Reproduce the A wave sphere that bounds rationally but not integrally
These original drawings accompany A wave sphere that bounds rationally but not integrally. The complete mathematical proof, all worked examples and complete solutions remain in that reader.
Download the unchanged original files
Keep make_figures.py at the root of an empty scratch directory and the PNG, SVG and geometry files in figures/. The two unchanged Markdown sources and CREDITS.txt/LICENSE.txt stay at the package root; they are downloads and are not generated by the renderer. The renderer creates figures/ during a fresh run.
- make_figures.py — 3960 bytes; SHA-256
D7FF3D2BB7CE501E46327DD4B2FC3CF54FD5C7ECE7935DD217EC941092DC59FC. - geometry.json — 780 bytes; SHA-256
4A401D2CA017240EFBB529130D0943506AFAE27EFD2482625AFE37F6ADCC39B1. - wave-cycle-torsion.png — 186943 bytes; SHA-256
A755A2ED26EB1BDD0E472686F3CAE5D0E57B21D4F9ECC5AF8A66CB861CAF83B7. - wave-cycle-torsion.svg — 115423 bytes; SHA-256
3BD8EF75D71E3DBB65090045A3340ABE6FA2752443D56205BA4E81AF448B5331. - wave-cycle-coefficients-learner.md — 9257 bytes; SHA-256
CA3FD87F570CE5395F66A8077877246FFD2EC9115A2A4F7F12CBCEC7F6DA737D. - wave-cycle-torsion-working-proof.md — 17446 bytes; SHA-256
4612625D62DC9B1526192BDE35AC089FB55F5D457D6322ACF12F158C24038BE0. - LICENSE.txt — 449 bytes; SHA-256
3A508E9D86CB2C5F92DF4862D97617A45F76910084967F87D4B0C0968B5C0E7D. - CREDITS.txt — 1697 bytes; SHA-256
938F39184D75E9D4688BFD58FECA6AB0A84E8D59E57661E355FD3A4EAB0C8F1B.
Run the renderer
The unchanged program requires Python, NumPy and Matplotlib. In the scratch directory run:
python -X utf8 make_figures.py
It writes the supplied PNG, SVG and exact geometry files. A fresh replay with Python 3.13.9, NumPy 2.4.4 and Matplotlib 3.10.9, using Matplotlib’s bundled DejaVu Sans font, matched every renderer/figure file byte for byte and all decoded PNG RGBA pixels. No SVG date or identifier normalization was performed. Byte identity is qualified by that observed software and font environment; different versions may change rendering bytes. The geometry files specify the mathematical objects independently of rendering.
Check the mathematics and credits
The three panels give the exact section v = iae₁ + √(1+a²)e₃ of q = 1, the half-phase section z = exp(iθ/2)e₃ and base exp(iθ), and the exact integral two-arc matrix [[1,1],[−1,1]]. The sphere is the actual nonzero order-two generator, proved by the complete retractions and integral chain calculation. Sections are not all six real dimensions; sampled functions are not proofs. WT4–WT20 and the complete caption retain the phase seam, outward/canonical orientations, eight ordered facets and 24 tetrahedra, exact boundary −2kout, characteristic-zero and mod-2 distinctions, and the two-dimensional reduced-degree-zero exception. The full formal proof follows the complete learner and all six solutions. Source coefficient convention, general integral torsion, novelty priority and general component constancy remain unresolved.
CREDITS.txt gives every human source and locator. Original prose and figures are CC0; copyrighted references are comparison credits and are not imported.
The original renderer, scene/geometry and diagrams are GPT-6.1 Sol (OpenAI), Ultra work, October 2026, dedicated under CC0. Matplotlib and the bundled DejaVu Sans font retain their own component terms already included with this course. The native geometry is explanatory; the full proof establishes the theorem.