The first rational-algebra-driven CAD-native tool built around Isotropic Vector Matrix (IVM) geometry — where 30 of 32 polyhedra are algebraically exact and the remaining 2 achieve machine precision.
abcd.earth eliminates this problem at the foundation. By working in Quadray ABCD coordinates with Rational Trigonometry (N.J. Wildberger), we replace:
| Traditional CAD | abcd.earth |
|---|---|
| Distance = √(dx² + dy²) | Quadrance = dx² + dy² (exact integer/rational — √ deferred to GPU) |
| Angle = arccos(dot product) | Spread = 1 − dot²/(Q₁·Q₂) (exact rational) |
| sin(θ), cos(θ) for rotation | Two reflections via algebraic line reflection (no trig) |
| π, sin, cos for polygon vertices | Wildberger tangent recurrence (1 √ for any N-gon, then pure algebra) |
| φ = (1+√5)/2, then φ² via multiplication | φ² = φ + 1 (algebraic identity, not floating-point multiply) |
| Floating-point vertex coordinates | Integer, rational, or phi-algebraic ABCD coordinates |
| XYZ → rotate → XYZ (accumulates error) | ABCD → ABCD (closed-form, no coordinate conversion) |
Overview
abcd.earth is a vector-native geometric modelling engine designed to generate planar, radial, and curved spaceframes with arithmetic precision. Unlike traditional CAD software that relies on floating-point Cartesian approximations, our software operates entirely in pure ABCD (quadray) coordinates — a tetrahedral coordinate system where space is described by four equiangular basis vectors rather than three orthogonal axes.
The result: geometry that is algebraically exact from math core to GPU shader, with no accumulated floating-point error, no trigonometric drift, and none of the inherent floating-point inaccuracies that result from operating always and only with Cartesian coordinate systems.
Whether you’re exploring new geometric forms or creating actionable construction documents, abcd.earth handles it all — from hobbyist experimentation to professional fabrication workflows.
The Precision Advantage: Why This Matters
Traditional CAD software stores geometry as floating-point XYZ coordinates. Every operation — rotation, scaling, boolean, subdivision — introduces rounding error. Over thousands of operations on complex geometries, especially when dealing with geodesic domes or spaceframes, these errors compound: vertices drift, edges misalign, and symmetry degrades silently.
The √2 that appears in Cartesian edge lengths is a Cartesian disease, not a geometric fact. In Quadray coordinates, the tetrahedron/cube edge ratio is 2:1 — pure integer. The irrational only appears when you project to XYZ for display.
Close-packed spheres and Fuller’s Frequency. The IVM (Isotropic Vector Matrix) is the geometry of close-packed spheres — the cuboctahedral packing where every sphere touches 12 neighbours at equal distance. In abcd.earth, the Frequency parameter F is the Quadray scale factor: cube_edge = 2F, and sphere radius = F (one Quadray unit). At integer frequencies (F=1, F=2, F=3…), all polyhedra land exactly on integer Quadray grid points — the vertices of the IVM lattice.
Polyhedral Precision Inventory: 32/32 Complete
Every polyhedron in the engine is characterized by its coordinate exactness — the algebraic field in which its ABCD vertex coordinates live. These are measured properties of the actual vertex data, not theoretical claims.
Platonic Solids – the foundation (6 solids)
| solid | vertices | coordinate field | edge quadrance | notes |
|---|---|---|---|---|
| Tetrahedron | 4 | ℤ (integer {0,1}) | 8 (exact) | [1,0,0,0] basis vectors |
| Tetrahedron dual | 4 | ℤ (integer {0,1}) | 8 (exact) | [0,1,1,1] complements |
| cube | 8 | ℤ (integer {0,1}) | 4 (exact) | Tet ∪ Dual Tet vertices |
| octahedron | 6 | ℤ (integer {0,1}) | 8 (exact) | [1,1,0,0] all pairs |
| icosahedron | 12 | ℚ (φ) (phi-algebraic) | 4 (exact) | Permutations of (0, 1, φ, φ²)/2 |
| dodecahedron | 20 | ℚ (φ) (phi-algebraic) | 4/φ² (exact) | Cube + phi-rectangle family |
Key insight: The first four Platonics use only {0, 1} integer coordinates. No floating-point at all. The icosahedral family uses golden ratio identities (φ² = φ + 1) — algebraic, never transcendental.
Precision Summary
| Precision Level | Count | coordinate field | Error |
|---|---|---|---|
| Integer | 4 | ℤ — ABCD ∈ {0, 1} | Zero |
| Rational | 10 | ℚ — exact rational ABCD | Zero |
| Phi-algebraic | 16 | ℚ(φ) — rational in golden ratio | Zero |
| Machine (f64) | 2 | ℚ(τ) — tribonacci field | ≤ 6×10⁻¹⁵ |
| Totals | 32 | 30 exact, 2 machine-precision |
30 of 32 polyhedra have zero floating-point error. Not “low error” — zero. The ABCD coordinates are exact algebraic values. Edge quadrance and face spread are exact rational numbers (or rational functions of φ). This is the payoff of building on Rational Trigonometry instead of classical trig.
Core Features
ivm lattice engine
- Generate tetrahedral, cuboctahedral, and isotropic lattices.
- 10 spatial matrix types: 5 planar (Cartesian Square/Hex, IVM Taxicab/Chebyshev/Hex) + 5 radial (Cartesian Cubic, IVM Tetrahedral/Octahedral/Cuboctahedral/RD).
- Per-type controls: independent Col/Row dimensions (1–24), spacing, rotation with algebraic fast paths for constructible angles.
- Maintains exact symmetry and integer relationships.
4d / quadray coordinates
- Native ABCD representation ensures arithmetic precision.
- ABCD-to-clip GPU pipeline — Tom Ace basis folded into camera matrix. Shader is one multiply + one add. No XYZ intermediate.
- Avoids floating-point drift common in Cartesian-based CAD.
rational trigonometry math engine
- Built on N.J. Wildberger’s algebraic framework: quadrance replaces distance, spread replaces angle.
- Wildberger N-gon tangent recurrence: any regular polygon from 1 √ + pure algebra.
- PurePhi (φ² = φ + 1), PureRadicals (cached √2/√3/√6), PureCubics (heptagon/nonagon algebraic roots).
- Quadray Rotor: RT-pure rotation in ABCD space (no quaternions, no Euler angles).
- over 1,000 automated tests covering geometry, topology, coordinate conversion, face winding, edge uniformity, and file round-trips.
32/32 polyhedra — complete platonic + archimedean + catalan families
- All 6 Platonics in pure Quadray ABCD — zero Cartesian construction.
- All 13 Archimedeans via general operations: truncate, rectify, cantellate, dual.
- All 13 Catalans via dual of Archimedean parents.
- Geodesic subdivision (all 6 Platonic bases), tetrahelixes (3 variants), ABCD-native convex hull.
- Parametric keystone deformation: mono and balanced modes preserving area/volume exactly.
- Stellation: variable-height face extrusion for all supported polyhedra.
Matrix Polyhedra — planar and radial arrays
- cubic, tetrahedral, octahedral spatial grids in pure Quadray ABCD — zero Cartesian construction.
- spaceframe generation with simple (vault) and compound (dome) curvature
- All 13 Catalans via dual of Archimedean parents.
- Parametric keystone and twerk deformation: mono, balanced and twerked modes preserving area/volume exactly.
- planar or space filling variants available
metal shader & material pipeline
- wgpu/Metal-native rendering at 120fps (Apple Silicon target).
- PBR material system: base color, metallic, roughness, per-face and per-strut materials.
- Texture array pipeline: runtime hot-reload, tiled/stretched UV mapping, rotation control.
- Standard material library (metals, woods, minerals, synthetics) with Preferences editor.
- Five render pipelines sharing shader + depth buffer: edges, faces, backface, opaque, billboard.
- Per-instance edge color with live animation interpolation.
architectural detailing
- Struts: 3 profiles (THS triangular, RHS rectangular, tube), with mitre cutting (single + compound), inset control, and matched-node sizing.
- Nodes: 7 geodesic shapes (Tet, Octa, Cube, Icosa 1F–4F), continuous 30–300mm diameter range, packed mode for close-packed spheres.
- Double-hull struts: inner/outer prism surfaces for hollow structural sections.
- Per-polyhedron rendering: 11 named idea slots with per-matrix overrides.
3d fold animation & net unfolding
- 3D fold overlay: parametric t-slider (0=flat net → 1=closed polyhedron) with RT-pure fold direction detection.
- Panel Thickness – all faces can be assigned thickness with dynamic bevelled unfolding integral to all functions and net drawings
- 2D net unfolding: BFS-based face tree, seed face selection, fold angle labels, dihedral readouts.
- Net View palette: interactive SVG preview with pan/zoom, glue tabs, hubcap connector diagrams, strut elevation drawings.
- SVG export: dimensioned shop drawings with scale bar, face labels, fold angles — ready for fabrication.
dynamic dimensioning & labelling
- Auto-dimensions: live viewport overlay + SVG export with perimeter chains, bounding extents, radial measurements.
- Edge Callout Labels: property readouts (edge length, quadrance, face spread, instance name, custom text) anchored to edges with perpendicular leader lines, auto-justified layout.
- Strut schedule: classified edge inventory with lengths, net lengths after inset, dihedral angles, total material length.
- DimensionStyle: shared font, line weight, units (mm/cm/m/in/ft), decimal places — one control governs all text overlays.
- Metric system: 1 ABCD = 1 metre, with auto-scale display (nm to m).
tags & knowledge graph
- Tag system: per-instance key-value metadata with OptionSet group filtering (show/hide layers).
- Edge Callout Labels: geometry-attached text overlays in screen space, auto-justified with kinked leader lines.
- Knowledge Graph: live 3D visualization of the 69K-LOC codebase — 63 modules as polyhedra on concentric shells (Ideas/Forms/Instances), 20 concept moons, dependency edges.
- Unified “Tags & Labels” sidebar: general labels first, KG controls nested under collapsible header.
view manager & animation
- Named view snapshots with delta compression (only changed fields stored).
- Camera-only and full scene transitions with smoothstep interpolation.
- Demo Reel: 51 scripted views across 7 acts + randomizer for endless exploration.
- Playback transport: play all, stop, per-view duration (1–5s).
- View export/import (JSON) + views persisted in scene files.
scene management
- Instance-based scene: place, select, move, rotate, scale, group, delete with full undo/redo.
- ConnectedLines (wired vertices between instances) + ConnectedPolygons.
- Gumball move/scale/rotate with RT-pure ABCD handles + XYZ mode.
- Grid snap (free, vertex, 0.5m, 1m, 2m) + object snap (vertex, edge, face).
- IK constraints between instances (auto-detected from coincident vertices).
- 7 camera presets (4 Quadray axis views + 3 Cartesian) with animated transitions.
construction-ready output
- Native JSON: full scene persistence — geometry, camera, app state, tags, labels, views.
- SVG export: projected 2D views with dimensions, labels, and leader lines (from View Manager or Papercut).
- Net SVG: dimensioned flat patterns with fold angles, glue tabs, strut elevations, hub drawings.
- Strut schedule: classified edge inventory for fabrication cut lists.
- All output derived from native ABCD coordinates — geometry integrity verified.
The Architecture: Algebraic from core to pixel
The rendering boundary has been pushed to a single matrix multiply. The Tom Ace basis is folded into the camera matrix on the CPU, so the vertex shader is clip = abcd_to_clip × quadray + clip_offset. No Cartesian conversion. No normalization. No XYZ intermediate. Everything upstream is algebraically exact.
Export / File Format Support
Implemented
- JSON Scene – full parametric scene: geometry instances, camera, app state, tags, labels, View Manager views. Backward-compatible deserialization (`#[serde(default)]` on all new fields).
- JSON views – View Manager view sequences (camera + state deltas) for import/export and demo reels.
Vector Drawing Export (Implemented):
- SVG – 2D projected views from Papercut or View Manager with geometry, auto-dimensions, edge callout labels, and leader lines. Physical mm sizing (A4/A3/Letter sheet).
- Net SVG – flat unfolded patterns with dimensioned edges, fold angles, glue tabs, strut elevation drawings, hubcap connector diagrams, and scale bars. Ready for laser cutting or fabrication.
Planned
- STL – 3D printing and mesh fabrication
- OBJ – widely supported mesh format
- DXF – 2D fabrication drawings
- GLTF – web viewers, CAD/BIM imports
- STEP – precise CAD interoperability
- CSV – node and member tables for structural analysis
All output is derived from native ABCD coordinates, ensuring exact geometry and structural integrity through the entire pipeline.
Who It’s For
- Architects & Structural Engineers – create complex spaceframes, domes, and lattice structures with exact geometry that won’t degrade through design iterations.
- Dome Builders – design geodesic domes with algebraically precise strut lengths and node angles. No more measuring discrepancies from accumulated CAD error — every chord factor is exact.
- Truss & Spaceframe Fabricators – export construction tables where node coordinates, member lengths, and connection angles carry algebraic precision. Cut lists you can trust.
- Hobbyists & Makers – generate printable 3D structures where every strut length is a known exact value. From tabletop tensegrity models to full-scale pavilions.
- STEM Educators – hands-on exploration of symmetry groups, polyhedral families, and coordinate systems. Visualize Euler’s formula (V−E+F=2), demonstrate dual relationships, and show students why geometry doesn’t need trigonometry.
- Mathematics & Geometry Researchers – explore Fuller’s Synergetics and Wildberger’s Rational Trigonometry with an interactive tool that actually implements the theory.
- Computational Geometry – a reference implementation of exact-arithmetic polyhedral construction. All 32 Platonic/Archimedean/Catalan solids with verified topology.
- Physics & Materials Science – close-packed lattice structures (FCC, BCC, HCP analogues) arise naturally from IVM spatial matrices.
- Chemists & Molecular Modellers – polyhedral cage geometries (fullerenes, clathrates, zeolite frameworks) with exact vertex coordinates.
- Fabrication Shops – receive dimensioned output with exact node positions and member lengths. Eliminate tolerance stack-up.
Why abcd.earth?
Traditional CAD platforms treat geometry as floating-point approximations from the start. Every vertex is a trio of imprecise decimals. Every operation adds error. Every large-scale structure amplifies that error.
abcd.earth starts from the opposite premise:
- 30 of 32 polyhedra are algebraically exact — integer, rational, or phi-algebraic ABCD coordinates with zero floating-point error.
- Operations preserve exactness — truncate, rectify, cantellate, dual all operate via Quadray lerp with rational parameters. Exact in, exact out.
- The GPU sees ABCD directly — no Cartesian conversion in the geometry pipeline. The only coordinate transform is a single matrix multiply at the shader boundary.
- Rational Trigonometry eliminates transcendentals — no π, no sin, no cos in geometry code. Quadrance and spread are algebraic, deterministic, and exact.
- 978 tests enforce precision — topology, edge uniformity, face winding, and coordinate exactness validated for every polyhedron.
abcd.earth doesn’t approximate geometry. It computes it exactly — then renders it.
Intellectual Foundations
- R. Buckminster Fuller (Synergetics) — nature coordinates tetrahedrally, not cubically.
- N.J. Wildberger (Divine Proportions) — geometry is algebraic, not transcendental.
- Tom Ace — Quadray coordinate machinery and the ABCD basis matrix.
- Kirby Urner — expanded Fuller’s work to Quadrays through Python explorations.
- Julian Barbour (The Janus Point) — the Big Bang as a temporal pivot, not a beginning.
We built the tool that lets you see what they proved — and extended it with original research that the tool itself made possible.
Original Research
abcd.earth is not just an application of existing mathematics — it is a research platform that has produced three peer-reviewable papers, each arising from questions that the Quadray-first architecture made visible.
Geometric Janus Inversion (Thomson & Thomson, 2026)
The question Cartesian coordinates never prompt: In Quadray coordinates, four tetrahedral basis vectors span all of 3D space using only non-negative values. Negative coordinates are never needed. So what does the all-negative state [−,−,−,−] represent?
The answer: When a tetrahedron contracts through the origin and re-expands with all signs flipped, it enters the dual tetrahedron — vertices become face-centers and vice versa. The positive and negative Quadray arenas are two complete geometric states connected at the origin, a Geometric Janus Point.
Three proven results:
- The involution P → −P maps tet vertices to dual tet vertices (the Janus correspondence)
- All metric invariants (quadrance, spread, edge lengths) are preserved — the dual is congruent
- The Rationality Reciprocity: tetrahedral and cubic edge lengths are related by √2. Granting rational coordinates to one system forces irrationality on the other.
“Geometric Janus Inversion — Tetrahedral Coordinates, the Inversion Manifold, and the Three-State Geometry of Inside-Outing” (v10.6, Feb 2026). DOI: 10.13140/RG.2.2.10492.19848
Spread-Quadray Rotors (Thomson, 2026)
Spread-Quadray Rotors lift to ℝ⁴ × ℤ₂ via full 4D Quadray coordinates, using Rational Trigonometry’s spread and cross measures instead of sin/cos. Key advantages over quaternions:
- Algebraic exactness for many useful rotations: basis-axis rotations by tetrahedral angles yield exact rational coefficients.
- Explicit double-cover: quaternions hide the q ≡ −q ambiguity. Quadray Rotors make it an explicit Janus polarity bit (ℤ₂).
- Native tetrahedral geometry: the 109.47° basis angles never align through any continuous motion in 4D — the tetrahedron is “precessionally complete”.
“Spread-Quadray Rotors — A Tetrahedral Alternative to Quaternions” (v3.0, Feb 2026). DOI: 10.13140/RG.2.2.23476.51846
The 4D± Prime Projection Conjecture (Thomson, 2026)
Non-constructible prime polygons (7, 11, 13, 19…) — which the Gauss-Wantzel theorem proves impossible to construct with compass and straightedge — emerge as convex hull projections of algebraically-defined polyhedra at exact rational spread angles.
The most striking result: a single polyhedron family — the geodesic tetrahedron at increasing subdivision frequency — produces all three non-constructible primes (7, 11, 13) at Tier 1 rational spreads. No golden ratio, no compound engineering, no transcendental functions.
“The 4D± Prime Projection Conjecture” (v5.1, Feb 2026). DOI: 10.13140/RG.2.2.23043.98089
abcd.earth — From pure geometric exploration to precise construction.
30 exact polyhedra. Zero floating-point disease. Four original research papers at ResearchGate.
abcd.earth is The future of complex lattice-based architectural modelling.
