Lissajous Curves and the "3-6-9" Claim: Rational Frequency Ratios, Not Digits, Determine Closure

A widely circulated claim holds that the numbers 3, 6 and 9 — attributed to Nikola Tesla — are structurallyprivileged in nature, and that this privilege is visible in the geometry of Lissajous curves. This note tests thatclaim directly. Six specific assertions are examined, of which five are false and one is true but vacuous. Theclosure condition for a Lissajous curve is shown to depend only on the rationality of the frequency ratio, with nodependence on the decimal digits of the numerator or denominator: ratios such as 2:5, 4:7 and 5:8 close exactlyas cleanly as 3:6 and 3:9. The underlying “vortex mathematics” is shown to be an artifact of base-10 positionalnotation, since the digital root of n is n mod 9 and 9 = 10 − 1; in base 16 the equivalent “privileged” set would be5, 10 and 15. The quotation on which the claim rests has no identified primary source in Tesla’s writings. The noteconcludes by identifying what does survive the test — the Pythagorean relationship between small integer ratiosand consonance — and by pointing to the rigorous theory of frequency locking (Arnold tongues, mode locking,KAM theory) where the original intuition finds genuine mathematical expression.