# Full Text: QuadMath: An Analytical Review of 4D and Quadray Coordinates

> Extracted from `2025_QuadMath.pdf`

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QuadMath: An Analytical Review of 4D and Quadray
Coordinates
Daniel Ari Friedman
ORCID: 0000-0001-6232-9096
Email: daniel@activeinference.institute
DOI: 10.5281/zenodo.16887800
August 16, 2025
Contents
1
Introduction
1
1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.3 4D Namespace Framework
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.5 Manuscript Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.6 Companion Code and Tests
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.7 Reproducibility and Data Availability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.8 Graphical Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
4D Namespaces: Coxeter.4D, Einstein.4D, Fuller.4D
5
2.1 Coxeter.4D (Euclidean E4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 Einstein.4D (Relativistic spacetime) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3 Fuller.4D (Synergetics / Quadrays)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3.1 Directions, not dimensions (language and models)
. . . . . . . . . . . . . . . . . . .
6
2.3.2 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3.3 Clarifying remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4 Practical usage guide
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3
Quadray Analytical Details and Methods
8
3.1 Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2.1 Framework Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2.2 Key Mathematical Principles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2.3 Implementation Strategy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.3 Framework Integration
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4 Fuller.4D Coordinates and Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.5 Conversions and Vector Operations: Quadray ↔Cartesian (Fuller.4D ↔Coxeter.4D/XYZ) . .
9
3.5.1 Integer-coordinate constructions (compact derivation box) . . . . . . . . . . . . . . .
10
3.5.2 Example vertex lists and volume checks (illustrative) . . . . . . . . . . . . . . . . . .
11
3.6 Integer Volume Quantization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.7 Distances and Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.8 XYZ determinant and S3 conversion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.8.1 Alternative Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.9 Fisher Geometry in Quadray Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.10Practical Methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
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3.11Tetravolumes with Quadrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.11.1Bridging and native tetravolume formulas . . . . . . . . . . . . . . . . . . . . . . . .
14
3.11.2Short Python snippets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.11.3Random tetrahedra in the IVM (integer volumes) . . . . . . . . . . . . . . . . . . . .
18
3.11.4Algebraic precision
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.11.5XYZ determinant and the S3 conversion . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.11.6D^3 vs R^3: 60° “closing the lid” vs orthogonal “cubing”
. . . . . . . . . . . . . . .
18
3.12Practical Implementation Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.12.11. Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.12.22. Implementation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.12.33. Testing and Validation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.12.44. Documentation and Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.12.55. Optimization and Extension
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.13Code methods (anchors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.1Core Quadray operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.2Volume calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.3Coordinate conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.4Linear algebra utilities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.5Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.13.6Information geometry methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.13.7Metrics and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.13.8Examples and utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.13.9Symbolic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.13.10Visualization and animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.13.11Path and file utilities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.13.12Additional Nelder–Mead components
. . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.13.13Discrete variational components . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.13.14Glossary generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.13.15Geometry utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.13.16Comprehensive test coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.14Reproducibility checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4
Optimization in 4D
26
4.1 Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2 Nelder–Mead on Integer Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3 Volume-Level Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.4 Quadray Lattice Optimization Pseudocode
. . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.4.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.5 Discrete Lattice Descent (Information-Theoretic Variant)
. . . . . . . . . . . . . . . . . . .
30
4.6 Convergence and Robustness
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.7 Information-Geometric View (Einstein.4D analogy in metric form)
. . . . . . . . . . . . . .
30
4.7.1 Fisher Information as Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . .
30
4.7.2 4D Framework Integration through Fisher Information . . . . . . . . . . . . . . . . .
31
4.7.3 Comprehensive Fisher Information Analysis: Figures 10 and 11 . . . . . . . . . . . .
31
4.7.4 Natural Gradient Descent: Geodesic Motion on Information Manifold . . . . . . . . .
31
4.7.5 Information-Theoretic Foundations and 4D Framework Coherence
. . . . . . . . . .
34
4.7.6 Quadray-Specific Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.7.7 Variational Free Energy and Active Inference Integration
. . . . . . . . . . . . . . .
34
4.7.8 Advanced 4D Framework Integration: Active Inference Context . . . . . . . . . . . .
37
4.8 Multi-Objective and Higher-Dimensional Notes (Coxeter.4D perspective)
. . . . . . . . . .
37
4.9 External validation and computational context . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.10Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5
Extensions of 4D and Quadrays
38
5.1 Multi-Objective Optimization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
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5.2 Machine Learning and Robustness
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.3 Computer Graphics and GPU Acceleration
. . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.4 Active Inference and Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.5 Complex Systems and Collective Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.6 Geospatial Intelligence and the World Game . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.7 Quadrays, Synergetics (Fuller.4D), and William Blake . . . . . . . . . . . . . . . . . . . . .
39
5.8 Pedagogy and Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.9 Higher Dimensions and Decompositions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.10Limitations and Future Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
6
Discussion
41
6.1 Fisher Information and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
6.2 Quadray Coordinates and 4D Structure (Fuller.4D vs Coxeter.4D vs Einstein.4D)
. . . . . .
41
6.3 Integrating FIM with Quadray Models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
6.4 Implications for Optimization and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . .
42
6.4.1 Clarifications on “frequency/time” dimensions
. . . . . . . . . . . . . . . . . . . . .
42
6.4.2 On distance-based tetravolume formulas (clarification) . . . . . . . . . . . . . . . . .
42
6.4.3 Symbolic analysis (bridging vs native) (Results linkage)
. . . . . . . . . . . . . . . .
42
6.5 Community Ecosystem and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
7
Resources
43
7.1 Core Concepts and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.1.1 Information Geometry and Optimization . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.1.2 Active Inference and Free Energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.1.3 Mathematical Foundations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.2 Quadrays and Synergetics (Core Starting Points)
. . . . . . . . . . . . . . . . . . . . . . .
43
7.2.1 Introductory Materials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.2.2 Historical and Background Materials
. . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.3 4dsolutions Ecosystem: Comprehensive Computational Framework
. . . . . . . . . . . . .
43
7.3.1 Core Computational Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.3.2 Primary Hub: School_of_Tomorrow (Python + Notebooks) . . . . . . . . . . . . . . .
44
7.3.3 Additional Repositories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
7.3.4 Educational Framework and Curricula . . . . . . . . . . . . . . . . . . . . . . . . . .
44
7.3.5 Media and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.4 Community Discussions and Collaborative Platforms
. . . . . . . . . . . . . . . . . . . . .
45
7.4.1 Active Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.4.2 Historical Archives
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.5 Related Projects and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.5.1 Tetrahedral Voxel Engines
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.5.2 Academic Publications
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
7.5.3 Context and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
8
Equations and Math Supplement (Appendix)
46
8.1 Volume of a Tetrahedron (Lattice) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
8.2 Expanded Ace 5×5 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
8.3 Cayley-Menger Determinant (Coxeter.4D)
. . . . . . . . . . . . . . . . . . . . . . . . . . .
46
8.4 Piero della Francesca Formula (PDF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
8.5 Gerald de Jong Formula (GdJ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
8.6 Fisher Information Matrix (FIM)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
8.7 Empirical Fisher Information Matrix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
8.8 Natural Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
8.9 Free Energy (Active Inference)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
8.10Quadray Normalization (Fuller.4D)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
8.11Distance (Embedding Sketch; Coxeter.4D slice)
. . . . . . . . . . . . . . . . . . . . . . . .
49
8.12Minkowski Line Element (Einstein.4D analogy)
. . . . . . . . . . . . . . . . . . . . . . . .
49
8.13High-Precision Arithmetic Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
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8.13.1Reproducibility artifacts and external validation
. . . . . . . . . . . . . . . . . . . .
49
8.14Namespaces summary (notation)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
9
Appendix: The Free Energy Principle and Active Inference
50
9.1 Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
9.2 Mathematical Formulation and Equation Callouts (Equations linkage) . . . . . . . . . . . .
50
9.3 Four-Fold Partition and Tetrahedral Mapping (Quadrays; Fuller.4D)
. . . . . . . . . . . . .
50
9.4 How the 4D namespaces relate here
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
9.5 Joint Optimization in the Tetrahedral Framework (Methods linkage) . . . . . . . . . . . . .
54
9.6 Implications for AI and Robust Computation . . . . . . . . . . . . . . . . . . . . . . . . . .
54
9.7 Code, Reproducibility, and Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . .
54
10Appendix: Symbols and Glossary
55
10.1Sets and Spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
10.2Quadray Coordinates and Geometry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
10.3Optimization and Algorithms
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
10.4Information Theory and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
10.5Embeddings and Distances
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
10.6Greek Letters (usage)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
10.7Notes (usage and cross-references) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
10.8Polyhedra and Synergetic Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
10.9Acronyms and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
10.10API Index (auto-generated; Methods linkage) . . . . . . . . . . . . . . . . . . . . . . . . . .
58
1
Introduction
1.1
Abstract
We review a unified analytical framework for four dimensional (4D) modeling and Quadray coordinates,
synthesizing geometric foundations, optimization on tetrahedral lattices, and information geometry.
Building on R. Buckminster Fuller’s Synergetics and the Quadray coordinate system, with extensive ref-
erence to Kirby Urner’s computational implementations across multiple programming languages (see the
comprehensive 4dsolutions ecosystem including Python, Rust, Clojure, and POV-Ray implementations),
we review how integer lattice constraints yield integer volume quantization of tetrahedral simplexes,
creating discrete “energy levels” that regularize optimization and enable integer-based optimization. We
adapt standard methods (e.g., Nelder–Mead method) to the quadray lattice, define Fisher information in
Quadray parameter space, and analyze optimization as geodesic motion on an information manifold via
the natural gradient. We review three distinct 4D namespaces — Coxeter.4D (Euclidean E4), Einstein.4D
(Minkowski spacetime), and Fuller.4D (synergetics/Quadrays) — develop analytical tools and equations,
and survey extensions and applications across AI, active inference, cognitive security, and complex
systems. The result is a cohesive, interpretable approach for robust, geometry-grounded computation in
4D. All source code for the manuscript is available at QuadMath.
Keywords: Quadray coordinates, 4D geometry, tetrahedral lattice, integer volume quantization, infor-
mation geometry, optimization, synergetics, active inference.
1.2
Overview
Quadray coordinates provide a tetrahedral basis for modeling space and computation, standing in con-
trast to Cartesian cubic frameworks. Originating in Buckminster Fuller’s Synergetics, quadray coordi-
nates enable the replacement of right-angle orthonormal assumptions, with 60-degree coordination and
a unit tetrahedron of volume 1. This reframing yields striking integer relationships among common poly-
hedra and provides a natural account of space via close-packed spheres and the isotropic vector matrix
(IVM).
This paper unifies three threads:
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• Foundations: Quadray coordinates and their relation to 4D modeling more generally, with explicit
namespace usage (Coxeter.4D, Einstein.4D, Fuller.4D) to maintain clarity.
• Optimization framework:
Leverages integer volume quantization on tetrahedral lattices to
achieve robust, discrete convergence.
• Information geometry: Tools (e.g., Fisher Information, free-energy minimization) for interpreting
optimization as geodesic motion on statistical manifolds.
1.3
4D Namespace Framework
In this synthetic review, we distinguish three internal meanings of “4D,” following a dot-notation that
avoids cross-domain confusion. For comprehensive details, see Section 2: 4D Namespaces.
• Coxeter.4D — four-dimensional Euclidean space (E4), as in classical polytope theory. Coxeter em-
phasizes that Euclidean 4D is not spacetime; see the Dover edition of Regular Polytopes (p. 119) for
a clear statement to this effect; background on lattice packings in four dimensions aligns with the
treatment in Conway & Sloane’s Sphere Packings, Lattices and Groups.
• Einstein.4D — Minkowski spacetime (3D + time) with an indefinite metric; appropriate for rela-
tivistic physics but distinct from Euclidean E4.
• Fuller.4D — synergetics’ tetrahedral accounting of space using Quadrays (four non-negative coordi-
nates with at least one zero after normalization) and the Isotropic Vector Matrix (IVM) = Cubic Close
Packing (CCP) = Face-Centered Cubic (FCC) correspondence. This treats the regular tetrahedron
as a natural unit container and emphasizes angle/shape relations independent of time/energy.
1.4
Contributions
The paper makes the following key contributions:
• Namespaces mapping:
Coxeter.4D (Euclidean E4), Einstein.4D (Minkowski spacetime), and
Fuller.4D (Quadrays/IVM) →analytical tools and examples.
• Quadray-adapted Nelder–Mead: Integer-lattice normalization and volume-level tracking.
• Equations and methods: Comprehensive supplement with guidance for high-precision computa-
tion using libquadmath.
• Discrete optimizer: Integer-valued variational descent over the IVM (discrete_ivm_descent) with ani-
mation tooling, connecting lattice geometry to information-theoretic objectives.
1.5
Manuscript Structure
• Introduction: motivates Quadrays, clarifies 4D namespaces, and summarizes contributions.
• Methods: details coordinate conventions, exact tetravolumes, conversions, and lattice-aware opti-
mization methods (Nelder–Mead and discrete IVM descent).
• Results: empirical comparisons and demonstrations are shown inline and saved under quadmath/
output/ (PNG/CSV/NPZ/MP4) for reproducibility.
• Discussion: interprets results, limitations, and implications; outlines future work.
• Appendices: equations, free-energy background, and a consolidated symbols/glossary with an auto-
generated API index.
1.6
Companion Code and Tests
The manuscript is accompanied by a fully-tested Python codebase under src/ with unit tests under tests/.
Key artifacts used throughout the paper:
• Quadray APIs: src/quadray.py (Quadray, integer_tetra_volume, ace_tetravolume_5x5).
• Determinant utilities: src/linalg_utils.py (bareiss_determinant_int).
• Length-based volume: src/cayley_menger.py (tetra_volume_cayley_menger, ivm_tetra_volume_cayley_menger).
• XYZ conversion: src/conversions.py (urner_embedding, quadray_to_xyz).
• Examples: src/examples.py (example_ivm_neighbors, example_volume, example_optimize).
5

## Page 6

For comprehensive background resources, computational implementations, and related work, see the
Resources section.
1.7
Reproducibility and Data Availability
• The manuscript Markdown and code to generate the PDF are available on the project repository
(QuadMath on GitHub, @docxology username). See the repository home page for source, figures, and
scripts: QuadMath repository. The repository is also archived with DOI 10.5281/zenodo.16887791
at Zenodo.
• The manuscript is licensed under the Apache License 2.0. See the LICENSE file for details.
• The manuscript is accompanied by a fully-tested Python codebase under src/ with unit tests under
tests/, complemented by extensive cross-validation against Kirby Urner’s reference implementations
in the 4dsolutions ecosystem. See the Resources section for comprehensive details on computational
implementations and validation.
• All figures referenced in the manuscript are generated by scripts under quadmath/scripts/ and saved
to quadmath/output/ with lightweight CSV/NPZ alongside images.
• Tests accompany all methods under src/ and enforce 100% coverage for src/.
• Symbols and notation are standardized across sections; see Appendix: Symbols and Glossary for a
consolidated table of variables and constants used throughout. Equation labels (e.g., Eq. (1) and
Eq. (8)) and figure labels are automatically numbered by LaTeX for consistent cross-referencing.
• The manuscript is a work in progress and will be updated as the project progresses. There may be
errors and missing references, check all methods and equations for consistency.
1.8
Graphical Abstract
Panel A shows Quadray axes (A,B,C,D) under a symmetric embedding with wireframe context. Panel B
shows close-packed spheres at the tetrahedron vertices (IVM/CCP/FCC, “twelve around one”).
6

## Page 7

Figure 1: Quadray coordinate system overview (graphical abstract). Panel A: Four Quadray axes
(A,B,C,D) rendered as colored directional arrows from the origin to the vertices of a regular tetrahedron
under the default symmetric embedding. Each axis is distinctly colored (A=blue, B=orange, C=green,
D=red) with axis labels positioned at the vertex endpoints. A light gray wireframe connects the four ver-
tices to emphasize the tetrahedral geometry underlying the coordinate system. This panel illustrates the
fundamental Fuller.4D direction-based structure where Quadrays represent four canonical directions in
tetrahedral space rather than orthogonal Cartesian dimensions. Panel B: The same tetrahedral vertices
shown as close-packed spheres with radius chosen so neighboring spheres kiss along tetrahedron edges,
emphasizing the connection to the Isotropic Vector Matrix (IVM), Cubic Close Packing (CCP), and Face-
Centered Cubic (FCC) arrangements. Each sphere is colored to match its corresponding axis from Panel
A, with light edge wireframes providing geometric context. This visualization demonstrates how Quadray
coordinates naturally align with dense sphere packing and the “twelve around one” coordination motif
central to synergetics and Fuller.4D modeling.
7

