Multidimensional Shape Equations

Novel shape emergence in geometric algebra universes

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Overview

A computational investigation of universes where every value/state is represented as a multidimensional shape — a multivector in geometric algebra $Cl(d, 0)$. Shapes interact via the geometric product, inherently producing novel grades and forms. All interactions compress into a single unified equation that encodes the universe's grade-level statistics.

Research paper: multidimensional_shape_equations.md

Key Findings

1. Novel shapes appear immediately from the first interaction (cycle 0)
2. Anomaly growth is linear ($R^2 > 0.99$) with no critical threshold
3. Structural novelty dominates (62.2%), grade transitions (35.3%), complexity spikes (2.5%)
4. Container geometry dictates information encoding — the unified equation's grade distribution varies dramatically across geometries
5. Trivectors and pseudoscalars have significant energy — confirming the full richness of $Cl(4, 0)$ is realized
6. Anomalies are information, not errors — each anomaly represents unpredictable novelty

Quick Start

# Run the experiment
python3 shape_sim/run_experiment.py

# Generate figures
python3 shape_sim/figures.py

Structure

.
├── multidimensional_shape_equations.md   # Research paper (4,300 words)
├── mathematical_framework.md             # Formal math: 18 theorems
├── shape_sim/
│   ├── mse.py                            # Full Cl(4,0) implementation (1,241 lines)
│   ├── run_experiment.py                 # Experiment runner
│   ├── figures.py                        # 8 publication-quality figures
│   └── mse_results.json                  # Experiment data
├── figures/                              # 8 figures (PNG + PDF, 300 DPI)
│   ├── fig01_anomaly_timeline.png
│   ├── fig02_anomaly_breakdown.png
│   ├── fig03_energy_distribution.png
│   ├── fig04_grade_evolution.png
│   ├── fig05_anomaly_ratio.png
│   ├── fig06_container_radar.png
│   ├── fig07_interaction_diagram.png
│   └── fig08_phase_space.png
└── README.md

Requirements

• Python 3.10+
• numpy
• matplotlib

pip install numpy matplotlib

Experimental Setup

• Algebra: $Cl(4, 0)$ — 16-component multivectors (scalar, 4 vectors, 6 bivectors, 4 trivectors, 1 pseudoscalar)
• Containers: cube, sphere, ellipsoid, torus, open, Sinai
• Cycles: 50 per container
• Interactions: 100 per cycle
• Selection pressure: 40% retention
• Seed: 42 (reproducible)

Container Comparison

Container	Shapes	Anomalies	G2 (bivector)	G4 (pseudo)
cube	4,767	6,895	42.7%	7.6%
sphere	4,751	6,943	39.1%	1.1%
ellipsoid	4,219	5,922	28.7%	7.5%
torus	4,531	6,510	38.5%	0.7%
open	2,576	3,618	40.7%	15.7%
sinai	4,768	7,091	53.1%	1.2%

Mathematical Framework

The formal mathematical framework is documented in mathematical_framework.md, including:

• 18 theorems with proofs
• Full $Cl(d, 0)$ definitions (geometric product, grades, involutions, duality)
• Anomaly theory (structural novelty, grade transition, complexity spike)
• Conformal GA container representation
• The unified equation as a weighted average (not lossless)

Related Work

• Photon Billiard Computation — Container geometry effects on photon dynamics (precursor research)
• Geometric Algebra — Hestenes, Doran & Lasenby, Porteous
• Hyperdimensional Computing — High-dimensional vector representation

Citation

@misc{multidimensional_shape_equations,
  title = {Multidimensional Shape Equations: Novel Shape Emergence in Geometric Algebra Universes},
  author = {Alex},
  year = {2026},
  url = {https://github.com/alexsysctrl/multidimensional-shape-equations}
}

License

MIT