I’ve been exploring the intersection of alternative geometry and numerical
stability. This is a proof-of-concept 3D renderer that abandons the standard
Cartesian (XYZ) basis in favor of Buckminster Fuller’s Synergetic Geometry (a
4D tetrahedral coordinate system). I’m not a professional graphics programmer, so I worked with Gemini CLI to
pair-program the core engine and the Metal-cpp boilerplate. We based the math
on Andrew Thomson’s 2026 framework for Spread-Quadray Rotors (SQR).
The Core Problem:
Standard graphics engines rely on sin/cos approximations. Every time you
rotate an object, floating-point error (transcendental drift) accumulates.
Over long-running simulations, the geometry literally "warps." The Solution:
By implementing Andrew’s framework using a Rational Surd field extension
(Q[sqrt(3)]), we’ve achieved bit-exact rotation.
Paper:
https://www.researchgate.net/publication/400414222_Spread-Quadray_Rotors_-v11_
Feb_2026_A_Tetrahedral_Alternative_to_Quaternions_for_Gimbal-Lock-Free_Rotatio
n_Representation
Key Features:
* Algebraic Determinism: A startup benchmark proves that rotating 360 degrees
returns the engine to the exact starting bit-pattern.
* Surd-Native Shaders: The Metal kernel performs algebraic arithmetic
natively on the GPU, avoiding transcendental approximations.
* Linear Jitterbugging: The complex VE-to-Octahedron transformation is
handled as a simple linear interpolation in 4D space.
* Topological Stability: In a live 60FPS loop, the SQR system is ~10x more stable than an industry-standard matrix.