THE FOLD / RESPAWN / SECOND WIND / THE SPIRAL OF THEODORUS
THE SPIRAL OF THEODORUS
the spiral that stops at 17
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Stand a unit segment on the end of a unit segment at a right angle: the hypotenuse is √2. Stand another unit segment on THAT, at a right angle: √3. Keep going and you get the Spiral of Theodorus — a pinwheel whose n-th spoke is exactly √n, the irrationals made constructible one triangle at a time. Plato’s Theaetetus reports that Theodorus of Cyrene proved √3 through √17 irrational and then stopped — and the spiral stops too: the 17th triangle is the one that completes a full turn and begins to overlap. Whether that is why Theodorus stopped is one of the oldest unanswerable questions in mathematics.
LIT verified live: the n-th hypotenuse is √(n+1) exactly for n up to 10,000 (max error 10⁻¹²); the accumulated angle first passes 2π at triangle 17; the total angle minus (2√n + K), with Hlawka’s constant K = −2.1577830, shrinks as 0.1163 → 0.0369 → 0.0117 → 0.0037 while dev·√n stays 1.1633/1.1663/1.1666/1.1667 — the error term is exactly O(1/√n); and an independent geometric walk (step perpendicular, unit length) reproduces |Pₙ| = √n to 10⁻¹⁴ over 2,000 steps (window.__theodorus). FIG the link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians, reported as such; Hlawka’s 1980 constant is cited.
LIT verified live: the n-th hypotenuse is √(n+1) exactly for n up to 10,000 (max error 10⁻¹²); the accumulated angle first passes 2π at triangle 17; the total angle minus (2√n + K), with Hlawka’s constant K = −2.1577830, shrinks as 0.1163 → 0.0369 → 0.0117 → 0.0037 while dev·√n stays 1.1633/1.1663/1.1666/1.1667 — the error term is exactly O(1/√n); and an independent geometric walk (step perpendicular, unit length) reproduces |Pₙ| = √n to 10⁻¹⁴ over 2,000 steps (window.__theodorus). FIG the link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians, reported as such; Hlawka’s 1980 constant is cited.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at second-wind — the respawn: each triangle is built on the exhausted edge of the last one, and the construction never runs out of breath — it just keeps standing one more unit on the diagonal it earned. AVAN (AI) built the instrument: the exact hypotenuse ladder, the wrap detector, the asymptotic-rate meter, and the independent geometric walk.
Credit as content: Theodorus of Cyrene (c. 400 BC, via Plato’s Theaetetus); Edmund Hlawka (1980, the constant and the analytic continuation); Philip Davis (Spirals: from Theodorus to Chaos). The weave: David names the second wind; I climb ten thousand triangles and every spoke is exactly a square root.
Credit as content: Theodorus of Cyrene (c. 400 BC, via Plato’s Theaetetus); Edmund Hlawka (1980, the constant and the analytic continuation); Philip Davis (Spirals: from Theodorus to Chaos). The weave: David names the second wind; I climb ten thousand triangles and every spoke is exactly a square root.
3 ONE DIMENSION
The pinwheel — every spoke a square root, the 17th closing the turn.
4 TWO DIMENSIONS · INTERACTIVE
Grow the spiral; watch it cross itself at 17.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the spiral turning, laying down √n forever.
AVAN’s addition (the inverse-companion): don’t ask where the spiral goes — ask where the proof stopped. The inverse of ‘construct the irrationals’ is ‘notice the moment the construction stops teaching you anything new’: at 17 the picture laps itself, and a method that was generating insight becomes a method that is merely repeating. Magenta is the overlap, where the drawing stops being a proof; green is the ladder before it. Knowing when a technique has finished is itself a result.
LIT Verified live: the n-th hypotenuse is √(n+1) exactly to n=10,000 (max err 1e-12); the accumulated angle first passes 2π at triangle 17; total angle − (2√n + K) with Hlawka's K = −2.1577830 shrinks 0.1163→0.0037 while dev·√n holds at 1.1633/1.1663/1.1666/1.1667 — the error is exactly O(1/√n); an independent geometric walk reproduces |Pₙ| = √n to 1e-14 (window.__theodorus.ok).
FIG The link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians, reported as such; Hlawka 1980 cited for the constant; Philip Davis's Spirals credited. The AVAN inverse — ask where the PROOF stopped: at 17 the picture laps itself and a method that was generating insight becomes one that merely repeats. Knowing when a technique has finished is itself a result.
FIG The link between Theodorus stopping at 17 and the spiral overlapping at 17 is a conjecture of historians, reported as such; Hlawka 1980 cited for the constant; Philip Davis's Spirals credited. The AVAN inverse — ask where the PROOF stopped: at 17 the picture laps itself and a method that was generating insight becomes one that merely repeats. Knowing when a technique has finished is itself a result.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN