UUON-VFG-001 · Visual Field Geometry Engine

A parametric topology engine that computes and renders geometric structure from mathematical first principles.Live, in the browser, with no dependencies.

Author: Phillip Aguilar Ruiz III
Organization: UUON Foundation Inc.
License: SAL-1.0
Version: 1.7
Provenance: G°centric v1.0 · Position 33 = 100%

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Table of Contents

• What This Is
• Live Demo
• Mathematical Architecture
• Parameters
• Controls
• Observation Lenses
• Trellis Overlay
• Installation
• Usage
• Version History
• License

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What This Is

UUON-VFG-001 is not a shape renderer. It is a parametric topology engine: a system where geometric structure is computed from a seed primitive, transformed through a mathematical pipeline, and rendered live as an animated field.

Every point on screen is a node with a precise mathematical address. Every line is a provable relationship between two nodes. The patterns that emerge are not designed... they are discovered. The engine reveals the eigenstates of whatever symmetry schema you dial in.

The visual resemblance to biological neural architecture, crystal lattices, and entoptic phenomena is not metaphor. They share the same underlying geometric grammar: rotational symmetry, recursive self-similarity, field propagation, and nodal correspondence.

Note

This engine renders entirely on Canvas 2D, no WebGL, no dependencies, no build step. A single HTML file.

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Live Demo

Open UUON-VFG-001v1-7.html directly in any modern browser.

Default state on load:

• Dark mode · Triangle seed · REFL + INV transforms active
• ROT ORDER 11 · REFL AXES −8 · CHIRALITY 1.0
• SCALE 3.0 · VERTICES 20 · DIMENSION 4.0
• DEPTH 7 · SCALE RATIO 0.670 · ROT DELTA 3.142

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Mathematical Architecture

The engine runs a five-stage pipeline on every animation frame:

SEED  →  SYMMETRY  →  RECURSION  →  FIELD  →  LENS  →  PROJECTION

Stage 1 · Seed Geometry

Eleven geometric primitives, each defined by closed-form parametric equations:

Seed	Description
PT	Point cluster — radially distributed nodes
CIR	Circle — 64-point uniform parametric ring
TRI	Triangle — 3-vertex Euclidean primitive
SQ	Square — 4-vertex rectilinear primitive
POLY	n-Polygon — vertex count driven by VERTICES parameter
SPRL	Archimedean spiral — 120-point parametric
HLX	Helix — 80-point 3D helical projection
KNOT	Torus knot — (2,3) parametric closed curve
LAT	Lattice — 11×11 rectilinear node grid
PHD	Polyhedron — icosahedral projection
FRAC	Fractal — Koch snowflake, depth-limited

Stage 2 · Symmetry Operations

Three parameters define the full symmetry group acting on the seed:

• ROT ORDER n — generates n rotationally symmetric copies (negative = counter-rotation)
• REFL AXES — applies bilateral reflection (negative = reversed frame)
• CHIRALITY — rotates the reflection frame, producing handedness

Stage 3 · Recursion

Activated by the REC transform toggle. At each node, a scaled and rotated copy of the full structure is placed:

• DEPTH — generation count (negative = outward expansion)
• SCALE RATIO — contraction factor between levels (negative = mirror-inversion)
• ROT DELTA — rotation applied at each depth level (negative = counter-rotation)

Important

Recursion must be activated via the REC button. DEPTH must be non-zero for the pipeline to engage.

Stage 4 · Field Displacement

A value noise field displaces all nodes continuously over time. All parameters operate on a bipolar field — negative values invert the displacement direction:

• COHERENCE — geometric order vs. field dominance
• ENTROPY — displacement magnitude
• PERSISTENCE — temporal memory of displacement

Stage 5 · Projection

Pseudo-3D projection via parametric depth mapping. Controlled by the DIMENSION parameter. Values above 2.0 engage depth projection; negative values invert the projection axis.

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Parameters

All parameters operate on bipolar fields: every slider accepts negative and positive values, inverting the mathematical operation rather than simply reversing a scalar.

Domain

Parameter	Range	Effect
SCALE	−3.0 → 3.0	Seed geometry scale; negative flips the seed axis
VERTICES	3 → 24	Vertex count for polygon-class seeds
DIMENSION	−4.0 → 4.0	Projection depth; values >2 engage 3D mapping

Symmetry

Parameter	Range	Effect
ROT ORDER	−12 → 12	Rotational symmetry copies; negative = counter-rotation
REFL AXES	−8 → 8	Reflection axes; negative reverses the frame direction
CHIRALITY	−1.0 → 1.0	Reflection frame rotation; produces handedness

Recursion

Parameter	Range	Effect
DEPTH	−7 → 7	Generation depth; negative = outward expansion
SCALE RATIO	−0.99 → 0.99	Contraction factor; negative mirror-inverts each copy
ROT DELTA	−π → π	Per-level rotation; negative = counter-rotation

Field

Parameter	Range	Effect
COHERENCE	−1.0 → 1.0	Order vs. field dominance; negative inverts field direction
ENTROPY	−1.0 → 1.0	Displacement magnitude; negative contracts instead of expands
PERSISTENCE	−1.0 → 1.0	Temporal field memory

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Controls

Control	Action
Drag	Pan the field
Scroll	Zoom in/out — infinite range, tracks cursor
Pinch	Zoom on touch devices
↺ Reset	Return camera to default state
⏸ Pause	Freeze animation
◑ Dark	Toggle dark/light mode
ℹ Info	Open engine description panel

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Observation Lenses

Six perceptual transforms applied over the constant geometry. The geometry does not change — only the observation frame changes.

Lens	Transform
GEO	Raw geometry — no modification
ENT	Entoptic · Particle noise displacement
PHO	Phosphene · Radial interference pattern
FORM	Form Constant · Angular symmetry distortion
PRED	Predictive · Wave projection field
EYE	Closed-Eye · Cortical field displacement

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Trellis Overlay

The ⊹ button (dim by default, user discovers it) activates a secondary dimensional lattice rendered behind the primary geometry.

Four counter-rotating harmonic grids at different scales, phase offsets, and twist rates reveal the spatial frequency scaffold underlying all geometric structure. Each layer corresponds to a different harmonic of the base spatial frequency.

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Installation

No build step. No dependencies. No package manager required.

# Clone the repository
git clone https://github.com/uuon-foundation/uuon-vfg-001.git

# Open directly in browser
open UUON-VFG-001v1-7.html

Or download the single HTML file and open it locally.

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Usage

As a standalone engine

Open any versioned HTML file directly in a browser. Each version is a frozen, self-contained artifact.

As a module (coming via npm)

// Future API — see @uuon/vfg on npmjs
import { VFGEngine } from '@uuon/vfg'

const engine = new VFGEngine({
  seed: 'triangle',
  rotOrd: 11,
  refAxes: -8,
  chiral: 1.0,
  depth: 7,
  scaleR: 0.670,
  rotD: Math.PI
})

engine.mount(document.getElementById('canvas'))
engine.start()

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Version History

Version	Changes
v1.7	Default state locked to screenshot parameters; dark mode on boot; REFL + INV active on load
v1.6	Recursion pipeline fixed; all parameters bipolar; UI 1.5× scale; INFO panel
v1.5	Speed multiplier; Trellis overlay (off by default)
v1.4	Rotation always on; mouse pan + infinite zoom; default HELIX state
v1.3	Two-row bottom HUD; all controls visible without scroll
v1.2	Full-canvas layout; drawer panels
v1.1	Drawer layout; collapsible panels
v1.0	Initial release; three-column layout

Note

Every version is a frozen artifact. No file is ever overwritten. Version naming follows UUON-VFG-001v1-N.html.

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License

SAL-1.0 — UUON Source Available License 1.0

This engine is source-available for study, research, and personal use. Commercial use, derivative products, and integration into other systems require a license from UUON Foundation Inc.

© 2026 Phillip Aguilar Ruiz III · UUON Foundation Inc.
G°centric v1.0 · Position 33 = 100%

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UUON-VFG-001 is one engine in the UUON Mathematical Operating System — a biological-architecture computational platform built on the formal system F=(P,E,M,R,C).