PAPP Bonnet Torus f⁺ | Ratio 9/8 (Major Second)

3D Model

High-fidelity analytical torus generated via the PAPP v3.1.0 system (Phase 7.5). This model represents the positive branch (f⁺) of a “Bonnet Pair”—a rare geometric configuration where two non-congruent surfaces share identical metrics (isometry) and mean curvature.

This iteration corresponds to the 9/8 harmonic ratio (Major Second). Based on the 2025 research by Bobenko, Hoffmann, and Sageman-Furnas, the surface is a compact torus in ℝ³ with numerically verified isometry against its twin branch (f⁻) at machine precision (rel. error ≈ 10⁻¹⁴).

Technical Specifications:

• Topology: Closed Manifold Torus (Euler Characteristic χ = 0).

• Resolution: 80×80 (6,400 vertices, 12,800 triangular faces).

• Parameters: Optimized at the exact closure value τ = 0.151986*.

• Material: PAPP Standard “Solar/Amber” Kd (0.700, 0.348, 0.175).

• Analysis: Validated via Laplacian Spectral Gap and Persistent Homology.

Generated by PAPP Framework.