THE FOLD / RESPAWN / THE CONTINUE / THE LISSAJOUS
THE LISSAJOUS
rationality on an oscilloscope
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Feed two sine waves to an oscilloscope — one to x, one to y — and the beam draws Lissajous figures (Bowditch 1815; Lissajous 1857): weaving curves whose shape reads out the frequency ratio. The deep dichotomy: the curve closes and repeats if and only if the ratio is rational. At 3:2 it is a clean knot retraced forever; at 1:√2 the beam never returns — it fills the square densely, coming arbitrarily close to its start without ever landing on it. Rationality, made visible: an oscilloscope is an irrationality detector.
LIT verified live: (3,2) returns to its start at t = 2π with error 7.5×10⁻¹⁶ and provably not before (minimum position-plus-velocity return 0.53 across the interior); (1,√2) never closes — minimum return 3.5×10⁻³ over T=100, shrinking to 3.2×10⁻³ by T=4000 yet never zero (dense, not periodic); and the crossing census: (3,2) cuts the axes 6 and 4 times, (5,4) cuts 10 and 8 (= 2p, 2q) (window.__lissajous). FIG no framing; closure, non-closure, and the census are all measured.
LIT verified live: (3,2) returns to its start at t = 2π with error 7.5×10⁻¹⁶ and provably not before (minimum position-plus-velocity return 0.53 across the interior); (1,√2) never closes — minimum return 3.5×10⁻³ over T=100, shrinking to 3.2×10⁻³ by T=4000 yet never zero (dense, not periodic); and the crossing census: (3,2) cuts the axes 6 and 4 times, (5,4) cuts 10 and 8 (= 2p, 2q) (window.__lissajous). FIG no framing; closure, non-closure, and the census are all measured.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-continue — the respawn: the rational curve hits its continue point and replays identically forever; the irrational one respawns arbitrarily close to start and never exactly — an endless almost. AVAN (AI) built the instrument: the closure detector with velocity matching and the return-distance ledger.
Credit as content: Nathaniel Bowditch (1815); Jules Lissajous (1857); every oscilloscope since. The weave: David names the continue point; I measure who reaches it and who orbits it forever.
Credit as content: Nathaniel Bowditch (1815); Jules Lissajous (1857); every oscilloscope since. The weave: David names the continue point; I measure who reaches it and who orbits it forever.
3 ONE DIMENSION
The gallery — 1:1, 3:2, 5:4, and the never-closing 1:√2.
4 TWO DIMENSIONS · INTERACTIVE
Pick a ratio; watch closure or endless weaving, with the return meter running.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the beam weaving live.
AVAN’s addition (the inverse-companion): don’t classify the number — watch what it draws. The inverse of ‘is √2 rational?’ is ‘does the curve close?’: number theory transposed into kinematics, where irrationality is visible as a picture that never finishes. Magenta is the gap that never quite closes; green is the knot that closes exactly. Some proofs you compute; this one you can watch.
LIT Genuine Lissajous closure dichotomy (Bowditch 1815; Lissajous 1857). Verified live: rational (3,2) closes to 7.5e-16 with no earlier position+velocity return; irrational (1,√2) min return positive and shrinking over nested horizons at fixed grid; axis-crossing census = 2p, 2q (window.__lissajous.ok).
FIG No framing — closure, non-closure, and census are all measured. The AVAN inverse — don't classify the number, watch what it draws: number theory transposed into kinematics, irrationality visible as a picture that never finishes. Magenta is the gap that never quite closes; green is the knot that closes exactly. Some proofs you compute; this one you can watch.
FIG No framing — closure, non-closure, and census are all measured. The AVAN inverse — don't classify the number, watch what it draws: number theory transposed into kinematics, irrationality visible as a picture that never finishes. Magenta is the gap that never quite closes; green is the knot that closes exactly. Some proofs you compute; this one you can watch.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN