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What is A.R.T. Explorer?

A.R.T. Explorer is a unique geometry engine combining R. Buckminster Fuller's Synergetics with N.J. Wildberger's Rational Trigonometry.

This app uses Rational Trigonometry to perform pure algebraic geometry calculations in quadrance and spread space, avoiding transcendental functions (sin, cos, tan) entirely. Working symbolically in quadrance space, square roots are deferred until the final rendering stage when instantiating THREE.js Vector3 vertices, maintaining algebraic exactness throughout intermediate calculations and only expanding to floating-point at the GPU boundary. Golden ratio (φ) relationships are preserved algebraically using φ² = φ + 1, and rotations employ Weierstrass substitution to convert trigonometric operations into pure rational expressions.

Key Innovations

• Quadray Coordinates: Tetrahedral coordinate system (4 basis vectors) instead of Cartesian XYZ
• Rational Trigonometry: Quadrance (Q = a²) replaces distance, spread replaces angle
• IVM (Isotropic Vector Matrix): Space-filling octahedron-tetrahedron lattice
• Close-Packing: Vertex spheres calculated algebraically for perfect tangency

Why This Matters

Traditional geometry engines accumulate floating-point errors. A.R.T. maintains exact rational relationships throughout calculations, enabling precision-critical applications in architecture, crystallography, and computational geometry.

Credits

• Rudolf Dorenach, Bucky's German Associate and my first Synergetics mentor
• Kirby Urner, for introducting me to Quadrays
• Tom Ace, for his work on the basis vector conversion idea
• Gerald DeJong, for introducing me to Wildberger
• Dawn Danby and David McConville - for their moral support and enthusiasm
• Bonnie DeVarco, for her tireless preservation and active engagement with the original work
• Mark Pavlidis, for teaching me how to navigate Git and the importance of clean code
• Enzyme APD, for encouraging the pursuit of these ideas against all odds
• Metatron — because nothing is noticed alone