Rational Algebraic Superformula (RAS)

Status: validated Domain: math Source: → projects/rational-superformula/

What We Know

Rational replacement for Gielis superformula, parameterised entirely by prime factorisation
Published on Zenodo: DOI 10.5281/zenodo.20512346 (Jun 2, 2026)
Title: “The Rational Algebraic Superformula: Prime Factorisation as Geometry over ℚ”
Give & Resist formalised: (1-t²)^{2p} resists (primes), (2t)^{2q} gives (composites)
Key insight: “The shape is the number. The number is the shape.”
Connects factorisation → boundary shape → resonance, validated against v3 experimental data
Operates entirely over ℚ (rationals) — avoids irrationals by construction
Blender 3D pipeline built: ras_blender_3d.py — ceramic primes vs cactus-green composites
Applied geometry explorations: gear meshing, propeller blades (documented but not pursued further)

Q-RAS-01: Can RAS predict v3/v4 resonance amplitudes quantitatively (not just qualitatively)?
Q-RAS-02: Does RAS geometry map to cymatics patterns — i.e., do RAS boundary shapes appear as Chladni figures?
Q-RAS-03: What happens to RAS shapes for very large primes vs highly composite numbers?
Q-RAS-04: Can RAS be extended to 3D (volumes) while staying rational?

[[prime-composite-duality]] — nature: depends-on — give/resist IS the prime/composite duality expressed geometrically
[[square-root-problem]] — nature: bridges — RAS was designed to avoid irrationals; it identifies √ as the persistent problem
[[v3-experimental-proof]] — nature: supports — validated against v3 data
[[tusk-series]] — nature: analogous-to — both map number theory to different geometric domains (shape vs frequency)
[[four-factor-theory]] — nature: extends — could provide the mathematical formula for “resonance quality”
[[gear-meshing-primes]] — nature: extends — RAS applied to mechanical geometry (null but interesting)
[[cymatics]] — nature: bridges — RAS shapes may correspond to physical wave patterns

If combined with [[cymatics]], could predict which wave patterns emerge from prime vs composite excitation
If combined with [[sopfr-binding-energy]], could connect geometric “give/resist” to nuclear stability
Triangulation: RAS boundary shape + v3 amplitude + Tusk Series spike height → three representations of the same number