive growth and spatial self-organization using a single irrational constant — φ (the golden ratio). Seeds are placed in a polar field where angle increments by the golden angle (≈137.507°) and radius scales by a tunable growth exponent. The result is the densest possible packing of a plane by discrete elements without periodic repetition.

This is not a visualization of Fibonacci numbers. It is the mechanism by which nature determines where new growth nodes appear before any vascular or structural system connects them. The Pythagorean Graph Engine (Layer 08) models the veins. This engine models the seeds that precede them.

Biological Role — clouud Layer 10

| Layer | System | Engine | |-------|--------|--------| | 01 | Spine / CNS | uuon-clouud routing | | 03 | Skeleton | Wave Field 3D Engine | | 04 | Proprioception | Propagation Engine | | 05 | Visual Cortex | Recursive Fractal Engine | | 06 | Decision Layer | Boundary State Engine | | 07 | Fractal Skeleton | Kleinian-Hybrid IFS Engine | | 08 | Vascular Branching | Pythagorean Graph Engine | | 10 | Reproductive System | Phyllotaxis Seed Engine |

F = (P, E, M, R, C)

P — Parameters (Seed Vector, 88 bytes)

| Symbol | Parameter | Default | Range | |--------|-----------|---------|-------| | n | Seed count | 4000 | 100–4000 | | Δθ | Angle offset | 0° | −8° to +8° | | c | Spread scale | 3.4 | 1–14 | | w | Warp power | 2.50 | 0.2–3.0 | | k | Arm count | 13 | 1–13 | | γ | Growth exponent | 0.50 | 0.1–1.0 | | τ | Arm twist | 0.68 | 0.5–3.0 | | Δφ | φ mutation | 0 | −0.05 to +0.05 | | offx | Field offset X | 0 | −200 to +200 | | offy | Field offset Y | 0 | −200 to +200 | | fscale | Field scale | 1.0 | 0.3–2.5 |

E — Encoding

r = pow(i/n, γ)^w · R_max
θ = i · (GOLD_ANGLE + Δθ) · τ
GOLD_ANGLE = 2π · (1 − 1/φ) ≈ 2.39996… rad ≈ 137.507°

γ=0.5 is classical √n packing (Vogel 1979). Any other value changes the density law structurally, not aesthetically.

M — Mapping

index → polar(r, θ) → Cartesian(x, y) → canvas vertex
x = CX + offx + r·cos(θ + rot)
y = CY + offy + r·sin(θ + rot)

Field offset and scale are first-class M-layer parameters, not rendering shortcuts.

R — Representation

• Canvas 2D (live, current)
• JSON P-vector (exportable via Save State button)
• SVG export (planned v1.1.0)
• GLB point cloud (planned v1.2.0)

C — Compression

P = 88 bytes (11 float64 parameters)
R = W × H × 4 bytes (pixel buffer)
C = R / P — computed structurally per frame, displayed live in sidebar

At 1920×1080: C ≈ 94,371 : 1. This is a structural property, not a marketing claim.

Three Laws

I. Law of Irrational Packing φ is used because it is maximally irrational — worst approximable by rationals. Any deviation from GOLD_ANGLE produces visible arm locking. The Δφ parameter makes this physically auditable in real time. Move Δφ away from zero and watch the arms lock.

II. Law of Parametric Provenance Every visual state must be reconstructible from P alone. The Save State button exports the complete P-vector as a signed JSON file. Load State restores the exact state and syncs every control. This is compliance, not a feature.

III. Law of Compression Fidelity C must be computed as a structural property, not estimated. The live C ratio in the sidebar reads the actual canvas dimensions and computes the ratio from first principles on every frame.

Render Layers

| Layer | Description | |-------|-------------| | SEEDS | Core dots with radial glow — brightness encodes index depth | | SPIRAL | Arm tracings connecting seeds in planting order | | LATTICE | Polar grid at golden-angle multiples — shows the underlying coordinate structure | | VORONOI | Proximity mesh — connects seeds within threshold distance |

Theme System

Default: warm paper (#f5f3ef), black geometry at graded opacity, monotone throughout. Dark: void black (#0a0a0a), white geometry at matching opacity grades. Toggle: ◐ INVERT button. Grain texture regenerates on switch to avoid inversion artifacts.

Textures: GRAIN (256×256 canvas noise), HATCH (diagonal ruled lines at 45°), NOISE (sparse pixel scatter).

P-Vector JSON Schema

{
  "engine": "phyllotaxis-seed-engine",
  "version": "1.0.0",
  "license": "USAL-1.0",
  "author": "Phillip Aguilar Ruiz III",
  "org": "UUON Foundation Inc.",
  "repo": "https://github.com/UUON-Foundation/phyllotaxis-seed-engine",
  "timestamp": "2026-08-05T00:00:00.000Z",
  "P": { "seeds": 4000, "angle": 0, "spread": 3.4, "warp": 2.5, "arms": 13 },
  "E": { "gamma": 0.5, "twist": 0.68, "phimut": 0 },
  "M": { "offx": 0, "offy": 0, "fscale": 1.0 },
  "R": { "radius": 4.2, "layers": ["dots","spiral","lattice","voronoi"],
         "textures": ["grain","hatch"], "fring": 1, "rings": 17 },
  "anim": { "speed": 0.0, "pulse": 1.0, "t": 0 }
}

npm

npm install @uuon-foundation/phyllotaxis-seed-engine

Prior Art

• Fibonacci sequence, Binet's formula: Jacques Philippe Marie Binet, 1843
• Phyllotaxis polar model: Helmut Vogel, 1979 — Mathematical Biosciences 44(3–4)
• Golden angle: public domain mathematics
• Sin-hash grain noise: public domain GLSL technique

The individual mathematics is prior art. The F=(P,E,M,R,C) architecture, the three-law compliance framework, the Δφ audit mechanism, and the clouud biological assignment are original to Phillip Aguilar Ruiz III / UUON Foundation Inc. under USAL-1.0.

API

See api/README.md for the planned server layer specification.

UUON Foundation Inc. · Phillip Aguilar Ruiz III · [email protected] USAL-1.0 · Commercial licensing: [email protected]