The Rational Algebraic Superformula: Prime Factorisation as Geometry over Q

The Gielis superformula (2003) unifies a wide family of natural and engineered shapes

under a single parametric equation, but relies on transcendental trigonometric functions, arbi-

trary real exponents, and fractional roots—none of which preserve rationality. We present the

Rational Algebraic Superformula (RAS), a rigorously rational replacement that parameterises

closed plane curves by the prime factorisation of natural numbers. The RAS substitutes the

Weierstrass half-angle parametrisation for trigonometric functions, derives all exponents from

number-theoretic functions (sopfr, Ω, ω), and works with quadrance (squared radius) to avoid

irrational roots. We prove that the resulting quadrance profile Qr (t; n) ∈ Q for all t∈ Q, and

verify this computationally with exact arithmetic. The framework cleanly separates primes

from composites geometrically: primes produce near-circular boundaries with concentrated

spectral energy, while composites develop lobed boundaries with mode-splitting proportional

to their factorisation complexity. We formalise Gielis’s “give and resist” insight, extend to

3D via spherical products, connect boundary geometry to Laplacian eigenvalues and reso-

nance theory, and show that the RAS predictions align with experimental measurements

from the v3 torsion ring (May 2026), where prime-ratio frequency networks outperformed

composite-ratio networks by +28% in peak amplitude and +22% in channel coherence.