## Page 8

2
4D Namespaces: Coxeter.4D, Einstein.4D, Fuller.4D
This section provides the definitive reference for the three 4D frameworks used throughout this
manuscript. Each namespace represents a distinct mathematical framework with specific applications
in our quadray-based computational system.
2.1
Coxeter.4D (Euclidean E4)
Definition: Standard E4 with orthogonal axes and Euclidean metric; the proper setting for classical
regular polytopes. As Coxeter notes (Regular Polytopes, Dover ed., p. 119), this Euclidean 4D is not
spacetime. Lattice/packing discussions connect to Conway & Sloane’s systematic treatment of higher-
dimensional sphere packings and lattices (Sphere Packings, Lattices and Groups (Springer)).
Usage: Embed Quadray configurations or compare alternative parameterizations when a strictly Eu-
clidean 4D setting is desired.
Simplexes: Simplex structures extend naturally to 4D and beyond (e.g., pentachora).
Mathematical context: This framework is appropriate for standard Euclidean geometry, including the
Cayley-Menger determinant for computing volumes from edge lengths.
2.2
Einstein.4D (Relativistic spacetime)
Definition: Minkowski spacetime with indefinite metric signature, representing the geometric frame-
work for special relativity.
This namespace provides the mathematical foundation for understanding
space-time relationships and relativistic phenomena.
Spacetime: Minkowski metric signature.
Line element (mostly-plus convention; see Minkowski space): see Eq. (12) in the equations appendix.
Optimization analogy: Metric-aware geodesics generalize to information geometry where the Fisher
metric replaces the physical metric. See Fisher information and natural gradient.
Important note: This namespace is used ONLY as a metric/geodesic analogy when discussing informa-
tion geometry. Physical constants G, c, Λ do not appear in Quadray lattice methods and should not be
mixed with IVM unit conventions.
2.3
Fuller.4D (Synergetics / Quadrays)
Definition: Tetrahedral coordinate system based on four non-negative components representing direc-
tions to the vertices of a regular tetrahedron from its center. This namespace embodies the synergetic
approach to geometry, emphasizing shape relationships and integer tetravolumes within the IVM frame-
work.
Basis: Four non-negative components A,B,C,D with at least one zero post-normalization, treated as a
vector (direction and magnitude), not merely a point. Overview: Quadray coordinates.
Geometry: Tetrahedral; unit tetrahedron volume = 1; integer lattice aligns with close-packed spheres
(IVM). Background: Synergetics.
Distances: Computed via appropriate projective normalization; edges align with tetrahedral axes. The
IVM = CCP = FCC shortcut allows working in 3D embeddings for visualization while preserving the
underlying Fuller.4D tetrahedral accounting.
Implementation heritage:
Extensive computational validation through Kirby Urner’s 4dsolutions
ecosystem. See the Resources section for comprehensive details on computational implementations and
educational materials.
8

## Page 9

2.3.1
Directions, not dimensions (language and models)
Vector-first framing: Treat Quadrays as four canonical directions (“spokes” to the vertices of a regular
tetrahedron from its center), not as four orthogonal dimensions. The methane molecule (CH4) and caltrop
shape are helpful mental models.
Origins outside Synergetics: Quadrays did not originate with Fuller; we adopt the coordinate system
within the IVM context. See Quadray coordinates.
Language games: Quadrays and Cartesian are parallel vector languages on the same Euclidean con-
tainer; teaching them together avoids oscillating between “points now, vectors later.”
2.3.2
Figures
In the previous figure, we show the twelve nearest IVM neighbors with coordination patterns and vector
equilibrium geometry; the current figure illustrates random Quadray clouds under several embeddings.
Vector equilibrium (cuboctahedron): The shell formed by the 12 nearest IVM neighbors is the cuboc-
tahedron, also called the vector equilibrium in synergetics.
All 12 vertices are equidistant from the
origin with equal edge lengths, modeling a balanced local packing. This geometry underlies the “twelve
around one” close-packing motif and appears in tensegrity discussions as a canonical balanced struc-
ture. See background: Cuboctahedron (vector equilibrium) and synergetics references. Computational
demonstrations include related visualizations in the 4dsolutions ecosystem. See the Resources section
for comprehensive details.
2.3.3
Clarifying remarks
“A time machine is not a tesseract.” KU on synergeo The tesseract is a Euclidean 4D object (Coxeter.4D),
while Minkowski spacetime (Einstein.4D) is indefinite and not Euclidean; conflating the two leads to
category errors. Fuller.4D, in turn, is a tetrahedral, mereological framing of ordinary space emphasizing
shape/angle relations and IVM quantization. Each namespace carries distinct assumptions and should
be used accordingly in analysis.
2.4
Practical usage guide
• Use Fuller.4D when working with Quadrays, integer tetravolumes, and IVM neighbors (native lat-
tice calculations).
• Use Coxeter.4D for Euclidean length-based formulas, higher-dimensional polytopes, or comparisons
in E4 (including Cayley–Menger).
• Use Einstein.4D as a metric analogy when discussing geodesics or time-evolution; do not mix with
synergetic unit conventions.
9

## Page 10

Figure 2: IVM neighbors and coordination patterns (2×2 panel layout).
Panel A: The twelve
nearest IVM neighbors plotted as blue points in 3D space under the default embedding, showing the
positions corresponding to permutations of the Quadray integer coordinates {2,1,1,0}.
These points
form the vertices of a cuboctahedron (vector equilibrium) centered at the origin with uniform radial
distances. Panel B: The same neighbor points with radial edges (light lines) connecting each neighbor
to the central origin, emphasizing the spoke-like radial symmetry and equal distances from center to
shell. Panel C: Twelve-around-one close-packed spheres configuration where each neighbor position
hosts a sphere with radius chosen so neighboring spheres kiss along cuboctahedron edges, illustrating
the fundamental CCP/FCC/IVM correspondence. The central gray sphere represents the “one” in Fuller’s
“twelve around one” motif. Panel D: Adjacency graph showing strut connections (solid lines) between
touching neighbor spheres, revealing the cuboctahedron’s edge structure, plus light radial cables to the
origin representing a stylized tensegrity interpretation of the vector equilibrium geometry.
10

## Page 11

Figure 3: Random Quadray point clouds under different embeddings (3-panel comparison). Each
panel shows 200 randomly sampled integer Quadray coordinates with components in {0,1,2,3,4,5} pro-
jected to 3D space using different embedding matrices. Left panel (Default embedding): Points (blue)
under the default symmetric embedding matrix showing the natural tetrahedral-symmetric distribution
of normalized Quadrays in 3D space. Center panel (Scaled embedding, 0.75×): The same Quadray
points (orange) under a uniformly scaled version of the default embedding, demonstrating how the point
cloud structure scales proportionally while preserving relative geometries. Right panel (Urner embed-
ding): The same points (purple) projected through the canonical Urner embedding matrix, illustrating
how different linear mappings from Fuller.4D to Coxeter.4D (3D slice) affect the spatial distribution while
preserving the underlying discrete lattice relationships. This comparison demonstrates the flexibility in
choosing embeddings for visualization and analysis while maintaining the fundamental Quadray coordi-
nate relationships.
3
Quadray Analytical Details and Methods
3.1
Overview
This section provides detailed analytical methods for working with Quadray coordinates, including co-
ordinate conventions, volume calculations, and optimization approaches. We emphasize the distinction
between different 4D frameworks and provide practical computational methods.
3.2
Mathematical Foundations
The methods presented here rest on three interconnected mathematical frameworks. For comprehensive
definitions, see Section 2: 4D Namespaces.
3.2.1
Framework Distinctions
• Coxeter.4D: Euclidean 4D geometry with standard metric tensor and volume forms
• Einstein.4D: Minkowski spacetime with indefinite metric for information geometry analogies
• Fuller.4D: Synergetics/Quadray coordinates with integer lattice constraints and IVM unit conven-
tions
3.2.2
Key Mathematical Principles
• Integer volume quantization: Lattice constraints ensure tetrahedral volumes are exact integers
in IVM units
• Coordinate system bridges: Linear transformations between Fuller.4D and Coxeter.4D preserve
geometric relationships
11

## Page 12

• Information geometry: Fisher metric provides Riemannian structure for optimization on parame-
ter manifolds
• Exact arithmetic: Bareiss algorithm ensures determinant calculations remain exact for integer
inputs
3.2.3
Implementation Strategy
All mathematical concepts are implemented with: - Exact arithmetic where possible (integer determi-
nants, symbolic computation) - Numerical stability for floating-point operations (ridge regularization,
condition number monitoring) - Cross-validation between different formulations (Ace 5×5 vs Cayley–
Menger, native vs bridging approaches) - Comprehensive testing ensuring mathematical correctness
across edge cases
3.3
Framework Integration
The three 4D frameworks serve distinct but complementary roles in our implementation:
• Coxeter.4D provides the container: Euclidean 4D space serves as the mathematical foundation
for volume calculations, distance metrics, and geometric transformations. When we compute vol-
umes via Cayley–Menger determinants or XYZ coordinate determinants, we operate in this frame-
work.
• Einstein.4D provides the analogy: The Minkowski metric structure inspires our information ge-
ometry approach, where the Fisher information matrix acts as a Riemannian metric on parameter
space. Natural gradient descent follows geodesics on this information manifold, analogous to how
particles follow geodesics in spacetime.
• Fuller.4D provides the constraints: The Quadray coordinate system and IVM lattice impose in-
teger constraints that enable exact arithmetic and discrete optimization. The synergetics unit con-
ventions (regular tetrahedron volume = 1) create a quantized geometry where volumes are exact
integers.
This multi-framework approach allows us to: 1. Use standard Euclidean methods for volume calculations
where relevant or already in use (Coxeter.4D) 2. Apply information geometry principles for geodesic
optimization (Einstein.4D analogy)
3. Maintain exact arithmetic in the Isotropic Vector Matrix (IVM) setting through integer lattice con-
straints (Fuller.4D) 4. Bridge and swap among frameworks via coordinate transformations and unit con-
versions
3.4
Fuller.4D Coordinates and Normalization
• Quadray vector q = (a,b,c,d), a,b,c,d ≥0, with at least one coordinate zero under normalization.
• Projective normalization can add/subtract (k,k,k,k) without changing direction; choose k to enforce
non-negativity and one zero minimum.
• Isotropic Vector Matrix (IVM): integer quadrays describe CCP sphere centers; the 12 permutations
of {2,1,1,0} form the cuboctahedron (vector equilibrium).
– Integer-coordinate models: assigning unit IVM tetravolume to the regular tetrahedron yields in-
teger coordinates for several familiar polyhedra (inverse tetrahedron, cube, octahedron, rhom-
bic dodecahedron, cuboctahedron) when expressed as linear combinations of the four quadray
basis vectors. See overview: Quadray coordinates.
3.5
Conversions and Vector Operations: Quadray ↔Cartesian (Fuller.4D ↔Cox-
eter.4D/XYZ)
• Embedding conventions determine the linear maps between Quadray (Fuller.4D) and Cartesian
XYZ (a 3D slice or embedding aligned with Coxeter.4D conventions).
• References: Urner provides practical conversion write-ups and matrices; see:
12

## Page 13

– Quadrays and XYZ: Urner – Quadrays and XYZ
– Introduction with examples: Urner – Quadray intro
• Implementation: choose a fixed tetrahedral embedding; construct a 3×4 matrix M that maps
(a,b,c,d) to (x,y,z), respecting A,B,C,D directions to tetra vertices. The inverse map can be defined up
to projective normalization (adding (k,k,k,k)). When comparing volumes, use the S3=\sqrt{9/8} scale
to convert XYZ (Euclidean) volumes to IVM (Fuller.4D) units.
• Vector view: treat q as a vector with magnitude and direction; define dot products and norms by
pushing to XYZ via M.
3.5.1
Integer-coordinate constructions (compact derivation box)
• Under the synergetics convention (unit regular tetrahedron has tetravolume 1), many familiar solids
admit Quadray integer coordinates.
For example, the octahedron at the same edge length has
tetravolume 4, and its vertices can be formed as integer linear combinations of the four axes A,B,C,D
subject to the Quadray normalization rule.
• The cuboctahedron (vector equilibrium) arises as the shell of the 12 nearest IVM neighbors given
by the permutations of (2, 1, 1, 0). The rhombic dodecahedron (tetravolume 6) is the Voronoi cell of
the FCC/CCP packing centered at the origin under the same embedding.
• See the following figure for a schematic summary of these relationships.
Object
Quadray construction (sketch)
IVM volume
Regular tetrahedron
Vertices o=(0,0,0,0), p=(2,1,0,1),
q=(2,1,1,0), r=(2,0,1,1)
1
Cube (same edge)
Union of 3 mutually orthogonal
rhombic belts wrapped on the
tetra frame; edges tracked by
XYZ embedding; compare the
following figure
3
Octahedron (same edge)
Convex hull of mid-edges of
the tetra frame (pairwise axis
sums normalized)
4
Rhombic dodecahedron
Voronoi cell of FCC/CCP
packing at origin (dual to
cuboctahedron)
6
Cuboctahedron (vector
equilibrium)
Shell of the 12 nearest IVM
neighbors: permutations of
(2,1,1,0)
20
Truncated octahedron
Archimedean solid with 6
square and 8 hexagonal faces;
space-filling tiling
20
Small coordinate examples (subset):
• Cuboctahedron neighbors (representatives): (2,1,1,0), (2,1,0,1), (2,0,1,1), (1,2,1,0); the full shell is
all distinct permutations.
• Tetrahedron: [(0,0,0,0), (2,1,0,1), (2,1,1,0), (2,0,1,1)].
Short scripts:
1 python3 quadmath/scripts/polyhedra_quadray_constructions.py
Programmatic check (neighbors, equal radii, adjacency):
1 import numpy as np
13

## Page 14

2 from examples import example_cuboctahedron_vertices_xyz
3
4 xyz = np.array(example_cuboctahedron_vertices_xyz())
5 r = np.linalg.norm(xyz[0])
6 assert np.allclose(np.linalg.norm(xyz, axis=1), r)
7
8 # Touching neighbors have separation 2r
9 touch = []
10 for i in range(len(xyz)):
11
for j in range(i+1, len(xyz)):
12
d = np.linalg.norm(xyz[i] - xyz[j])
13
if abs(d - 2*r) / (2*r) < 0.05:
14
touch.append((i, j))
15 assert len(touch) > 0
3.5.2
Example vertex lists and volume checks (illustrative)
The following snippets use canonical IVM neighbor points (permutations of (2, 1, 1, 0)) to illustrate simple
decompositions consistent with synergetics volumes. Each tetra volume is computed via ace_tetravolume_5x5
and summed.
Octahedron (V = 4) as four unit IVM tetras around the origin:
1 from quadray import Quadray, ace_tetravolume_5x5
2
3 o = Quadray(0,0,0,0)
4 T = [
5
(Quadray(2,1,0,1), Quadray(2,1,1,0), Quadray(2,0,1,1)),
6
(Quadray(1,2,0,1), Quadray(1,2,1,0), Quadray(0,2,1,1)),
7
(Quadray(1,1,2,0), Quadray(1,0,2,1), Quadray(0,1,2,1)),
8
(Quadray(2,0,1,1), Quadray(1,2,0,1), Quadray(0,1,2,1)),
# representative variant
9 ]
10 V_oct = sum(ace_tetravolume_5x5(o, a, b, c) for (a,b,c) in T)
Cube (V = 3) as three unit IVM tetras (orthant-like around the origin):
1 from quadray import Quadray, ace_tetravolume_5x5
2
3 o = Quadray(0,0,0,0)
4 triples = [
5
(Quadray(2,1,0,1), Quadray(2,1,1,0), Quadray(2,0,1,1)),
6
(Quadray(1,2,0,1), Quadray(1,2,1,0), Quadray(0,2,1,1)),
7
(Quadray(1,1,2,0), Quadray(1,0,2,1), Quadray(0,1,2,1)),
8 ]
9 V_cube = sum(ace_tetravolume_5x5(o, a, b, c) for (a,b,c) in triples)
Notes.
• These decompositions are illustrative and use canonical IVM neighbor triples that produce unit
tetras under ace_tetravolume_5x5. Other equivalent tilings are possible.
• Volumes are invariant to adding (𝑘, 𝑘, 𝑘, 𝑘) to each vertex of a tetra (projective normalization), which
the 5×5 determinant respects.
3.6
Integer Volume Quantization
For a tetrahedron with vertices P₀..P₃ in the Quadray integer lattice (Fuller.4D), see Eq. (1) in the equa-
tions appendix.
• With integer coordinates, the determinant is integer; lattice tetrahedra yield integer volumes.
14

## Page 15

• Unit conventions: regular tetrahedron volume = 1 (synergetics).
Notes.
• 𝑃0, … , 𝑃3 are tetrahedron vertices in Quadray coordinates.
• 𝑉is the Euclidean volume measured in IVM tetra-units; the 1/6 factor converts the parallelepiped
determinant to a tetra volume.
• Background and variations are discussed under Tetrahedron volume formulas: Tetrahedron – vol-
ume.
Tom Ace 5×5 determinant (tetravolume directly from quadrays), see Eq. (2) in the equations appendix.
This returns the same integer volumes for lattice tetrahedra. See the implementation ace_tetravolume_5x5.
Notes.
• Rows correspond to the Quadray 4-tuples of the four vertices with a final affine column of ones; the
last row enforces projective normalization.
• The factor 1
4 returns tetravolumes in IVM units consistent with synergetics. See also Quadray coor-
dinates.
Equivalently, define the 5×5 matrix of quadray coordinates augmented with an affine 1 as shown in Eq.
(3) in the equations appendix.
Points vs vectors: subtracting points is shorthand for forming edge vectors. We treat quadray 4-tuples
as vectors from the origin; differences like (𝑃1 −𝑃0) mean “edge vectors,” avoiding ambiguity between
“points” and “vectors.”
Equivalently via Cayley–Menger determinant (Coxeter.4D/Euclidean lengths) (Cayley–Menger determi-
nant), see Eq. (5) in the equations appendix.
References: Cayley–Menger determinant, lattice tetrahedra discussions in geometry texts; see also Tetra-
hedron – volume. Code: integer_tetra_volume, ace_tetravolume_5x5.
Notes.
• Pairwise distances: 𝑑𝑖𝑗are Euclidean distances between vertices 𝑃𝑖and 𝑃𝑗.
• Length-only formulation: Cayley–Menger provides a length-only formula for simplex volumes,
here specialized to tetrahedra; see the canonical reference above.
Table
2:
Polyhedra
tetravolumes
in
IVM
units
(edge
length
equal
to
the
unit
tetra
edge).
{#tbl:polyhedra_volumes}
Polyhedron (edge = tetra edge)
Volume (tetra-units)
Regular Tetrahedron
1
Cube
3
Octahedron
4
Rhombic Dodecahedron
6
Cuboctahedron (Vector Equilibrium)
20
Truncated Octahedron
20
3.7
Distances and Metrics
Distance definitions depend on the chosen embedding and normalization. For cross-references to infor-
mation geometry, see Eq. (FIM) and natural gradient in the Equations appendix.
15

## Page 16

3.8
XYZ determinant and S3 conversion
Given XYZ coordinates of tetrahedron vertices (x_i, y_i, z_i), the Euclidean volume is computed as shown
in Eq. (4) in the equations appendix.
Convert to IVM units via 𝑉𝑖𝑣𝑚= 𝑆3 ⋅𝑉𝑥𝑦𝑧with 𝑆3 = √9/8. See background discussion under Tetrahe-
dron – volume.
3.8.1
Alternative Volume Formulas
• Piero della Francesca (PdF) formula, which consumes edge lengths and returns Euclidean vol-
umes. See Eq. (6) in the equations appendix.
Convert to IVM units via 𝑉𝑖𝑣𝑚= 𝑆3 ⋅𝑉𝑥𝑦𝑧with 𝑆3 = √9/8. See background discussion under
Tetrahedron – volume.
• Gerald de Jong (GdJ) formula, which natively returns tetravolumes. See Eq. (7) in the equations
appendix.
In Quadray coordinates, one convenient native form uses edge-vector differences and an integer-
preserving determinant (agreeing with Ace 5×5):
where each column is formed from Quadray component differences of 𝑃1 −𝑃0, 𝑃2 −𝑃0, 𝑃3 −𝑃0
projected to a 3D slice consistent with the synergetics convention; integer arithmetic is exact and
the factor 1
4 produces IVM tetravolumes. See de Jong’s Quadray notes and Urner’s implementations
for derivations (Quadray coordinates).
• Euclidean embedding distance via appropriate linear map from quadray to R³.
• Information geometry metric: Fisher Information Matrix (FIM)
– FIM[𝑖, 𝑗] = 𝔼[ 𝜕𝜃𝑖log 𝑝(𝑥; 𝜃) 𝜕𝜃𝑗log 𝑝(𝑥; 𝜃) ]
– Acts as Riemannian metric; natural gradient uses FIM⁻¹ ∇θ L. See Fisher information.
3.9
Fisher Geometry in Quadray Space
• Symmetries of quadray lattices often induce near block-diagonal FIM.
• Determinant and spectrum characterize conditioning and information concentration.
3.10
Practical Methods
3.11
Tetravolumes with Quadrays
• The tetravolume of a tetrahedron with vertices given as Quadrays a,b,c,d can be computed directly
from their 4-tuples via the Tom Ace 5×5 determinant; see Eq. (2) for the canonical form.
• Unit regular tetrahedron from origin: with o=(0,0,0,0), p=(2,1,0,1), q=(2,1,1,0), r=(2,0,1,1), we have
V_ivm(o,p,q,r)=1. Doubling each vector scales volume by 8, as expected.
• Equivalent length-based formulas agree with the 5×5 determinant:
– Cayley–Menger: 288 𝑉2 = det
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
1
1
1
1
1
0
𝑑2
01
𝑑2
02
𝑑2
03
1
𝑑2
10
0
𝑑2
12
𝑑2
13
1
𝑑2
20
𝑑2
21
0
𝑑2
23
1
𝑑2
30
𝑑2
31
𝑑2
32
0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
.
– Piero della Francesca (PdF) Heron-like formula (converted to IVM via 𝑆3 = √9/8).
16

## Page 17

Let edge lengths meeting at a vertex be 𝑎, 𝑏, 𝑐, and the opposite edges be 𝑑, 𝑒, 𝑓. The Euclidean
volume satisfies
Convert to IVM units via 𝑉𝑖𝑣𝑚= 𝑆3 ⋅𝑉𝑥𝑦𝑧with 𝑆3 = √9/8. See background discussion under
Tetrahedron – volume.
• Gerald de Jong (GdJ) formula, which natively returns tetravolumes.
In Quadray coordinates, one convenient native form uses edge-vector differences and an integer-
preserving determinant (agreeing with Ace 5×5):
where each column is formed from Quadray component differences of 𝑃1 −𝑃0, 𝑃2 −𝑃0, 𝑃3 −𝑃0
projected to a 3D slice consistent with the synergetics convention; integer arithmetic is exact and
the factor 1
4 produces IVM tetravolumes. See de Jong’s Quadray notes and Urner’s implementations
for derivations (Quadray coordinates).
3.11.1
Bridging and native tetravolume formulas
• Lengths (bridging): PdF and Cayley–Menger (CM) consume Cartesian lengths (XYZ) and produce
Euclidean volumes; convert to IVM units via 𝑆3 = √9/8.
• Quadray-native: Gerald de Jong (GdJ) returns IVM tetravolumes directly (no XYZ bridge). Tom
Ace’s 5×5 coordinate formula is likewise native IVM. All agree numerically with CM+S3 on shared
cases.
References and discussion: Urner – Flickr diagram. For computational implementations and educational
materials, see the Resources section.
Figure:
automated comparison (native Ace 5×5 vs CM+S3) across small examples (see script
sympy_formalisms.py). The figure and source CSV/NPZ are in quadmath/output/.
3.11.2
Short Python snippets
1 from quadray import Quadray, ace_tetravolume_5x5
2
3 o = Quadray(0,0,0,0)
4 p = Quadray(2,1,0,1)
5 q = Quadray(2,1,1,0)
6 r = Quadray(2,0,1,1)
7 assert ace_tetravolume_5x5(o,p,q,r) == 1
# unit IVM tetra
1 import numpy as np
2 from cayley_menger import ivm_tetra_volume_cayley_menger
3
4 # Example: regular tetrahedron with edge length 1 (XYZ units)
5 d2 = np.ones((4,4)) - np.eye(4)
# squared distances
6 V_ivm = ivm_tetra_volume_cayley_menger(d2)
# = 1/8 in IVM tetra-units
1 # SymPy implementation of Tom Ace 5×5 (symbolic determinant)
2 from sympy import Matrix
3
4 def qvolume(q0, q1, q2, q3):
5
M = Matrix([
6
q0 + (1,),
7
q1 + (1,),
8
q2 + (1,),
9
q3 + (1,),
10
[1, 1, 1, 1, 0],
11
])
12
return abs(M.det()) / 4
17

## Page 18

Figure 4: Validation of bridging vs native tetravolume formulations across canonical exam-
ples. This bar chart compares IVM tetravolumes computed via two independent methods: the “bridg-
ing” approach using Cayley–Menger determinants on Euclidean edge lengths converted to IVM units
via the synergetics factor 𝑆3 = √9/8, versus the “native” approach using Tom Ace’s 5×5 determinant
formula that operates directly on Quadray coordinates without XYZ intermediates. Test cases: Unit
tetrahedron (V=1), 2× edge scaling (V=8), mixed coordinate tetrahedron, centered tetrahedron (V=3),
and large mixed tetrahedron, all using integer Quadray coordinates.
Results: The overlapping bars
demonstrate numerical agreement at machine precision between the length-based Coxeter.4D approach
(Cayley–Menger + S3 conversion) and the coordinate-based Fuller.4D approach (Ace 5×5), confirming
the mathematical equivalence of these formulations under synergetics unit conventions. Raw numerical
data saved as bridging_vs_native.csv for reproducibility and further analysis.
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## Page 19

Figure 5: Tetrahedron volume scaling relationships: Euclidean vs IVM unit conventions. This
plot demonstrates the mathematical relationship between edge length scaling and tetravolume under
both Euclidean (XYZ) and IVM (synergetics) unit conventions. X-axis: Edge length scaling factor (0 to
5.0). Y-axis: Tetrahedron volume in respective units. Blue line (Euclidean): Volume scales as the
cube of edge length, following the standard 𝑉=
√
2
12 ⋅𝐿3 relationship for regular tetrahedra. Orange
line (IVM): Volume scales as the cube of edge length but in IVM tetra-units, following 𝑉𝑖𝑣𝑚= 1
8 ⋅𝐿3
where the regular tetrahedron with unit edge has volume 1/8. Key insight: The ratio between these
two scaling laws is the synergetics factor 𝑆3 = √9/8 ≈1.06066, which converts between Euclidean and
IVM volume conventions. Mathematical foundation: This scaling relationship demonstrates how both
conventions preserve the cubic scaling relationship, but with different fundamental units reflecting the
different geometric assumptions of Coxeter.4D (Euclidean) versus Fuller.4D (synergetics) frameworks.
The plot provides the theoretical foundation for understanding volume conversions and scaling behavior
in the IVM system.
19

## Page 20

Figure 6: Synergetic polyhedra volume relationships in the Quadray/IVM framework (compre-
hensive visualization). This figure combines 3D polyhedra visualizations with an extended network
diagram showing integer volume relationships among key synergetic polyhedra. Left panel (3D visual-
izations): Color-coded polyhedra including regular tetrahedron (V=1, fundamental unit), cube (V=3), oc-
tahedron (V=4), rhombic dodecahedron (V=6), cuboctahedron (V=20), and truncated octahedron (V=20),
all constructed with consistent edge lengths and proper geometric faces. Right panel (network dia-
gram): Extended volume relationships showing fundamental shapes (V=1,3,4,6), complex constructions
(V=20), and scaling relationships (2× edge length →8× volume). Additional polyhedra: Includes trun-
cated octahedron (V=20) and scaled variants demonstrating the “third power” volume scaling law V ∝
L³ in IVM units. Geometric constructions: Edge-union relationships, truncation operations, dual poly-
hedra, and Voronoi cell constructions. Fuller.4D significance: These integer volume ratios reflect the
quantized nature of space-filling in synergetics, where the regular tetrahedron provides a natural unit
container and other polyhedra emerge as integer multiples, supporting discrete geometric computation
and exact lattice-based optimization methods. All constructions respect the IVM unit convention where
the regular tetrahedron has tetravolume 1.
20

## Page 21

1 # Symbolic variant with SymPy (exact radicals)
2 from sympy import Matrix, sqrt, simplify
3 from symbolic import cayley_menger_volume_symbolic, convert_xyz_volume_to_ivm_symbolic
4
5 d2 = Matrix([[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]])
6 V_xyz_sym = cayley_menger_volume_symbolic(d2)
# sqrt(2)/12
7 V_ivm_sym = simplify(convert_xyz_volume_to_ivm_symbolic(V_xyz_sym))
# 1/8
3.11.3
Random tetrahedra in the IVM (integer volumes)
• The 12 CCP directions are the permutations of (2, 1, 1, 0). Random walks on this move set generate
integer-coordinate Quadrays; resulting tetrahedra have integer tetravolumes.
1 from itertools import permutations
2 from random import choice
3 from quadray import Quadray, ace_tetravolume_5x5
4
5 moves = [Quadray(*p) for p in set(permutations((2,1,1,0)))]
6
7 def random_walk(start: Quadray, steps: int) -> Quadray:
8
cur = start
9
for _ in range(steps):
10
m = choice(moves)
11
cur = Quadray(cur.a+m.a, cur.b+m.b, cur.c+m.c, cur.d+m.d)
12
return cur
13
14 A = random_walk(Quadray(0,0,0,0), 1000)
15 B = random_walk(Quadray(0,0,0,0), 1000)
16 C = random_walk(Quadray(0,0,0,0), 1000)
17 D = random_walk(Quadray(0,0,0,0), 1000)
18 V = ace_tetravolume_5x5(A,B,C,D)
# integer
3.11.4
Algebraic precision
• Determinants via floating-point introduce rounding noise.
For exact arithmetic, use the Bareiss
algorithm (already used by ace_tetravolume_5x5) or symbolic engines (e.g., sympy). For large random-
walk examples with integer inputs, volumes are exact integers.
• When computing via XYZ determinants, high-precision floats (e.g., gmpy2.mpfr) or symbolic matrices
avoid vestigial errors; round at the end if the underlying result is known to be integral.
3.11.5
XYZ determinant and the S3 conversion
• Using XYZ coordinates of the four vertices: see Eq. (4) for the determinant form and the S3 conver-
sion to IVM units.
3.11.6
D^3 vs R^3: 60° “closing the lid” vs orthogonal “cubing”
• IVM (D^3) heuristic: From a 60–60–60 corner, three non-negative edge lengths 𝐴, 𝐵, 𝐶along
quadray directions enclose a tetrahedron by “closing the lid.” In synergetics, the tetravolume scales
as the simple product 𝐴𝐵𝐶under IVM conventions (unit regular tetra has volume 1). By contrast,
in the orthogonal (R^3) habit, one constructs a full parallelepiped (12 edges); the tetra occupies
one-sixth of the triple product of edge vectors. The IVM path is more direct for tetrahedra.
21

## Page 22

• Pedagogical note: Adopt a vector-first approach. Differences like (𝑃𝑖−𝑃0) denote edge vectors;
Quadrays and Cartesian can be taught in parallel as vector languages on the same Euclidean con-
tainer.
Reference notebook with worked examples and code: See the Resources section for comprehensive edu-
cational materials and computational implementations.
See implementation: tetra_volume_cayley_menger.
• Lattice projection: round to nearest integer quadray; renormalize to maintain non-negativity and a
minimal zero.
3.12
Practical Implementation Workflow
The methods described in this document follow a unified workflow that ensures mathematical correctness,
computational efficiency, and reproducibility:
3.12.1
1. Mathematical Formulation
• Theoretical foundations: Establish mathematical relationships between frameworks (Coxeter.4D
↔Fuller.4D ↔Einstein.4D)
• Coordinate transformations: Define linear maps between coordinate systems with exact arith-
metic
• Volume formulations: Implement multiple approaches (Ace 5×5, Cayley–Menger, symbolic) for
cross-validation
3.12.2
2. Implementation Strategy
• Exact arithmetic: Use Bareiss algorithm for integer determinants, symbolic computation for exact
results
• Numerical stability: Implement ridge regularization, condition number monitoring, and error han-
dling
• Cross-validation: Ensure different formulations produce identical results on test cases
3.12.3
3. Testing and Validation
• Unit tests: 100% coverage of all mathematical functions with edge case handling
• Integration tests: Validate coordinate transformations and volume calculations across frameworks
• Numerical validation: Compare floating-point implementations with exact symbolic results
3.12.4
4. Documentation and Reproducibility
• Code-documentation sync: All mathematical concepts have corresponding implementations in
src/
• Figure generation: Scripts in quadmath/scripts/ generate all figures from source code
• Data export: CSV/NPZ files accompany all figures for complete reproducibility
3.12.5
5. Optimization and Extension
• Lattice constraints: Integer volume quantization enables discrete optimization methods
• Information geometry: Fisher metric guides natural gradient descent on parameter manifolds
• Active inference: Free energy minimization drives both perception and action updates
This workflow ensures that mathematical theory, computational implementation, and practical applica-
tion remain coherent and verifiable throughout the development process.
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3.13
Code methods (anchors)
3.13.1
Core Quadray operations
Quadray
Source: src/quadray.py — Quadray vector class with non-negative components and at least one
zero (Fuller.4D).
DEFAULT_EMBEDDING
Source: src/quadray.py — canonical 3×4 symmetric embedding matrix for Quadray
to XYZ conversion.
to_xyz
Source: src/quadray.py — map quadray to R³ via a 3×4 embedding matrix (Fuller.4D →Coxeter.4D
slice).
magnitude
Source: src/quadray.py — return Euclidean magnitude ||q|| under the given embedding (vector
norm).
dot
Source: src/quadray.py — return Euclidean dot product <q1,q2> under the given embedding.
3.13.2
Volume calculations
integer_tetra_volume
Source: src/quadray.py — integer 3×3 determinant for lattice tetravolume.
ace_tetravolume_5x5
Source: src/quadray.py — Tom Ace 5×5 determinant in IVM units.
tetra_volume_cayley_menger
Source: src/cayley_menger.py — length-based formula (XYZ units).
ivm_tetra_volume_cayley_menger
Source: src/cayley_menger.py — Cayley–Menger volume converted to
IVM units.
3.13.3
Coordinate conversions
urner_embedding
Source: src/conversions.py — canonical XYZ embedding.
quadray_to_xyz
Source: src/conversions.py — apply embedding matrix to map Quadray to XYZ.
3.13.4
Linear algebra utilities
bareiss_determinant_int
Source: src/linalg_utils.py — exact integer Bareiss determinant.
3.13.5
Optimization methods
nelder_mead_quadray
Source: src/nelder_mead_quadray.py — Nelder–Mead optimization adapted to the in-
teger quadray lattice.
discrete_ivm_descent
Source:
src/discrete_variational.py — greedy integer-valued descent over the
IVM using canonical neighbor moves; returns a DiscretePath with visited Quadrays and objective values.
neighbor_moves_ivm
Source: src/discrete_variational.py — return the 12 canonical IVM neighbor moves
as Quadray deltas.
23

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apply_move
Source: src/discrete_variational.py — apply a lattice move and normalize to the canonical
representative.
3.13.6
Information geometry methods
fisher_information_matrix
Source: src/information.py — empirical outer-product estimator.
fisher_information_quadray
Source: src/information.py — compute Fisher information matrix in both
Cartesian and Quadray coordinates.
natural_gradient_step
Source: src/information.py — damped inverse-Fisher step.
free_energy
Source: src/information.py — discrete-state variational free energy.
expected_free_energy
Source: src/information.py — expected free energy for Active Inference with prior
preferences.
active_inference_step
Source: src/information.py — joint perception-action update step in Active Infer-
ence.
information_geometric_distance
Source:
src/information.py — compute information-geometric dis-
tance between two points.
perception_update
Source:
src/information.py — continuous-time perception update:
dμ/dt = D μ -
dF/dμ.
action_update
Source: src/information.py — continuous-time action update: da/dt = -dF/da.
finite_difference_gradient
Source: src/information.py — compute numerical gradient of a scalar func-
tion via central differences.
3.13.7
Metrics and analysis
shannon_entropy
Source: src/metrics.py — Shannon entropy H(p) for a discrete distribution.
information_length
Source: src/metrics.py — path length in information space via gradient-weighted
arc length.
fim_eigenspectrum
Source: src/metrics.py — eigen-decomposition of a Fisher information matrix.
fisher_condition_number
Source: src/metrics.py — compute the condition number of the Fisher infor-
mation matrix.
fisher_curvature_analysis
Source: src/metrics.py — comprehensive analysis of Fisher information ma-
trix curvature.
24

## Page 25

fisher_quadray_comparison
Source:
src/metrics.py — compare Fisher information matrices between
coordinate systems.
3.13.8
Examples and utilities
example_ivm_neighbors
Source: src/examples.py — return the 12 nearest IVM neighbors as permutations
of {2,1,1,0}.
example_cuboctahedron_neighbors
Source: src/examples.py — return twelve-around-one IVM neighbors
(vector equilibrium shell).
example_cuboctahedron_vertices_xyz
Source: src/examples.py — return XYZ coordinates for the twelve-
around-one neighbors.
example_partition_tetra_volume
Source: src/examples.py — construct a tetrahedron from the four-fold
partition and return tetravolume.
3.13.9
Symbolic computation
cayley_menger_volume_symbolic
Source: src/symbolic.py — return symbolic Euclidean tetrahedron vol-
ume from squared distances.
convert_xyz_volume_to_ivm_symbolic
Source: src/symbolic.py — convert a symbolic Euclidean volume
to IVM tetravolume via S3.
3.13.10
Visualization and animation
animate_discrete_path
Source: src/visualize.py — animate a DiscretePath to MP4; saves CSV/NPZ tra-
jectory to quadmath/output/.
plot_ivm_neighbors
Source: src/visualize.py — scatter the 12 IVM neighbor points in 3D.
plot_partition_tetrahedron
Source: src/visualize.py — plot the four-fold partition as a labeled tetra-
hedron in 3D.
animate_simplex
Source: src/visualize.py — animate simplex evolution across iterations.
plot_simplex_trace
Source: src/visualize.py — plot per-iteration diagnostics for Nelder–Mead.
3.13.11
Path and file utilities
get_repo_root
Source: src/paths.py — heuristically find repository root by walking up from start.
get_output_dir
Source: src/paths.py — return quadmath/output path at the repo root and ensure it exists.
get_data_dir
Source: src/paths.py — return quadmath/output/data path and ensure it exists.
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get_figure_dir
Source: src/paths.py — return quadmath/output/figures path and ensure it exists.
3.13.12
Additional Nelder–Mead components
SimplexState
Source: src/nelder_mead_quadray.py — optimization trajectory state containing vertices, val-
ues, volume, and history.
order_simplex
Source: src/nelder_mead_quadray.py — sort vertices by objective value ascending and return
paired lists.
centroid_excluding
Source: src/nelder_mead_quadray.py — integer centroid of three vertices, excluding
the specified index.
project_to_lattice
Source: src/nelder_mead_quadray.py — project a quadray to the canonical lattice rep-
resentative via normalize.
compute_volume
Source: src/nelder_mead_quadray.py — integer IVM tetra-volume from the first four ver-
tices.
3.13.13
Discrete variational components
DiscretePath
Source: src/discrete_variational.py — optimization trajectory on the integer quadray lat-
tice.
OptionalMoves
Source: src/discrete_variational.py — lightweight protocol for optional typing of moves
parameter.
3.13.14
Glossary generation
build_api_index
Source: src/glossary_gen.py — build API index from source directory.
generate_markdown_table
Source: src/glossary_gen.py — generate markdown table from API entries.
inject_between_markers
Source:
src/glossary_gen.py — inject payload between markers in markdown
text.
3.13.15
Geometry utilities
minkowski_interval
Source: src/geometry.py — return the Minkowski interval squared ds² (Einstein.4D).
Relevant tests (tests/):
• test_quadray.py (unit IVM tetra, divisibility-by-4 scaling, Ace vs. integer method)
• test_quadray_cov.py (Ace determinant basic check)
• test_cayley_menger.py (regular tetra volume in XYZ units)
• test_linalg_utils.py (Bareiss determinant behavior)
• test_examples.py, test_examples_cov.py (neighbors, examples)
• test_metrics.py, test_metrics_cov.py, test_information.py, test_paths.py, test_paths_cov.py
26

## Page 27

3.13.16
Comprehensive test coverage
The test suite provides 100% coverage of all source modules with the following focus areas:
Core functionality tests
• test_quadray.py: Quadray class operations, normalization, volume calculations, vector operations
• test_cayley_menger.py: Cayley–Menger determinant validation, IVM conversion accuracy
• test_linalg_utils.py: Bareiss algorithm correctness, edge cases, numerical stability
Optimization and variational methods
• test_nelder_mead_visual.py: Nelder–Mead adaptation to quadray lattice, simplex evolution
• test_discrete_variational.py: IVM neighbor moves, discrete descent algorithms, path tracking
• test_active_inference.py: Free energy minimization, perception-action updates
Information geometry and metrics
• test_information.py: Fisher information matrix computation, natural gradient steps
• test_metrics.py: FIM eigendecomposition, curvature analysis, coordinate system comparisons
• test_information_cov.py: Extended coverage of information geometry functions
Examples and utilities
• test_examples.py: IVM neighbor constructions, polyhedra volume relationships
• test_examples_cov.py: Extended example function coverage
• test_paths.py: Repository structure utilities, output directory management
Visualization and symbolic computation
• test_visualize.py: Plotting functions, animation generation, data export
• test_visualize_cov.py: Extended visualization coverage, edge case handling
• test_symbolic.py: SymPy symbolic volume calculations, exact arithmetic
• test_symbolic_cov.py: Symbolic computation error handling
• test_sympy_formalisms.py: End-to-end symbolic workflow validation
API and documentation generation
• test_glossary_gen.py: Automatic API index generation, markdown table formatting
• test_glossary_gen_cov.py: Extended glossary generation coverage
3.14
Reproducibility checklist
• Complete implementation coverage: All formulas and methods discussed in the paper are imple-
mented in src/ modules and verified by comprehensive test suites in tests/.
• Exact arithmetic for integer inputs: Determinants are computed using the Bareiss algorithm
for exact integer arithmetic; floating-point paths are used only where appropriate and results are
converted (e.g., via S3) as specified.
• Deterministic random experiments: Random-walk experiments use fixed seeds and produce in-
teger volumes; Ace 5×5 determinant agrees with length-based methods across all test cases.
• Volume tracking and convergence: Integer simplex volume monitoring detects convergence
plateaus; face/edge analyses interpret sensitivity along edges and enable subspace searches across
faces.
• Test-driven development: All source code follows TDD principles with 100% coverage require-
ments enforced via .coveragerc configuration.
• Figure and data generation: All figures referenced in documentation are generated by scripts in
quadmath/scripts/ that import from src/ modules, ensuring code-documentation coherence.
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• Cross-platform compatibility: Headless matplotlib backend (MPLBACKEND=Agg) ensures CI
compatibility; deterministic RNG seeds guarantee reproducible outputs.
• Dependency management: All dependencies managed through uv and pyproject.toml with exact
version pinning via uv.lock.
• Path resolution: No hardcoded paths; all output directories resolved via paths.py utilities for cross-
platform compatibility.
• Data export formats: Figures saved alongside CSV/NPZ data for complete reproducibility; all
generation scripts print output paths for manifest collection.
All source code, tests, and documentation are available in the docxology/QuadMath repository, ensuring
complete transparency and reproducibility of the methods described herein.
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4
Optimization in 4D
4.1
Overview
This section describes optimization methods adapted to the integer Quadray lattice, emphasizing discrete
convergence and information-geometric approaches. The methods leverage the IVM’s natural quantiza-
tion and extend to higher-dimensional spaces via Coxeter.4D embeddings.
4.2
Nelder–Mead on Integer Lattice
• Adaptation: standard Nelder–Mead simplex operations with projection to integer Quadray coordi-
nates.
• Projection: after each reflection/expansion/contraction, snap to nearest integer lattice point via
projective normalization.
• Volume tracking: monitor integer tetravolume as convergence diagnostic; discrete steps create
stable plateaus.
4.2.1
Parameters
• Reflection 𝛼≈1
• Expansion 𝛾≈2
• Contraction 𝜌≈0.5
• Shrink 𝜎≈0.5
References: original Nelder–Mead method and common parameterizations in optimization texts and sur-
vey articles; see overview: Nelder–Mead method.
4.3
Volume-Level Dynamics
• Simplex volume decreases in discrete integer steps, creating stable plateaus (“energy levels”).
• Termination: when volume stabilizes at a minimal level and function spread is below tolerance.
• Monitoring: track integer simplex volume and the objective spread at each iteration for convergence
diagnostics.
4.4
Quadray Lattice Optimization Pseudocode
1 while not converged:
2
order vertices by objective
3
centroid of best three
4
propose reflected (then possibly expanded/contracted) point
5
project to integer quadray; renormalize with (k,k,k,k)
6
accept per standard tests; else shrink toward best
7
update integer volume and function spread trackers
4.4.1
Figures
As shown in the following figure, the discrete Nelder–Mead converges on plateaus.
Raw artifacts: the full trajectory animation simplex_animation.mp4 and per-frame vertices (simplex_animation_vertices
.csv/.npz) are available in quadmath/output/. The full optimization trajectory is provided as an animation
(MP4) in the repository’s output directory.
29

## Page 30

Figure 7: Discrete Nelder–Mead optimization trajectory on the integer Quadray lattice.
This
time-series plot tracks key diagnostic quantities across 12 optimization iterations for a simple quadratic
objective function defined on the integer Quadray lattice. X-axis: Optimization iteration (0 through 12).
Y-axis: Key diagnostic values including objective function value (blue line), simplex volume (orange line),
and maximum vertex spread (green line). Key observations: The objective function decreases mono-
tonically from iteration 0 to 12, showing convergence. The simplex volume (orange) exhibits discrete
plateaus characteristic of integer-lattice optimization, where the Nelder–Mead algorithm can only move
to integer coordinate positions. The maximum vertex spread (green) decreases as the simplex contracts
around the optimum, indicating that the four vertices of the optimization tetrahedron are converging to a
tight cluster. Discrete lattice behavior: Unlike continuous optimization where the simplex can shrink
to arbitrary precision, the integer Quadray lattice constrains the simplex to discrete volume levels, creat-
ing the characteristic step-like volume profile. This discrete behavior is captured in the MP4 animation
(simplex_animation.mp4) and the diagnostic traces in the following figure. The final simplex volume is minimal
on the integer lattice, representing a stable “energy level” where further discrete moves do not improve
the objective function.
30

## Page 31

Figure 8: Nelder-Mead simplex evolution on integer Quadray lattice (2×2 panel). This comprehen-
sive visualization shows the simplex optimization process at key iterations (0, 3, 6, 9) to demonstrate the
discrete convergence behavior. Top-left (Iteration 0): Initial simplex configuration with four vertices
forming a tetrahedron in 3D embedding space, starting from widely dispersed positions. Top-right (It-
eration 3): Early optimization state showing initial simplex contraction and vertex repositioning toward
the optimal region. Bottom-left (Iteration 6): Mid-optimization with vertices converging toward the
optimum at coordinates (2,2,2). Bottom-right (Iteration 9): Final converged state where all vertices
have collapsed to the optimal point (2,2,2), representing successful convergence to the global minimum.
Key features: Each subplot shows the tetrahedral simplex with vertices as red spheres and edges as blue
lines connecting the vertices. The objective function values and vertex spread are displayed in each sub-
plot title, showing the monotonic decrease in both quantities. Discrete lattice behavior: The step-wise
convergence demonstrates how the integer Quadray lattice constrains optimization to discrete volume
levels, creating the characteristic plateau behavior seen in the diagnostic traces.
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## Page 32

Figure 9: Complete simplex optimization trace visualization. This 3D plot shows the complete tra-
jectory of all four simplex vertices across all optimization iterations, providing a comprehensive view
of the optimization path. Vertex traces: Each vertex follows a distinct colored path (red, blue, green,
orange) from its initial position to the final converged point at (2,2,2). Key iteration markers: Large
markers at iterations 0, 3, 6, and 9 highlight critical stages in the optimization process. Convergence
point: The black star at (2,2,2) marks the final converged state where all vertices meet at the global opti-
mum. Optimization insights: The trace reveals how the simplex contracts systematically, with vertices
moving in coordinated patterns that respect the integer lattice constraints. The discrete nature of the
optimization is evident in the step-wise vertex movements, which can only occur to valid integer Quadray
coordinates. This visualization complements the 2×2 panel view by showing the complete optimization
trajectory in a single, interpretable plot.
32

