The Synergetics Cookbook

Description

The methods behind this project are catalogued in The Synergetics Cookbook — a practitioner’s reference covering the metric tensor, rotation algebra, convex hull, and projection pipeline that make Quadray coordinates self-sufficient. Every method was found by attending to what the tetrahedral basis already encodes, not translated from Cartesian originals.

The forward was written at the end of a long working session on the abcd.earth codebase — addressed to every reader, human or machine, who opens the document intending to build something with it.

Contents: Foundations & Coordinate System — Operations (Translation, Scaling, Mirrors) — Metric Tensor $M = 4I – J$ — Tetrahedral Pythagoras — Pseudo-Inverse $B^\dagger$ — Height Functions & Plane Constraints — Integer Polyhedra Catalogue (61+) — F, G, H Circulant Rotations — ABCD-to-Clip Projection — Convex Hull — Computational Techniques — CCP Frequency & Volumetric Grid — Geodesic Subdivision — Anti-Patterns & Common Mistakes

Format and versioning:

PDF, 45+ pages, LaTeX-typeset. Living document — current version v0.7 (April 2026). Purchasers receive updates.

The ABCD.EARTH codebase proves these methods work. This document is why they work — and the specification to build your own.

The Quadray Bible catalogues 40+ formulae across 13 sections. The crown jewels:

1. Metric tensor $M = 4I – J$ — computes Cartesian Quadrance directly from ABCD coordinates; integer arithmetic on integer polyhedra, no conversion, no radicals
2. Tetrahedral Pythagoras — the +2 cross term is the algebraic signature of Tetrahedral closure, bridging Fuller’s Synergetics and Wildberger’s Rational Trigonometry in one equation
3. F, G, H circulant rotation — native ABCD rotation matrices; at 120° collapses to pure integer permutation, exact rationals at all Tetrahedral symmetry angles
4. ABCD-to-clip — single matrix multiply from tetrahedral coordinates to GPU clip space, no XYZ anywhere in the pipeline
5. Geodesic “one √” — reduces the per-vertex radical count from many (classical methods) to exactly one: the geometrically irreducible scaling from flat subdivision to curved surface

The Synergetics Cookbook is not a textbook — it’s a book of recipes. With 40+ formulae across 13 sections, we define a complete rational geometry engine from coordinates to clip space. The source code shows that it works. This document shows why it works — and gives you everything you need to build your own implementation, in any language, on any hardware, including hardware that has never existed: integer-arithmetic geometry engines with no floating-point unit required for the core Platonic and Archimedean families. $25 is a screaming deal for a specification that took decades to derive.

Use Cases

1. Build Your Own Geometry Engine The Quadray Bible is a complete specification — coordinate system, metric tensor, rotation algebra, convex hull, projection pipeline, all the way to GPU clip space. Implement it in Rust, Python, C, Swift, JavaScript, or any language from this document alone. The open-source ABCD.EARTH codebase shows that it works. This document shows why, and gives you everything you need to build your own.

2. Avoid the Mistakes We Already Made Six documented anti-patterns distill decades of wrong turns into two pages — the Cartesian roundtrip, premature radicals, naive interpolation bugs, normalisation traps, and more. Each one names the specific mistake, explains why it fails, and gives the correct ABCD-native solution. This section alone can save months of debugging, helping resist and correct the classical but irrational methods most AI-coding agents have been trained on.

3. Design Integer-Arithmetic Hardware The four core Platonic solids use coordinates in {0, 1}. The metric tensor is multiply-and-add on small integers. A 120° rotation is a pure permutation — zero ALU cycles. Most polyhedral geometry in this framework needs no floating-point unit at all. This is the specification for a new class of rational math engine: FPGA geometry cores, embedded CNC/robotics controllers, RISC-V chips doing geodesic computation without an FPU, or WebAssembly modules that avoid f64 entirely.

4. Benchmark and Validate Rational Geometry Tools 61+ polyhedra generators, each classified by arithmetic precision — integer, rational, golden-algebraic, or float. 1,107+ tests proving the Cartesian roundtrip is unnecessary, not just avoidable. If you are building or evaluating rational geometry software, this is the reference catalogue to measure against.

5. Bridge Two Mathematical Traditions Fuller’s Synergetics (tetrahedral coordinate thinking) and Wildberger’s Rational Trigonometry (algebraic exactness) have never been formally joined in a single computational framework. The metric tensor M = 4I − J is that bridge. Researchers, educators, and practitioners in either tradition gain a documented pathway to the other — with working code and verified formulae, not just theory.

*Telemetry, Tracking, Guidance, Target Acquisition. Integer and rational arithmetic eliminates cumulative floating-point drift — the class of error behind the 1991 Patriot/Scud tracking failure (0.34s clock drift → 500m gate error). ABCD geometry computation is exact, deterministic, and bit-reproducible across platforms. Tracking spreads and intercept quadrances computed via the metric tensor carry zero rounding error at any operation count.

Licence:  © Andy Ross Thomson, M.Arch, OAA 2026 ~ ALL RIGHTS RESERVED