Gear Meshing & Primes

Status: null-result Domain: physics Source: → projects/rational-superformula/RAS_APPLICATIONS_NOTES.md

What We Know

• RAS applied to gear meshing geometry — tested whether shared prime factors help gears mesh
• Result: GCD vs uniformity r = −0.265 — shared factors DON’T help; gears need complements, not matches
• Propeller blades: prime profiles = smooth/efficient, composite profiles = lobed/aggressive (like whale tubercles)
• Blender 3D pipeline: projects/rational-superformula/ras_blender_3d.py
• Not pursuing further — documented for future reference

What We Don’t Know

• Q-GM-01: Does the “complements not matches” finding have implications for prime-ratio network design?
• Q-GM-02: Is the propeller finding (prime=smooth, composite=aggressive) universal across mechanical applications?

Relationships

• [[ras]] — nature: extends — RAS applied to mechanical geometry
• [[prime-composite-duality]] — nature: supports — even in mechanics, primes and composites play different structural roles
• [[coprimality]] — nature: contradicts (partially) — in gears, coprimality hurts rather than helps

Bridging Potential

• The negative result is informative: prime structure isn’t universally “better” — context matters
• “Complements not matches” may apply to resonance networks too (untested)

Key Evidence

• r = −0.265 correlation (GCD vs uniformity)
• Application notes: projects/rational-superformula/RAS_APPLICATIONS_NOTES.md
• Blender visualizations: projects/rational-superformula/ras_blender_3d.py