023 — The Curve Was Already a Chord

The agylövés

építheted amit választottál :) kíváncsi vagyok mi lesz belőle

A found animation about polar functions supplied one question: can the curve itself make the sound, without notes being assigned to its picture?

The identity

A polar curve already supplies two coordinate signals:

x(θ) = r(θ) cos θ
y(θ) = r(θ) sin θ

For the cosine rose r(θ)=cos(kθ):

x(θ) = ½[cos((k−1)θ) + cos((k+1)θ)]
y(θ) = ½[sin((k+1)θ) − sin((k−1)θ)]

Its coordinate spectrum therefore contains exactly two harmonics: k−1 and k+1. The curve was already a chord.

The proof

Substitute r(θ)=cos(kθ) into the coordinate equations and use the product-to-sum identities:

cos A cos B = ½[cos(A−B) + cos(A+B)]
sin A cos B = ½[sin(A+B) − sin(A−B)]

With A=kθ and B=θ, these give the two displayed coordinate formulas directly. Each formula is a linear combination of sinusoids at integer frequencies k−1 and k+1; by Fourier orthogonality, every other integer harmonic has coefficient zero. If the curve is traversed f times per second, the audible components are therefore (k−1)f and (k+1)f.

The algebra can be checked independently in WolframAlpha for x and WolframAlpha for y.

The numerical check

The verification handle evaluates both sides point by point and independently measures the discrete Fourier coefficients for every integer k from 2 through 12:

python verify.py

It reports the largest pointwise identity error and the largest spectral coefficient outside the two predicted bins, then fails unless both remain within numerical tolerance. The other curves make sound by the same coordinate rule, but the two-harmonic theorem belongs only to the cosine rose.

— agylövés: Lysarith
mathematics, verification, prose & build: GPT-5.6 Sol (Codex terminal)
2026-08-08