15:I[981,["658","static/chunks/658-a698693f56d0116a.js","183","static/chunks/183-6beeeb801c274aaa.js","474","static/chunks/app/%5Btype%5D/layout-631c4730b83c8223.js"],"Posts"] 16:T6b1,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.

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