## Page 33

4.5
Discrete Lattice Descent (Information-Theoretic Variant)
• Integer-valued descent over the IVM using the 12 neighbor moves (permutations of {2,1,1,0}), snap-
ping to the canonical representative via projective normalization.
• Objective can be geometric (e.g., Euclidean in an embedding) or information-theoretic (e.g., local
free-energy proxy); monotone decrease is guaranteed by greedy selection.
• API: discrete_ivm_descent in src/discrete_variational.py.
Animation helper:
animate_discrete_path in src/
visualize.py.
Short snippet (paper reproducibility):
1 from quadray import Quadray, DEFAULT_EMBEDDING, to_xyz
2 from discrete_variational import discrete_ivm_descent
3 from visualize import animate_discrete_path
4
5 def f(q: Quadray) -> float:
6
x, y, z = to_xyz(q, DEFAULT_EMBEDDING)
7
return (x - 0.5)**2 + (y + 0.2)**2 + (z - 0.1)**2
8
9 path = discrete_ivm_descent(f, Quadray(6,0,0,0))
10 animate_discrete_path(path)
4.6
Convergence and Robustness
• Discrete steps reduce numerical drift; improved stability vs. unconstrained Cartesian.
• Natural regularization from volume quantization; fewer wasted evaluations.
• Compatible with Gauss–Newton/Natural Gradient guidance using FIM for metric-aware steps
(Amari, natural gradient).
4.7
Information-Geometric View (Einstein.4D analogy in metric form)
The Fisher Information Matrix (FIM) provides a fundamental bridge between the three 4D frameworks,
establishing a Riemannian metric on parameter space that guides optimization through information ge-
ometry. This section demonstrates how the FIM connects Coxeter.4D (Euclidean parameter space), Ein-
stein.4D (information-geometric flows), and Fuller.4D (tetrahedral structure) in a unified optimization
framework.
4.7.1
Fisher Information as Riemannian Metric
The empirical Fisher Information Matrix 𝐹𝑖𝑗quantifies the local curvature of the log-likelihood surface
around parameter estimates, providing a natural metric for parameter space geometry. This fundamental
concept in information geometry establishes a Riemannian structure on the statistical manifold, where
distances and angles are measured according to the intrinsic geometry of the probability distributions
rather than the extrinsic Euclidean geometry of the parameter space.
For a model with parameters w = (𝑤0, 𝑤1, 𝑤2) and loss function 𝐿(w), the FIM is estimated as the
expected outer product of score functions (see Eq. (9) in the equations appendix).
where 𝐿𝑛represents the loss for individual data samples. This matrix captures both parameter sensitivity
(diagonal elements) and parameter interactions (off-diagonal elements), revealing the intrinsic geometry
of the optimization landscape.
The Fisher Information Matrix serves as the natural metric tensor 𝑔𝑖𝑗= 𝐹𝑖𝑗on the statistical mani-
fold, replacing the Euclidean metric 𝛿𝑖𝑗with a data-dependent metric that reflects the actual curvature
structure of the objective function. This geometric interpretation enables the application of differential
geometry concepts to optimization problems, where geodesics (locally distance-minimizing paths) follow
the natural gradient direction 𝐹−1∇𝐿rather than the standard gradient ∇𝐿.
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The theoretical foundation of this approach stems from the work of Rao (1945) and Amari (1985), who
established information geometry as a framework for analyzing statistical models through differential
geometry. The FIM naturally arises as the Hessian of the Kullback-Leibler divergence between nearby
probability distributions, making it the canonical choice for measuring distances on the statistical mani-
fold.
In the context of optimization, the FIM provides several key advantages:
1. Invariance to parameterization: The natural gradient 𝐹−1∇𝐿is invariant to smooth, invertible
parameter transformations, unlike the standard gradient which depends on the choice of coordinate
system.
2. Optimal step sizing: The FIM automatically determines appropriate step sizes in different param-
eter directions, scaling updates according to local curvature.
3. Geometric consistency: Optimization follows geodesics on the statistical manifold, respecting the
intrinsic geometry of the parameter space rather than imposing an artificial Euclidean structure.
This geometric approach to optimization is particularly powerful in the context of the 4D frameworks,
where it provides a unified mathematical language for describing optimization dynamics across different
geometric paradigms.
4.7.2
4D Framework Integration through Fisher Information
Coxeter.4D (Euclidean): In standard Euclidean parameter space, the metric tensor is simply 𝛿𝑖𝑗, pro-
viding uniform scaling in all directions. The FIM 𝐹𝑖𝑗generalizes this to capture the actual curvature
structure of the objective function.
Einstein.4D (Minkowski analogy): The Fisher metric replaces the spacetime metric, where geodesics
follow 𝐹−1∇𝐿instead of straight lines. This creates optimal parameter update paths that respect the
intrinsic geometry of the statistical manifold. The natural gradient update rule Δw = −𝜂𝐹−1∇𝐿im-
plements geodesic motion on the information manifold, analogous to how particles follow geodesics in
relativistic spacetime.
Fuller.4D (Synergetics): The tetrahedral structure of Quadray coordinates naturally encodes the four-
fold partition of optimization problems, while the FIM provides the metric structure for efficient naviga-
tion through this space. The discrete nature of the IVM lattice creates natural quantization effects that
can be exploited for computational efficiency.
4.7.3
Comprehensive Fisher Information Analysis: Figures 10 and 11
The following figures demonstrate the comprehensive nature of Fisher Information analysis, showing
both the matrix structure and its eigenspectrum interpretation. This analysis reveals the anisotropic
nature of parameter space and guides the design of efficient optimization strategies.
Figure 10: Fisher Information Matrix (FIM) with 4D Framework Context. This comprehensive
three-panel visualization demonstrates the empirical Fisher information matrix and its deep connections
to the three 4D mathematical frameworks through code-grounded analysis.
Figure 11: Comprehensive Fisher Information Eigenspectrum with Curvature Analysis. This de-
tailed three-panel visualization provides comprehensive analysis of the parameter space geometry within
the 4D framework context, including tetrahedral parameter space visualization.
4.7.4
Natural Gradient Descent: Geodesic Motion on Information Manifold
The Fisher Information Matrix enables natural gradient descent, which implements geodesic motion on
the information manifold. Unlike standard gradient descent that follows straight lines in parameter space,
natural gradient descent follows curved paths that respect the intrinsic geometry defined by the FIM.
The natural gradient update rule is given by:
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## Page 35

Figure 10: Fisher Information Matrix (FIM) with 4D Framework Context.
This comprehensive
three-panel visualization demonstrates the empirical Fisher information matrix and its deep connections
to the three 4D mathematical frameworks through code-grounded analysis. Left panel: Linear regres-
sion model visualization showing the misspecified quadratic model 𝑦= 𝑤0 + 𝑤1𝑥+ 𝑤2𝑥2 with true
parameters 𝑤𝑡𝑟𝑢𝑒= [1.0, −2.0, 0.5] and estimated parameters 𝑤𝑒𝑠𝑡= [0.3, −1.2, 0.0]. The panel displays
data points, true model fit (green line), estimated model fit (red dashed line), and diagnostic informa-
tion including Mean Squared Error (MSE). This visualization grounds the Fisher Information analysis in
the actual model that generates the parameter gradients. Center panel: The 3×3 Fisher information
matrix 𝐹𝑖𝑗estimated from per-sample gradients of the misspecified linear regression model, displayed
as a heatmap with precise value annotations. The matrix structure reveals the local curvature of the
log-likelihood surface, where brighter colors indicate higher information content. Matrix interpreta-
tion: Diagonal elements 𝐹𝑖𝑖quantify the sensitivity of the objective to changes in parameter 𝑤𝑖, while
off-diagonal elements 𝐹𝑖𝑗capture parameter interactions and potential redundancy. Right panel: 3D
tetrahedral visualization of the 4D framework integration, showing how Coxeter.4D (Euclidean), Ein-
stein.4D (Minkowski), and Fuller.4D (Synergetics) frameworks connect through the tetrahedral structure.
Mathematical foundation: The FIM is computed according to Eq. (9) where gradients are computed
with respect to parameters 𝑤0, 𝑤1, 𝑤2 from the misspecified model. Coxeter.4D (Euclidean): Standard
3D parameter space with Euclidean metric 𝛿𝑖𝑗. Einstein.4D (Minkowski): Fisher metric 𝐹𝑖𝑗replaces
spacetime metric; geodesics follow Δ𝑤= 𝐹−1∇𝐿for optimal parameter updates. Fuller.4D (Synerget-
ics): Tetrahedral coordinate system with IVM quantization. Information content: Diagonal dominance
shows each parameter contributes independently to the model’s predictive power, while off-diagonal
elements reveal parameter interactions and potential redundancy. This FIM structure guides natural
gradient descent by weighting parameter updates according to local curvature, leading to more efficient
convergence than standard gradient descent.
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Figure 11: Comprehensive Fisher Information Eigenspectrum with Curvature Analysis.
This
detailed three-panel visualization provides comprehensive analysis of the parameter space geometry
within the 4D framework context, including tetrahedral parameter space visualization. Left panel: Bar
chart showing the eigenvalue decomposition of the empirical Fisher information matrix, with eigenval-
ues sorted in descending order and color-coded for visual clarity. Each bar is precisely annotated with
its numerical value, revealing the principal curvature directions of the parameter space. Center panel:
Comprehensive curvature analysis providing key metrics, eigenvalue interpretation, and 4D framework
connections. Key metrics: Condition number (anisotropy measure), anisotropy index (normalized di-
rectional variation), and total curvature (trace of F). Eigenvalue interpretation: Each eigenvalue 𝜆𝑖
represents the curvature strength in the corresponding principal direction. Large eigenvalues indicate
directions of high curvature (tight constraints) where the objective function changes rapidly with param-
eter changes, while small eigenvalues indicate directions of low curvature (loose constraints) where the
objective function is relatively flat. Right panel: 3D tetrahedral visualization of the parameter space
structure based on the Fisher Information eigenvectors and eigenvalues. The tetrahedron vertices rep-
resent the origin and the three principal curvature directions, scaled by the square root of eigenvalues
to show the anisotropic structure. 4D framework connection: The eigenvalues reveal the anisotropic
nature of the parameter space, explaining why natural gradient descent (which scales updates by 𝐹−1)
converges more efficiently than standard gradient descent. Coxeter.4D: The eigenvalues quantify the
Euclidean geometry of parameter space in different directions. Einstein.4D: The Fisher metric geometry
creates curved geodesics that respect the intrinsic parameter space structure. Fuller.4D: The tetrahe-
dral structure provides a natural coordinate system for representing the four-fold partition of optimization
problems, with the parameter space tetrahedron directly reflecting the curvature structure. Optimiza-
tion implications: Natural gradient descent scales parameter updates by 𝐹−1, creating anisotropic
scaling that improves convergence on ill-conditioned problems. The tetrahedral visualization shows how
the parameter space anisotropy creates natural directions for efficient optimization. This geometric un-
derstanding is crucial for designing effective optimization strategies and understanding model behavior
in the context of information geometry.
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where 𝜂is the learning rate, 𝐹is the Fisher Information Matrix from Eq. (9), and ∇𝐿is the standard
gradient of the loss function. This update rule implements geodesic motion on the statistical manifold,
where the metric tensor 𝑔𝑖𝑗= 𝐹𝑖𝑗determines the local geometry.
The theoretical foundation of natural gradient descent was established by Amari (1998) in the context
of information geometry. The key insight is that the natural gradient 𝐹−1∇𝐿is the steepest descent
direction when distances are measured using the Fisher metric rather than the Euclidean metric. This
makes natural gradient descent invariant to smooth, invertible parameter transformations, a property
that standard gradient descent lacks.
In the context of the 4D frameworks, natural gradient descent provides a unified approach to optimization
that respects the intrinsic geometry of each framework:
• Coxeter.4D: The natural gradient respects the actual curvature structure of the objective function
rather than imposing artificial Euclidean geometry.
• Einstein.4D: The Fisher metric replaces the spacetime metric, creating geodesic flows that follow
the intrinsic geometry of the parameter space.
• Fuller.4D: The tetrahedral structure provides natural coordinate systems where the FIM can exhibit
beneficial structural properties.
The efficiency of natural gradient descent comes from its ability to automatically adapt step sizes to local
curvature. In directions of high curvature (large eigenvalues of 𝐹), the natural gradient takes smaller
steps, while in directions of low curvature (small eigenvalues), it takes larger steps. This anisotropic
scaling leads to faster convergence and better numerical stability compared to standard gradient descent.
4.7.5
Information-Theoretic Foundations and 4D Framework Coherence
The Fisher Information approach provides several key advantages that integrate naturally with the 4D
framework structure:
1. Geometric Consistency: The FIM ensures that optimization respects the intrinsic geometry of the
parameter space, maintaining consistency across all three 4D frameworks.
2. Anisotropic Scaling: Natural gradient descent automatically adapts step sizes to local curvature,
improving convergence efficiency on problems with strong parameter space anisotropy.
3. Framework Bridging: The FIM serves as a mathematical bridge between Coxeter.4D (Euclidean
geometry), Einstein.4D (information-geometric flows), and Fuller.4D (tetrahedral structure).
4. Quantitative Analysis: The eigenspectrum provides quantitative measures of parameter space
structure, enabling principled optimization strategy design.
4.7.6
Quadray-Specific Considerations
Under Quadray parameterizations, the FIM often exhibits block-structured and symmetric patterns that
simplify matrix inversion for natural-gradient steps. This structural regularity arises from the tetrahedral
symmetry of the IVM lattice and can be exploited for computational efficiency.
The discrete nature of the IVM lattice also influences the FIM structure, as parameter updates are con-
strained to integer coordinate positions. This creates a natural regularization effect that can improve
optimization stability and convergence.
4.7.7
Variational Free Energy and Active Inference Integration
The Fisher Information framework naturally extends to variational inference and active inference, where
the free energy principle guides both perception and action through information-geometric optimization.
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Figure 12: Natural Gradient Trajectory: Geodesic Motion on Information Manifold. This visu-
alization demonstrates the parameter trajectory of natural gradient descent, showing how information-
geometric optimization creates optimal paths through parameter space. Trajectory: The blue line with
markers traces the parameter evolution from initial guess to final optimum, revealing the path taken
through the 2D parameter space. Markers: Each marker represents one optimization step, with spac-
ing indicating the step size and convergence rate. Start/End markers: Green circle marks the initial
parameter values, red circle marks the converged optimum. 4D Framework Connection: This trajec-
tory demonstrates geodesic motion on the information manifold, where the Fisher metric (Einstein.4D
analogy) replaces the physical metric. The natural gradient follows Eq. (10), creating optimal paths
through parameter space that respect the intrinsic geometry. Convergence behavior: The trajectory
shows smooth, direct convergence to the optimum, characteristic of natural gradient descent on well-
conditioned objectives. Comparison with standard gradient descent: Natural gradient descent typ-
ically produces more direct trajectories than standard gradient descent, especially on ill-conditioned
problems where the parameter space has strong anisotropy. This efficiency comes from the FIM-based
scaling that adapts step sizes to local curvature. The trajectory demonstrates how information-geometric
optimization leverages the intrinsic geometry of the parameter space to achieve faster, more stable con-
vergence than naive gradient methods. Grid overlay: Added for better readability and to emphasize the
discrete nature of the optimization steps.
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## Page 39

Figure 13: Variational Free Energy Landscape with 4D Framework Integration. This visualization
shows the variational free energy ℱ= −log 𝑃(𝑜|𝑠) + KL[𝑄(𝑠)||𝑃(𝑠)] (see Eq. (11)) as a function of
the variational distribution parameter, demonstrating the geometry of the variational manifold. X-axis:
Variational parameter 𝑞(state = 0) controlling the distribution over the two discrete states. Y-axis: Free
energy value ℱin natural units. Curve interpretation: The free energy exhibits a clear minimum at the
optimal variational distribution, representing the best approximation to the true posterior given the con-
straints of the variational family. 4D Framework Connection: The free energy landscape represents
the geometry of the variational manifold, where optimization follows geodesics defined by the Fisher met-
ric (Einstein.4D analogy). In active inference frameworks, minimizing free energy drives both perception
and action, analogous to how geodesics minimize proper time in relativistic spacetime. KL divergence
component: The free energy balances data fit (first term) with regularization (KL divergence from prior),
preventing overfitting while maintaining good predictive performance. Optimization geometry: The
smooth, convex shape of the free energy landscape makes optimization straightforward using natural
gradient descent, which respects the intrinsic geometry of the parameter space. This variational frame-
work provides a principled approach to approximate inference in complex models where exact posterior
computation is intractable, while maintaining connections to the broader 4D mathematical frameworks.
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## Page 40

4.7.8
Advanced 4D Framework Integration: Active Inference Context
The integration of Fisher Information with Active Inference demonstrates the full power of the 4D frame-
work approach, where Coxeter.4D provides exact geometry, Einstein.4D supplies information-geometric
flows, and Fuller.4D offers the tetrahedral structure for representing the four-fold partition of perception-
action systems.
For comprehensive Active Inference visualizations including 4D natural gradient trajectories and free
energy landscapes, see Section 9: Free Energy and Active Inference.
• Quadray relevance: block-structured and symmetric patterns often arise under quadray parame-
terizations, simplifying F inversion for natural-gradient steps.
4.8
Multi-Objective and Higher-Dimensional Notes (Coxeter.4D perspective)
• Multi-objective: vertices encode trade-offs; simplex faces approximate Pareto surfaces; integer vol-
ume measures solution diversity.
• Higher dimensions: decompose higher-dimensional simplexes into tetrahedra; sum integer volumes
to extend quantization.
4.9
External validation and computational context
The optimization methods developed here build upon and complement the extensive computational frame-
work in Kirby Urner’s 4dsolutions ecosystem. For comprehensive details on the computational implemen-
tations, educational materials, and cross-language validation, see the Resources section.
4.10
Results
• The simplex-based optimizer exhibits discrete volume plateaus and converges to low-spread config-
urations; see the simplex figures above and the MP4/CSV artifacts in quadmath/output/.
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5
Extensions of 4D and Quadrays
Here we review some extensions of the Quadray 4D framework, including multi-objective optimization,
machine learning, computer graphics and GPU acceleration, active inference, complex systems, peda-
gogy, and implementations, with an emphasis on cognitive security.
5.1
Multi-Objective Optimization
• Simplex faces encode trade-offs; integer volume measures solution diversity.
• Pareto front exploration via tetrahedral traversal.
5.2
Machine Learning and Robustness
• Geometric regularization: Quadray-constrained weights/topologies yield structural priors and im-
proved stability.
• Adversarial robustness: Discrete lattice projection reduces vulnerability to gradient-based adver-
sarial perturbations by limiting directions.
• Ensembles: Tetrahedral vertex voting and consensus improve robustness.
References: see Fisher information, Natural gradient, and quadray conversion notes by Urner for embed-
ding choices.
5.3
Computer Graphics and GPU Acceleration
• Quadray visualization acceleration: GPU-accelerated rendering of tetrahedral coordinate sys-
tems enables real-time exploration of 4D geometric structures. The parallel nature of GPU archi-
tectures naturally maps to the four-basis vector representation of quadrays, allowing simultaneous
computation of vertex positions, edge connections, and face tessellations across thousands of tetra-
hedra.
• Integer arithmetic optimization: GPU compute shaders excel at integer-based volume calcula-
tions and determinant computations using the Bareiss algorithm. The discrete lattice structure of
quadray coordinates benefits from parallel integer arithmetic units, achieving significant speedups
over CPU implementations for large-scale geometric computations.
• Dynamic programming acceleration: GPU-accelerated dynamic programming algorithms lever-
age CUDA Dynamic Parallelism for adaptive parallel computation of recursive geometric algorithms.
This approach enables efficient handling of varying computational workloads in tetrahedral decom-
position and optimization problems, as demonstrated in applications like the Mandelbrot set compu-
tation where dynamic parallelism manages computational complexity effectively.
• Parallel geometric algorithms: Implementation of GPU-optimized versions of algorithms like
QuickHull for convex hull computation in quadray space achieves substantial performance improve-
ments. The tetrahedral lattice structure naturally supports parallel prefix sum operations and effi-
cient neighbor queries, enabling real-time visualization of complex 4D geometric transformations.
• Memory bandwidth optimization: The structured memory access patterns of quadray coordinates
align well with GPU memory hierarchies, enabling efficient coalesced memory access for large-scale
geometric datasets. This optimization is particularly beneficial for applications requiring real-time
rendering of complex polyhedral structures and dynamic tessellations.
References: GPU-accelerated geometry processing techniques (arxiv.org), CUDA Dynamic Parallelism
for adaptive computation (developer.nvidia.com), and parallel scan algorithms for optimization (devel-
oper.nvidia.com).
5.4
Active Inference and Free Energy
• Free energy ℱ= −log 𝑃(𝑜∣𝑠) + KL[𝑄(𝑠) ‖ 𝑃(𝑠)] (see Eq. (11) in the equations appendix); back-
ground: Free energy principle and overviews connecting to predictive coding and control.
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• Belief updates follow steepest descent in Fisher geometry using the natural gradient (see Eq. (10)
in the equations appendix); quadray constraints improve stability/interpretability.
• Links to metabolic efficiency and biologically plausible computation.
• For more information, see the Appendix: The Free Energy Principle and Active Inference.
5.5
Complex Systems and Collective Intelligence
• Tetrahedral interaction patterns support distributed consensus and emergent behavior.
• Resource allocation and network flows benefit from geometric constraints.
• Cognitive security: Applying cognitive security can safeguard distributed consensus mechanisms
from manipulation, preserving the reliability of emergent behaviors in complex systems. Incorpo-
rating cognitive security measures can protect the integrity of belief updates and decision-making
processes, ensuring that actions are based on accurate and unmanipulated information.
5.6
Geospatial Intelligence and the World Game
• Spatial data integration:
Quadray tetrahedral frameworks provide natural tessellations for
geospatial data analysis, where the Dymaxion projection’s minimal distortion aligns with Fuller’s
World Game objectives of holistic global perspective.
The tetrahedral lattice supports efficient
spatial indexing and neighbor queries for distributed geospatial intelligence operations.
• Resource allocation optimization: The World Game’s goal of “making the world work for 100%
of humanity” translates to multi-objective optimization problems where tetrahedral simplex faces
encode trade-offs between population centers, resource distribution, and ecological constraints. In-
teger volume quantization ensures discrete, interpretable solutions for global resource allocation.
• Cognitive security in distributed sensing: Geospatial intelligence networks benefit from tetra-
hedral consensus mechanisms that resist manipulation of spatial data streams. The geometric con-
straints of Fuller.4D provide natural validation frameworks for detecting anomalous spatial patterns
and maintaining data integrity across distributed sensor networks.
• Tetrahedral tessellations for global modeling: The World Game’s emphasis on interconnected
global systems maps naturally to tetrahedral decompositions of the Dymaxion projection, where
each tetrahedron represents a coherent region for local optimization while maintaining global con-
nectivity through shared faces and edges.
5.7
Quadrays, Synergetics (Fuller.4D), and William Blake
• Quadrays (tetrahedral coordinates) instantiate Fuller’s Synergetics emphasis on the tetrahedron as
a structural primitive; in this manuscript’s terminology this corresponds to Fuller.4D. Tetrahedral
frames support part–whole reasoning and efficient decompositions used throughout.
• William Blake’s “fourfold vision” (single, twofold, threefold, fourfold) provides a historical metaphor
for multiscale perception and inference. Read through Fisher geometry and natural gradient dynam-
ics, it parallels multilayer predictive processing and counterfactual simulation. For background, see
a concise overview of Blake’s visionary psycho‑topographies in British Art Studies (visionary art
analysis) and the Active Inference Institute’s MathArt Stream #8 (Active Inference & Blake).
• Juxtaposing Blake and Fuller foregrounds “comprehensivity”: holistic design and sensemaking via
geometric primitives. Context: (Fuller & Blake: Lives in Juxtaposition) and pedagogical antecedents
in experimental design education at Black Mountain College (Diaz, Chance and Design at Black
Mountain College – PDF).
• Implications for Quadray practice: four‑facet summaries of models/trajectories, tetrahedral consen-
sus in ensembles, and stigmergic annotation patterns for cognitive security and distributed sense-
making.
5.8
Pedagogy and Implementations
Kirby Urner’s comprehensive 4dsolutions ecosystem provides extensive educational resources and cross-
platform implementations for Quadray computation and visualization. For comprehensive details on ed-
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ucational frameworks, cross-language implementations, historical context, and community development,
see the Resources section.
5.9
Higher Dimensions and Decompositions
• Decompose higher-dimensional simplexes into tetrahedra; sum integer volumes to maintain quanti-
zation.
• Tessellations support parallel/distributed implementations.
5.10
Limitations and Future Work
• Benchmark breadth: extend beyond convex/quadratic toys to real tasks (registration, robust regres-
sion, control) with ablations.
• Distance sensitivity: compare embeddings and their effect on optimizer trajectories; document rec-
ommended defaults.
• Hybrid schemes: study schedules that interleave continuous proposals with lattice projection.
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6
Discussion
Quadray geometry (Fuller.4D) offers an interpretable, quantized view of geometry, topology, informa-
tion, and optimization.
Integer volumes enforce discrete dynamics, acting as a structural prior that
can regularize optimization, reduce overfitting, prevent numerical fragility, and enable integer-based ac-
celerated methods. Information geometry provides a right language for optimization in the synergetic
tradition: optimization proceeds not through arbitrary parameter-space moves in continuous space, but
along geodesics defined by information content (see Eq. (8) and Eq. (10) in the equations appendix;
overview: Natural gradient).
Limitations and considerations:
• Embeddings and distances: Mapping between quadray and Euclidean coordinates must be se-
lected carefully for distance calculations.
• Hybrid strategies: Some problems may require hybrid strategies (continuous steps with periodic
lattice projection).
• Benchmarking: Empirical benchmarking remains important to quantify benefits across domains.
In practical analysis and simulation, numerical precision matters.
Integer-volume reasoning is exact
in theory, but empirical evaluation (e.g., determinants, Fisher Information, geodesics) can benefit from
high-precision arithmetic. When double precision is insufficient, quad-precision arithmetic (binary128)
via GCC’s libquadmath provides the __float128 type and a rich math API for robust computation. See the
official documentation for details on functions and I/O: GCC libquadmath.
6.1
Fisher Information and Curvature
The Fisher Information Matrix (FIM) defines a Riemannian metric on parameter space and quantifies
local curvature of the statistical manifold. High curvature directions (large eigenvalues of F) indicate
parameters to which the model is most sensitive; small eigenvalues indicate sloppy directions. Our eigen-
spectrum visualization (see the Fisher Information Matrix eigenspectrum figure above) highlights these
scales. Background: Fisher information.
Implication: curvature-aware steps using Eq. (10) in the equations appendix adaptively scale updates by
the inverse metric, improving conditioning relative to vanilla gradient descent.
A curious connection unites geodesics in information geometry, the physical principle of least action,
and Buckminster Fuller’s tensegrity geodesic domes (Fuller.4D). On statistical manifolds, geodesics are
shortest paths under the Fisher metric, and natural-gradient flows approximate least-action trajectories
by minimizing an information-length functional constrained by curvature (Eqs.
(8), (10) in the equa-
tions appendix). In tensegrity domes, geodesic lines on triangulated spherical shells distribute stress
nearly uniformly while the network balances continuous tension with discontinuous compression, attain-
ing maximal stiffness with minimal material. Both systems exemplify constraint-balanced minimalism:
an extremal path emerges by trading off cost (action or information length) against structure (metric cur-
vature or tensegrity compatibility). The shared economy—optimal routing through low-cost directions—
links geodesic shells in architecture to geodesic flows in parameter spaces; see background on tensegri-
ty/geodesic domes @Web.
6.2
Quadray Coordinates and 4D Structure (Fuller.4D vs Coxeter.4D vs Ein-
stein.4D)
Quadray coordinates provide a tetrahedral basis with projective normalization, aligning with close-
packed sphere centers (IVM). Symmetries common in quadray parameterizations often yield near
block-diagonal structure in F, simplifying inversion and preconditioning. Overview: Quadray coordinates
and synergetics background. We stress the namespace boundaries: (i) Fuller.4D for lattice and integer
volumes, (ii) Coxeter.4D for Euclidean embeddings, lengths, and simplex families, (iii) Einstein.4D for
metric analogies only — not for interpreting synergetic tetravolumes.
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6.3
Integrating FIM with Quadray Models
Applying the FIM within quadray-parameterized models ties statistical curvature to tetrahedral structure.
Practical takeaways:
• Use fisher_information_matrix to estimate F from per-sample gradients; inspect principal directions via
fim_eigenspectrum.
• Exploit block patterns induced by quadray symmetries to stabilize metric inverses and reduce com-
pute.
• Combine integer-lattice projection with natural-gradient steps to balance discrete robustness and
curvature-aware efficiency.
• Purely discrete alternatives (e.g., discrete_ivm_descent) provide monotone integer-valued descent when
gradients are unreliable; hybrid schemes can interleave discrete steps with curvature-aware contin-
uous proposals.
6.4
Implications for Optimization and Estimation
6.4.1
Clarifications on “frequency/time” dimensions
• Fuller’s discussions often treat frequency/energy as an additional organizing dimension distinct from
Euclidean coordinates. In our manuscript, we keep the shape/angle relations (Fuller.4D) separate
from time/energy bookkeeping; when temporal evolution is needed, we use explicit trajectories and
metric analogies (Einstein.4D) without conflating with Euclidean 4D objects (Coxeter.4D). This sep-
aration avoids category errors while preserving the intended interpretability.
6.4.2
On distance-based tetravolume formulas (clarification)
• When volumes are computed from edge lengths, PdF and Cayley–Menger operate in Euclidean length
space and are converted to IVM tetravolumes via the S3 factor. In contrast, the Gerald de Jong
formula computes IVM tetravolumes natively, agreeing numerically with PdF/CM after S3 without
explicit XYZ intermediates. Tom Ace’s 5×5 determinant sits in the same native camp as de Jong’s
method. See references under the methods section for links to Urner’s code notebooks and discus-
sion.
6.4.3
Symbolic analysis (bridging vs native) (Results linkage)
• Exact (SymPy) comparisons confirm that CM+S3 and Ace 5×5 produce identical IVM tetravolumes
on canonical small integer-quadray examples. See the bridging vs native comparison figure above
and the manifest sympy_symbolics.txt alongside bridging_vs_native.csv in quadmath/output/.
• Curvature-aware optimizers: Kronecker-factored approximations (K-FAC) leverage structure in F to
accelerate training and improve stability; see K-FAC (arXiv:1503.05671). Similar ideas apply when
quadray structure induces separable blocks.
• Model selection: eigenvalue spread of F provides a lens on parameter identifiability; near-zero modes
suggest redundancies or over-parameterization.
• Robust computation: lattice normalization in quadray space yields discrete plateaus that comple-
ment FIM-based scaling for numerically stable trajectories.
6.5
Community Ecosystem and Validation
The extensive computational ecosystem around Quadrays and synergetic geometry provides validation,
pedagogical context, and practical implementations that complement and extend the methods developed
in this manuscript.
Cross-language implementations serve as independent verification of algorithmic
correctness while educational materials demonstrate practical applications across diverse computational
environments.
See the Resources section for comprehensive details on the 4dsolutions organization,
cross-language implementations, educational frameworks, and community platforms.
45

## Page 46

7
Resources
This section provides comprehensive resources for learning about and working with Quadrays, synerget-
ics, and the computational methods discussed in this manuscript.
7.1
Core Concepts and Background
7.1.1
Information Geometry and Optimization
• Fisher information: Fisher information (reference) — see also Eq. (8) in the equations appendix
• Natural gradient: Natural gradient (reference) — see also Eq. (10) in the equations appendix
7.1.2
Active Inference and Free Energy
• Active Inference Institute: Welcome to Active Inference Institute
• Comprehensive review: Active Inference — recent review (UCL Discovery, 2023)
7.1.3
Mathematical Foundations
• Tetrahedron volume formulas: length-based Cayley–Menger determinant and determinant-based
expressions on vertex coordinates (see Tetrahedron – volume)
• Exact determinants: Bareiss algorithm, used in our integer tetravolume implementations
• Optimization baseline: the Nelder–Mead method, adapted here to the Quadray lattice
7.2
Quadrays and Synergetics (Core Starting Points)
7.2.1
Introductory Materials
• Quadray coordinates (intro and conversions): Urner – Quadray intro, Urner – Quadrays and
XYZ
• Quadrays and the Philosophy of Mathematics: Urner – Quadrays and the Philosophy of Mathe-
matics
• Synergetics background and IVM: Synergetics (Fuller, overview)
• Quadray coordinates overview: Quadray coordinates (reference)
7.2.2
Historical and Background Materials
• RW Gray projects — Synergetics text: rwgrayprojects.com (synergetics)
• Fuller FAQ: C. J. Fearnley’s Fuller FAQ
• Synergetics resource list: C. J. Fearnley’s resource page
• Wikieducator: Synergetics hub
• Quadray animation: Quadray.gif (Wikimedia Commons)
• Fuller Institute: BFI — Big Ideas: Synergetics
7.3
4dsolutions Ecosystem: Comprehensive Computational Framework
The 4dsolutions organization provides the most extensive computational framework for Quadrays and
synergetic geometry, spanning 29+ repositories with implementations across multiple programming lan-
guages.
7.3.1
Core Computational Modules
Primary Python Libraries
• Math for Wisdom (m4w): m4w (repo)
– Quadray vectors and conversions: qrays.py (Qvector, SymPy-aware)
46

## Page 47

– Synergetic tetravolumes and modules: tetravolume.py with PdF-CM vs native IVM and BEAST
algorithms
Cross-Language Validation
• Rust implementation: rusty_rays (performance-oriented)
– Sources: Rust library implementation, Rust command-line interface
• Clojure implementation: synmods (functional paradigm)
– Sources: qrays.clj, ramping_up.clj
7.3.2
Primary Hub: School_of_Tomorrow (Python + Notebooks)
Repository: School_of_Tomorrow
Core Modules
• qrays.py: Quadray implementation with normalization, conversions, and vector ops (source)
• quadcraft.py: POV-Ray scenes for CCP/IVM arrangements, animations, and tutorials (source)
• flextegrity.py: Polyhedron framework, concentric hierarchy, POV-Ray export (source)
• Additional modules: polyhedra.py, identities.py, smod_play.py (synergetic modules)
Key Notebooks
• Qvolume.ipynb: Tom Ace 5×5 determinant with random-walk demonstrations (source)
• VolumeTalk.ipynb: Comparative analysis of bridging vs native tetravolume formulations (source)
• QuadCraft_Project.ipynb: 1,255 lines of interactive CCP navigation and visualization tutorials (source)
• Additional notebooks: TetraBook.ipynb, CascadianSynergetics.ipynb, Rendering_IVM.ipynb, SphereVolumes.ipynb
(visual and curricular materials)
7.3.3
Additional Repositories
Tetravolumes (Algorithms and Pedagogy)
• Repository: tetravolumes
• Code: tetravolume.py
• Notebooks: Atoms R Us.ipynb, Computing Volumes.ipynb
Visualization and Rendering
• BookCovers: VPython for interactive educational animations (repo)
– Examples: bookdemo.py, stickworks.py, tetravolumes.py
7.3.4
Educational Framework and Curricula
Oregon Curriculum Network (OCN)
• OCN portal: OCN portal
• Python for Everyone: pymath page
Historical Documentation
• Python5 notebooks: Polyhedrons 101.ipynb
• Historical variants: qrays.py also appears in Python5 (archive)
• Python edu-sig archives: Python edu-sig archives tracing 25+ years of development
47

## Page 48

7.3.5
Media and Publications
• YouTube demonstrations: Synergetics talk 1, Synergetics talk 2, Additional
• Academia profile: Kirby Urner at Academia.edu
7.4
Community Discussions and Collaborative Platforms
7.4.1
Active Platforms
• Math4Wisdom Knowledge Engineering: Collaborative platform with various art, resources, and
cross-reference materials
• synergeo discussion archive: Groups.io platform with ongoing community discussions and tech-
nical exchanges
7.4.2
Historical Archives
• GeodesicHelp threads: GeodesicHelp computations archive (Google Groups) documenting com-
putational approaches and problem-solving techniques
7.5
Related Projects and Applications
7.5.1
Tetrahedral Voxel Engines
• QuadCraft: Tetrahedral voxel engine using Quadrays
7.5.2
Academic Publications
• Flextegrity: Generating the Flextegrity Lattice (academia.edu)
7.5.3
Context and Integration
These materials popularize the IVM/CCP/FCC framing of space, integer tetravolumes, and projective
Quadray normalization. They inform the methods in this paper and complement the src/ implementations
(see quadray.py, cayley_menger.py, linalg_utils.py).
The ecosystem provides extensive validation, pedagogical context, and practical implementations that
complement and extend the methods developed in this manuscript.
Cross-language implementations
serve as independent verification of algorithmic correctness while educational materials demonstrate
practical applications across diverse computational environments.
48

## Page 49

8
Equations and Math Supplement (Appendix)
8.1
Volume of a Tetrahedron (Lattice)
𝑉= 1
6 |det [ 𝑃1 −𝑃0, 𝑃2 −𝑃0, 𝑃3 −𝑃0 ]|
(1)
Notes.
• 𝑃0, … , 𝑃3 are vertex coordinates; the determinant computes the volume of the parallelepiped
spanned by edge vectors, with the 1/6 factor converting to tetra volume.
Tom Ace 5×5 tetravolume (IVM units):
𝑉𝑖𝑣𝑚= 1
4
∣
∣
∣
∣
det
⎛
⎜
⎜
⎜
⎜
⎝
𝑎0
𝑎1
𝑎2
𝑎3
1
𝑏0
𝑏1
𝑏2
𝑏3
1
𝑐0
𝑐1
𝑐2
𝑐3
1
𝑑0
𝑑1
𝑑2
𝑑3
1
1
1
1
1
0
⎞
⎟
⎟
⎟
⎟
⎠
∣
∣
∣
∣
(2)
Notes.
• Rows correspond to Quadray 4-tuples of the vertices; the last row encodes the affine constraint.
Division by 4 returns IVM tetravolume.
8.2
Expanded Ace 5×5 Matrix
The expanded form of the Ace 5×5 matrix with explicit Quadray coordinates:
𝑀(𝑞0, 𝑞1, 𝑞2, 𝑞3) =
⎡
⎢⎢
⎣
𝑞01
𝑞02
𝑞03
𝑞04
1
𝑞11
𝑞12
𝑞13
𝑞14
1
𝑞21
𝑞22
𝑞23
𝑞24
1
𝑞31
𝑞32
𝑞33
𝑞34
1
1
1
1
1
0
⎤
⎥⎥
⎦
,
𝑉𝑖𝑣𝑚= 1
4 ∣det 𝑀(𝑞0, 𝑞1, 𝑞2, 𝑞3)∣
(3)
Notes.
• Matrix structure: Each row represents a vertex with its Quadray coordinates plus affine coordinate
1.
• Last row: Enforces projective normalization constraint.
• Volume computation: Determinant divided by 4 gives IVM tetravolume.
XYZ determinant volume and S3 conversion:
𝑉𝑥𝑦𝑧= 1
6 ∣det ⎛
⎜
⎜
⎝
𝑥𝑎
𝑦𝑎
𝑧𝑎
1
𝑥𝑏
𝑦𝑏
𝑧𝑏
1
𝑥𝑐
𝑦𝑐
𝑧𝑐
1
𝑥𝑑
𝑦𝑑
𝑧𝑑
1
⎞
⎟
⎟
⎠
∣,
𝑉𝑖𝑣𝑚= 𝑆3 𝑉𝑥𝑦𝑧,
𝑆3 = √9
8
(4)
Notes.
• Homogeneous determinant in Cartesian coordinates for tetra volume; conversion to IVM units uses
𝑆3 = √9/8 as used throughout.
8.3
Cayley-Menger Determinant (Coxeter.4D)
For tetrahedron volume from edge lengths (Coxeter.4D approach):
49

## Page 50

288 𝑉2 = det
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
1
1
1
1
1
0
𝑑2
01
𝑑2
02
𝑑2
03
1
𝑑2
10
0
𝑑2
12
𝑑2
13
1
𝑑2
20
𝑑2
21
0
𝑑2
23
1
𝑑2
30
𝑑2
31
𝑑2
32
0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(5)
Notes.
• Pairwise distances: 𝑑𝑖𝑗are Euclidean distances between vertices 𝑃𝑖and 𝑃𝑗.
• Length-only formulation: Cayley–Menger provides a length-only formula for simplex volumes,
here specialized to tetrahedra.
• Conversion to IVM: Use 𝑉𝑖𝑣𝑚= 𝑆3 ⋅𝑉𝑥𝑦𝑧with 𝑆3 = √9/8 to convert to IVM units.
8.4
Piero della Francesca Formula (PDF)
For tetrahedron volume from edge lengths meeting at a vertex:
144 𝑉2
𝑥𝑦𝑧= 4𝑎2𝑏2𝑐2−𝑎2 (𝑏2+𝑐2−𝑓2)2−𝑏2 (𝑐2+𝑎2−𝑒2)2−𝑐2 (𝑎2+𝑏2−𝑑2)2+(𝑏2+𝑐2−𝑓2)(𝑐2+𝑎2−𝑒2)(𝑎2+𝑏2−𝑑2)
(6)
Notes.
• Edge lengths: 𝑎, 𝑏, 𝑐are edges meeting at a vertex, 𝑑, 𝑒, 𝑓are opposite edges.
• Conversion to IVM: Use 𝑉𝑖𝑣𝑚= 𝑆3 ⋅𝑉𝑥𝑦𝑧with 𝑆3 = √9/8.
8.5
Gerald de Jong Formula (GdJ)
Native Quadray formula for tetrahedron volume:
𝑉𝑖𝑣𝑚= 1
4 ∣det (
𝑎1 −𝑎0
𝑎2 −𝑎0
𝑎3 −𝑎0
𝑏1 −𝑏0
𝑏2 −𝑏0
𝑏3 −𝑏0
𝑐1 −𝑐0
𝑐2 −𝑐0
𝑐3 −𝑐0
)∣
(7)
Notes.
• Quadray differences: Each column represents edge vectors 𝑃1 −𝑃0, 𝑃2 −𝑃0, 𝑃3 −𝑃0 in Quadray
coordinates.
• Native IVM: Returns tetravolume directly in IVM units without conversion.
• Integer arithmetic: Exact for integer Quadray coordinates.
See code: tetra_volume_cayley_menger. For tetrahedron volume background, see Tetrahedron – volume. Exact
integer determinants in code use the Bareiss algorithm. External validation: these formulas align with
implementations in the 4dsolutions ecosystem. See the Resources section for comprehensive details.
8.6
Fisher Information Matrix (FIM)
Background: Fisher information.
𝐹𝑖,𝑗= 𝔼[𝜕log 𝑝(𝑥; 𝜃)
𝜕𝜃𝑖
𝜕log 𝑝(𝑥; 𝜃)
𝜕𝜃𝑗
]
(8)
Notes.
50

## Page 51

• Defines the Fisher information matrix as the expected outer product of score functions; see Fisher
information.
Figure: empirical estimate shown in the FIM heatmap figure. See code: fisher_information_matrix.
See src/information.py — empirical outer-product estimator (fisher_information_matrix).
8.7
Empirical Fisher Information Matrix
For empirical estimation from data, the Fisher Information Matrix is computed as:
𝐹𝑖,𝑗= 1
𝑁
𝑁
∑
𝑛=1
𝜕log 𝑝(𝑥𝑛; 𝜃)
𝜕𝜃𝑖
𝜕log 𝑝(𝑥𝑛; 𝜃)
𝜕𝜃𝑗
(9)
Notes.
• Empirical estimate of the FIM from 𝑁data samples; converges to the theoretical FIM as 𝑁→∞.
• Used in natural gradient descent and information geometry applications.
8.8
Natural Gradient
Background: Natural gradient (Amari).
𝜃←𝜃−𝜂𝐹(𝜃)−1 ∇𝜃𝐿(𝜃)
(10)
Explanation.
• Natural gradient update: right-precondition the gradient by the inverse of the Fisher metric (Amari);
see Natural gradient.
See code: natural_gradient_step.
See src/information.py — damped inverse-Fisher step (natural_gradient_step).
8.9
Free Energy (Active Inference)
ℱ= −log 𝑃(𝑜∣𝑠) + KL[𝑄(𝑠) ‖ 𝑃(𝑠)]
(11)
Explanation.
• Partition: variational free energy decomposes into expected negative log-likelihood and KL between
approximate posterior and prior; see Free energy principle.
See code: free_energy.
See src/information.py — discrete-state variational free energy (free_energy).
Note: The main figures demonstrating natural gradient trajectories and free energy landscapes are
shown in Section 4: Optimization in 4D. The appendix focuses on unique figures specific to mathematical
formulations and validation.
8.10
Quadray Normalization (Fuller.4D)
Given 𝑞= (𝑎, 𝑏, 𝑐, 𝑑), choose 𝑘= min(𝑎, 𝑏, 𝑐, 𝑑) and set 𝑞′ = 𝑞−(𝑘, 𝑘, 𝑘, 𝑘) to enforce at least one zero
with non-negative entries.
51

## Page 52

8.11
Distance (Embedding Sketch; Coxeter.4D slice)
Choose linear map 𝑀from quadray to ℝ3 (or ℝ4) consistent with tetrahedral axes; then 𝑑(𝑞1, 𝑞2) =
‖𝑀(𝑞1) −𝑀(𝑞2)‖2.
8.12
Minkowski Line Element (Einstein.4D analogy)
𝑑𝑠2 = −𝑐2 𝑑𝑡2 + 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2
(12)
Background: Minkowski space.
8.13
High-Precision Arithmetic Note
When evaluating determinants, FIMs, or geodesic distances for sensitive problems, use quad precision
(binary128) via GCC’s libquadmath (__float128, functions like expq, sqrtq, and quadmath_snprintf).
See GCC
libquadmath. Where possible, it is useful to use symbolic math libraries like SymPy to compute exact
values.
8.13.1
Reproducibility artifacts and external validation
• This manuscript’s artifacts: Raw data in quadmath/output/ for reproducibility and downstream anal-
ysis:
– fisher_information_matrix.csv / .npz: empirical Fisher matrix and inputs
– fisher_information_eigenvalues.csv / fisher_information_eigensystem.npz: eigenspectrum and eigenvec-
tors
– natural_gradient_path.png with natural_gradient_path.csv / .npz: projected trajectory and raw coordi-
nates
– ivm_neighbors_data.csv / ivm_neighbors_edges_data.npz: neighbor coordinates (Quadray and XYZ)
– polyhedra_quadray_constructions.png: synergetics volume relationships schematic
• External validation resources: The 4dsolutions ecosystem provides extensive cross-validation.
See the Resources section for comprehensive details on computational implementations and valida-
tion.
8.14
Namespaces summary (notation)
• Coxeter.4D: Euclidean E4; regular polytopes; not spacetime (cf. Coxeter, Regular Polytopes, Dover
ed., p. 119). Connections to higher-dimensional lattices and packings as in Conway & Sloane.
• Einstein.4D: Minkowski spacetime; indefinite metric; used here only as a metric analogy when dis-
cussing geodesics and information geometry.
• Fuller.4D: Quadrays/IVM; tetrahedral lattice with integer tetravolume; unit regular tetrahedron has
volume 1; synergetics scale relations (e.g., S3).
52

## Page 53

9
Appendix: The Free Energy Principle and Active Inference
9.1
Overview
The Free Energy Principle (FEP) posits that biological systems maintain their states by minimizing vari-
ational free energy, thereby reducing surprise via prediction and model updating. Active Inference ex-
tends this by casting action selection as inference under prior preferences. Background: see the concise
overview on the Free energy principle and the monograph Active Inference (MIT Press).
This appendix emphasizes relationships among:
(i) the four-fold partition of Active Inference, (ii)
Quadrays (Fuller.4D) as a geometric scaffold for mapping this partition, and (iii) information-geometric
flows (Einstein.4D analogy) that underpin perception–action updates. For the naming of 4D namespaces
used throughout—Coxeter.4D (Euclidean E4), Einstein.4D (Minkowski spacetime analogy), Fuller.4D
(Synergetics/Quadrays)—see 02_4d_namespaces.md.
9.2
Mathematical Formulation and Equation Callouts (Equations linkage)
• Variational free energy (discrete states) — see Eq. (11) in the equations appendix, implemented by
free_energy.
• Fisher Information Matrix (FIM) as metric — see Eq.
(8) in the equations appendix and
fisher_information_matrix.
• Natural gradient descent under information geometry — see Eq. (10) in the equations appendix and
natural_gradient_step; overview: Natural gradient.
Figures: The following Active Inference figures demonstrate the integration of natural gradient descent
with Active Inference principles and the 4D framework context.
Discrete variational optimization on the quadray lattice:
discrete_ivm_descent greedily descends a free-
energy-like objective over IVM moves, yielding integer-valued trajectories. See the path animation arti-
fact discrete_path.mp4 in quadmath/output/.
9.3
Four-Fold Partition and Tetrahedral Mapping (Quadrays; Fuller.4D)
Active Inference partitions the agent–environment system into four coupled states:
• Internal (𝜇) — agent’s internal states
• Sensory (𝑠) — observations
• Active (𝑎) — actions
• External (𝜓) — latent environmental causes
See, for an overview of this partition and generative process formulations, the Active Inference review
and the general entry on Active inference.
Tetrahedral mapping via Quadrays (Fuller.4D): assign each state to a vertex of a tetrahedron, using
Quadray coordinates (A,B,C,D) with non-negative components and at least one zero after normalization.
One canonical mapping is
A \leftrightarrow Internal (\mu),
B \leftrightarrow Sensory (s),
C \leftrightarrow
Active (a), D \leftrightarrow External (\psi).
The edges capture the pairwise couplings (e.g., \mu\text{--}s
for perceptual inference; a\text{--}\psi for control).
Integer tetravolume then quantifies the “coupled
capacity” region spanned by jointly feasible states in a time slice; see Quadray and tetravolume methods
in 03_quadray_methods.md.
Interpretation note: this Quadray-based mapping is a didactic geometric scaffold. It is not standard in
the Active Inference literature, which typically develops the four-state partition in probabilistic graphical
terms. Our use highlights structural symmetries and discrete volumetric quantities available in Fuller.4D,
building on the computational foundations developed in the 4dsolutions ecosystem for tetrahedral model-
ing and volume calculations. See the Resources section for comprehensive details on the computational
implementations.
53

## Page 54

Figure 14: Active Inference four-fold partition mapped to a Quadray tetrahedron in Fuller.4D.
This 3D tetrahedral visualization demonstrates the geometric embedding of Active Inference’s fundamen-
tal four-fold partition within the Quadray coordinate system. Tetrahedral structure: The four vertices
of the regular tetrahedron represent the four components of the Active Inference framework: percep-
tion, action, internal states, and external states.
Partition mapping: Each face of the tetrahedron
corresponds to a specific partition of the four-fold system, with the edges representing the relationships
and interactions between different components. Fuller.4D significance: This geometric representa-
tion leverages the tetrahedral nature of Quadray coordinates to provide an intuitive visualization of the
Active Inference framework’s structure. The tetrahedron serves as a natural container for the four-fold
partition, emphasizing the interconnected nature of perception, action, and state representation in active
inference. Optimization context: The tetrahedral geometry also suggests natural optimization strate-
gies that respect the four-fold structure, potentially leading to more efficient inference algorithms that
leverage the geometric relationships between different components. This visualization demonstrates how
the Fuller.4D framework can provide insights into complex systems like Active Inference through geo-
metric intuition.
54

## Page 55

Figure 15: 4D Natural Gradient Trajectory with Active Inference Context. This comprehensive
visualization demonstrates natural gradient descent operating within the Active Inference framework,
showing how information-geometric optimization drives perception-action dynamics.
3D Trajectory:
The main panel shows the 4D parameter evolution in 3D space with time encoded as color, representing
the four-fold partition of Active Inference: perception (μ), action (a), internal states (s), and external
causes (ψ). Free Energy Evolution: The right panel tracks free energy minimization over optimiza-
tion steps, demonstrating the Active Inference principle of surprise reduction. Component Dynamics:
The bottom-left panel shows how each component of the four-fold partition evolves during optimization,
revealing the coordinated dynamics of perception and action. Optimization Diagnostics: The bottom-
center panel displays step sizes and gradient norms, providing insights into the convergence behavior
and numerical stability of the natural gradient algorithm. Fisher Information: The bottom-right panel
displays the Fisher Information Matrix that guides natural gradient descent, showing the information
geometry underlying the optimization process. This figure demonstrates how natural gradient descent
implements geodesic motion on the information manifold, analogous to how particles follow geodesics
in Einstein.4D spacetime, while operating within the tetrahedral structure of Fuller.4D coordinates. The
optimization now shows stable convergence in just 11 steps with final parameter errors below 0.015,
demonstrating the effectiveness of information-geometric optimization in Active Inference frameworks.
55

## Page 56

Figure 16: Free Energy Landscape in 4D Active Inference Framework. This comprehensive visu-
alization explores the variational free energy surface over perception and action parameters. 3D land-
scape: The surface plot shows the free energy as a function of two variational parameters, revealing the
complex topology that Active Inference optimization must navigate. Contour analysis: 2D contours pro-
vide detailed information about parameter sensitivity and optimization paths. Cross-sectional analysis:
Multiple cross-sections at different parameter values demonstrate how free energy varies with respect
to individual parameters, revealing the landscape’s structure. Four-fold partition visualization: The
text panel explains how Active Inference maps to tetrahedral structures in Fuller.4D, with the four compo-
nents (μ, s, a, ψ) representing internal states, sensory observations, actions, and external causes. Infor-
mation geometry metrics: Local curvature analysis reveals the Fisher information structure, showing
how the information manifold’s geometry influences optimization dynamics. Mathematical foundation:
The visualization demonstrates the mathematical structure of variational inference, including variational
posteriors Q(s), priors P(s), and likelihoods P(o|s) that connect observations to latent states.
56

## Page 57

Code linkage (no snippet): see example_partition_tetra_volume in src/examples.py and the partition tetrahedron
figure above.
9.4
How the 4D namespaces relate here
• Fuller.4D (Quadrays): geometric embedding of the four-state partition on a tetrahedron; integer
tetravolumes and IVM moves provide discrete combinatorial structure.
• Coxeter.4D (Euclidean E4): exact Euclidean measurements (e.g., Cayley–Menger determinants) for
tetrahedra underlying volumetric comparisons and scale relations.
• Einstein.4D (Minkowski analogy): information-geometric flows (natural gradient, metric-aware up-
dates) supply a continuum picture for perception–action dynamics.
The three roles are complementary: Fuller.4D encodes partition structure, Coxeter.4D provides exact
metric geometry for static comparisons, and Einstein.4D guides dynamical descent.
9.5
Joint Optimization in the Tetrahedral Framework (Methods linkage)
• Perception: update 𝜇to minimize prediction error on 𝑠under the generative model (descending
∇𝜇𝐹).
• Action: select 𝑎that steers 𝜓toward preferred outcomes (descending ∇𝑎𝐹).
Continuous-time flows (Einstein.4D analogy for metric/geodesic intuition):
see perception_update and
action_update in src/information.py. Discrete Quadray moves connect to these flows via greedy descent on a
local free-energy-like objective; see discrete_ivm_descent in src/discrete_variational.py and the path artifacts
in quadmath/output/.
9.6
Implications for AI and Robust Computation
FEP/Active Inference provide algorithms that unify perception and action under uncertainty, offering
biologically plausible alternatives to standard RL with adaptive exploration and robust decision-making.
See applications in AI (arXiv:1907.03876).
9.7
Code, Reproducibility, and Cross-References
– Equation references: Eq.
(Free Energy), Eq.
(FIM), Eq.
(Natural Gradient) in 08_equations_appendix
.md.
– Code anchors (for readers who want to run experiments):
free_energy, fisher_information_matrix,
natural_gradient_step,
perception_update,
action_update, and discrete_ivm_descent in src/information.py and src/
discrete_variational.py.
Demo and figures generated by quadmath/scripts/information_demo.py and quadmath/scripts/active_inference_figures
.py output to quadmath/output/:
• Active
Inference
Visualizations:
figure_13_4d_trajectory.png,
figure_14_free_energy_landscape.png
demonstrating 4D framework integration
• Information Geometry Visualizations: fisher_information_matrix.png, fisher_information_eigenspectrum.
png, natural_gradient_path.png, free_energy_curve.png, partition_tetrahedron.png
• Raw data: figure_13_data.npz, figure_14_data.npz, fisher_information_matrix.csv, fisher_information_matrix.npz
(F, grads, X, y, w_true, w_est), fisher_information_eigenvalues.csv, fisher_information_eigensystem.npz
• External validation: Cross-reference with volume calculations and tetrahedral modeling tools from
the 4dsolutions ecosystem. See the Resources section for comprehensive details.
57

## Page 58

10
Appendix: Symbols and Glossary
This appendix consolidates the symbols, variables, and constants used throughout the manuscript.
10.1
Sets and Spaces
Symbol
Name
ℝ𝑛
Euclidean space
IVM
Isotropic Vector Matrix
Coxeter.4D
Euclidean 4D (E4)
Einstein.4D
Minkowski spacetime (3+1)
Fuller.4D
Synergetics/Quadray tetrahedral space
Descriptions:
• ℝ𝑛: 𝑛-dimensional real vector space.
• IVM: Quadray integer lattice (CCP sphere centers).
• Coxeter.4D: Four-dimensional Euclidean geometry (not spacetime); see Coxeter, Regular Polytopes
(Dover ed., p. 119); related lattice/packing background in Conway & Sloane.
• Einstein.4D: Relativistic spacetime with Minkowski metric.
• Fuller.4D: Quadrays with projective normalization and IVM unit conventions.
10.2
Quadray Coordinates and Geometry
Symbol
Name
Description
𝑞= (𝑎, 𝑏, 𝑐, 𝑑)
Quadray point
Non-negative coordinates with
at least one zero after
normalization
𝐴, 𝐵, 𝐶, 𝐷
Quadray axes
Canonical tetrahedral axes
mapped by the embedding
𝑘
Normalization offset
𝑘= min(𝑎, 𝑏, 𝑐, 𝑑) used to set
𝑞′ = 𝑞−(𝑘, 𝑘, 𝑘, 𝑘)
𝑞′
Normalized Quadray
Canonical representative with
at least one zero and
non-negative entries
𝑃0, … , 𝑃3
Tetrahedron vertices
Vertices used in volume
formulas
𝑑𝑖𝑗
Pairwise distances
Distance between vertices 𝑃𝑖
and 𝑃𝑗(squared in CM matrix)
det(⋅)
Determinant
Determinant of a matrix
|⋅|
Magnitude
Absolute value (determinant
magnitude)
𝑉𝑖𝑣𝑚
Tetravolume (IVM)
Tetrahedron volume in
synergetics/IVM units; unit
regular tetra has 𝑉𝑖𝑣𝑚= 1
𝑉𝑥𝑦𝑧
Tetravolume (XYZ)
Euclidean tetrahedron volume
𝑆3
Scale factor
𝑆3 = √9/8 with
𝑉𝑖𝑣𝑚= 𝑆3 𝑉𝑥𝑦𝑧(synergetics
unit convention)
Coxeter.4D
Namespace
Euclidean E4; regular
polytopes
58

## Page 59

Symbol
Name
Description
Einstein.4D
Namespace
Minkowski spacetime (metric
analogy only here)
Fuller.4D
Namespace
Quadrays/IVM; integer
tetravolume
Eq. (lattice_det)
Lattice determinant
Integer-lattice volume via 3x3
determinant
Eq. (ace5x5)
Tom Ace 5x5
Direct IVM tetravolume from
Quadrays
Eq. (cayley_menger)
Cayley–Menger
Length-based formula: 288
V^2 = det(·)
10.3
Optimization and Algorithms
Symbol
Name
𝛼
Reflection coefficient
𝛾
Expansion coefficient
𝜌
Contraction coefficient
𝜎
Shrink coefficient
𝑉𝑖𝑣𝑚
Integer volume monitor
Descriptions:
• 𝛼, 𝛾, 𝜌, 𝜎: Nelder–Mead parameters (typical values 1, 2, 0.5, 0.5).
• 𝑉𝑖𝑣𝑚: Tracks simplex volume across iterations.
10.4
Information Theory and Geometry
Symbol
Name
Description
log
Natural logarithm
Logarithm base 𝑒
𝔼[⋅]
Expectation
Mean with respect to a
distribution
𝐹𝑖𝑗
Fisher Information Matrix
𝔼[𝜕𝜃𝑖log 𝑝⋅𝜕𝜃𝑗log 𝑝]; Eq. (8) in
the equations appendix
ℱ
Variational free energy
−log 𝑃(𝑜∣
𝑠) + KL[𝑄(𝑠) ‖ 𝑃(𝑠)]; Eq. (11)
in the equations appendix
KL[𝑄‖ 𝑃]
Kullback–Leibler divergence
∑𝑄log(𝑄/𝑃); information
distance
∇𝜃𝐿
Natural gradient
𝐹(𝜃)−1∇𝜃𝐿(𝜃); Eq. (10) in the
equations appendix
𝜂
Step size
Learning-rate scalar used in
updates
𝜃
Parameters
Model parameter vector;
indices 𝜃𝑖
𝑑𝑠2
Minkowski line element
−𝑐2 𝑑𝑡2 + 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2; Eq.
(12) in the equations appendix
𝑐
Speed of light
Physical constant appearing in
Minkowski metric
59

## Page 60

10.5
Embeddings and Distances
Symbol
Name
Description
𝑀
Embedding matrix
Linear map from Quadray to
ℝ3 (Urner-style unless noted)
‖⋅‖2
Euclidean norm
√𝑥2
1 + ⋯+ 𝑥2𝑛
𝑅, 𝐷
Edge scales
Cube edge 𝑅and Quadray
edge 𝐷with 𝐷= 2𝑅(common
convention)
10.6
Greek Letters (usage)
Symbol
Name
Description
𝛼, 𝛾, 𝜌, 𝜎
NM coefficients
Nelder–Mead parameters
(reflection, expansion,
contraction, shrink)
𝜃
Theta
Parameter vector in models
and metrics
𝜇
Mu
Internal states (Active
Inference)
𝜓
Psi
External states (Active
Inference)
𝜂
Eta
Step size / learning rate
10.7
Notes (usage and cross-references)
• Figures referenced: In-text references use LaTeX’s automatic figure numbering for consistent
cross-referencing.
• Equation references: Use labels defined in the text (e.g., Eq. (1) in the equations appendix).
• Namespaces: We use Coxeter.4D, Einstein.4D, Fuller.4D consistently to designate Euclidean E4,
Minkowski spacetime, and Quadray/IVM synergetics, respectively.
This avoids conflation of Eu-
clidean 4D objects (e.g., tesseracts) with spacetime constructs and synergetic tetravolume conven-
tions.
• External validation: Cross-reference implementations from the 4dsolutions ecosystem for algorith-
mic verification and performance comparison baselines. See the Resources section for comprehen-
sive details.
10.8
Polyhedra and Synergetic Shapes
Symbol
Name
Description
Tetrahedron
Regular tetrahedron
Fundamental unit with V=1 in
IVM units
Cube
Regular hexahedron
V=3 in IVM units; orthogonal
space-filling
Octahedron
Regular octahedron
V=4 in IVM units;
edge-midpoint construction
Rhombic Dodecahedron
12-faced solid
V=6 in IVM units; Voronoi cell
of FCC packing
Cuboctahedron
Vector equilibrium
V=20 in IVM units; shell of 12
IVM neighbors
60

## Page 61

Symbol
Name
Description
Truncated Octahedron
Archimedean solid
V=20 in IVM units;
space-filling tiling
10.9
Acronyms and abbreviations
Acronym
Meaning
CM
Cayley–Menger (determinant-based tetrahedron
volume)
PdF
Piero della Francesca (Heron-like tetrahedron
volume)
GdJ
Gerald de Jong (Quadray-native tetravolume
expression)
K-FAC
Kronecker-Factored Approximate Curvature
(optimizer using structured Fisher)
CCP
Cubic Close Packing (same centers as FCC)
FCC
Face-Centered Cubic (same centers as CCP)
E4
Four-dimensional Euclidean space (Coxeter.4D)
NM
Nelder–Mead (simplex optimization algorithm)
4dsolutions
Kirby Urner’s GitHub organization with
extensive Quadray implementations
BEAST
Synergetic modules (B, E, A, S, T) in Fuller’s
hierarchical system
OCN
Oregon Curriculum Network (educational
framework integrating Quadrays)
POV-Ray
Persistence of Vision Raytracer (used in
quadcraft.py visualizations)
10.10
API Index (auto-generated; Methods linkage)
The table below enumerates public symbols from src/ modules.
Module
Symbol
Kind
Signature
Summary
cayley_menger
ivm_tetra_volume_cayley_menger
function
(d2)
Compute IVM
tetravolume from
squared
distances via
Cayley–Menger.
cayley_menger
tetra_volume_cayley_menger
function
(d2)
Compute
Euclidean
tetrahedron
volume from
squared
distances
(Coxeter.4D).
61

## Page 62

Module
Symbol
Kind
Signature
Summary
conversions
quadray_to_xyz
function
(q, M)
Map a Quadray to
Cartesian XYZ
via a 3x4
embedding
matrix (Fuller.4D
-> Coxeter.4D
slice).
conversions
urner_embedding
function
(scale)
Return a 3x4
Urner-style
symmetric
embedding
matrix (Fuller.4D
-> Coxeter.4D
slice).
discrete_variational
DiscretePath
class
| Optimization
trajectory on the
integer quadray
lattice. | | `
discrete_variational
` | `OptionalMoves`
| class |
discrete_variational
apply_move
function
(q, delta)
Apply a lattice
move and
normalize to the
canonical
representative.
discrete_variational
discrete_ivm_descent
function
(objective, start,
moves=, max_iter=,
on_step=)
Greedy discrete
descent over the
quadray integer
lattice.
discrete_variational
neighbor_moves_ivm
function
()
Return the 12
canonical IVM
neighbor moves
as Quadray
deltas.
examples
example_cuboctahedron_neighbors
function
()
Return twelve-
around-one IVM
neighbors (vector
equilibrium
shell).
examples
example_cuboctahedron_vertices_xyz
function
()
Return XYZ
coordinates for
the twelve-
around-one
neighbors.
examples
example_ivm_neighbors function
()
Return the 12
nearest IVM
neighbors as
permutations of
{2,1,1,0}
(Fuller.4D).
62

## Page 63

Module
Symbol
Kind
Signature
Summary
examples
example_optimize
function
()
Run
Nelder–Mead
over integer
quadrays for a
simple convex
objective
(Fuller.4D).
examples
example_partition_tetra_volume
function
(mu, s, a, psi)
Construct a
tetrahedron from
the four-fold
partition and
return
tetravolume
(Fuller.4D).
examples
example_volume
function
()
Compute the unit
IVM tetrahedron
volume from
simple quadray
vertices
(Fuller.4D).
geometry
minkowski_interval
function
(dt, dx, dy, dz, c)
Return the
Minkowski
interval squared
ds^2
(Einstein.4D).
63

## Page 64

Module
Symbol
Kind
Signature
Summary
glossary_gen
ApiEntry
class
| | | `glossary_gen
` | `
build_api_index` |
function | `(
src_dir)` | | | `
glossary_gen` | `
generate_markdown_table
` | function | `(
entries)` | | | `
glossary_gen` | `
inject_between_markers
` | function | `(
markdown_text,
begin, end, payload
)` | | | `
information` | `
action_update` |
function | `(action
, free_energy_fn,
step_size, epsilon)
` | Continuous-time
action update: da/
dt = - dF/da. | | `
information` | `
active_inference_step
` | function | `(mu
, action,
free_energy_fn,
derivative_operator
, step_size,
epsilon)` | Joint
perception-action
update step in
Active Inference. |
| `information` |
`
expected_free_energy
` | function | `(
log_p_o_given_s, q,
p, log_p_o)` |
Expected free
energy for Active
Inference with
prior preferences.
| | `information` |
`
finite_difference_gradient
` | function | `(
function, x,
epsilon)` | Compute
numerical gradient
of a scalar
function via
central differences
. | | `information`
| `
fisher_information_matrix
` | function | `(
gradients,
64

## Page 65

Module
Symbol
Kind
Signature
Summary
nelder_mead_quadray
centroid_excluding
function
(vertices,
exclude_idx)
Integer centroid
of three vertices,
excluding the
specified index.
nelder_mead_quadray
compute_volume
function
(vertices)
Integer IVM
tetra-volume
from the first
four vertices.
nelder_mead_quadray
nelder_mead_quadray
function
(f,
initial_vertices,
alpha, gamma, rho,
sigma, max_iter,
tol, on_step)
Nelder–Mead on
the integer
quadray lattice.
nelder_mead_quadray
order_simplex
function
(vertices, f)
Sort vertices by
objective value
ascending and
return paired
lists.
nelder_mead_quadray
project_to_lattice
function
(q)
Project a quadray
to the canonical
lattice
representative
via normalize.
paths
get_data_dir
function
()
Return quadmath/
output/data path
and ensure it
exists.
paths
get_figure_dir
function
()
Return quadmath/
output/figures path
and ensure it
exists.
paths
get_output_dir
function
()
Return
quadmath/output
path at the repo
root and ensure
it exists.
paths
get_repo_root
function
(start)
Heuristically find
repository root
by walking up
from start.
quadray
DEFAULT_EMBEDDING
constant
| | | `quadray` | `
Quadray` | class |
Quadray vector
with
non-negative
components and
at least one zero
(Fuller.4D).
quadray
ace_tetravolume_5x5
function
(p0, p1, p2, p3)
Tom Ace 5x5
determinant in
IVM units
(Fuller.4D).
65

## Page 66

Module
Symbol
Kind
Signature
Summary
quadray
dot
function
(q1, q2, embedding)
Return Euclidean
dot product
<q1,q2> under
the given
embedding.
quadray
integer_tetra_volume
function
(p0, p1, p2, p3)
Compute integer
tetra-volume
using det[p1-p0,
p2-p0, p3-p0]
(Fuller.4D).
quadray
magnitude
function
(q, embedding)
Return Euclidean
magnitude ||q||
under the given
embedding
(vector norm).
quadray
to_xyz
function
(q, embedding)
Map quadray to
R^3 via a 3x4
embedding
matrix (Fuller.4D
-> Coxeter.4D
slice).
symbolic
cayley_menger_volume_symbolic
function
(d2)
Return symbolic
Euclidean
tetrahedron
volume from
squared
distances.
symbolic
convert_xyz_volume_to_ivm_symbolic
function
(V_xyz)
Convert a
symbolic
Euclidean
volume to IVM
tetravolume via
S3.
visualize
animate_discrete_path function
(path, embedding,
save)
Animate a point
moving along a
discrete quadray
path.
visualize
animate_simplex
function
(vertices_list,
embedding, save)
Animate simplex
evolution across
iterations.
visualize
plot_ivm_neighbors
function
(embedding, save)
Scatter the 12
IVM neighbor
points in 3D.
visualize
plot_partition_tetrahedron
function
(mu, s, a, psi,
embedding, save)
Plot the four-fold
partition as a
labeled
tetrahedron in
3D.
visualize
plot_simplex_trace
function
(state, save)
Plot per-iteration
diagnostics for
Nelder–Mead.
66


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*Extraction method: pymupdf*