THE FOLD / RESPAWN / EVENT HORIZON / THE DOTTIE
THE DOTTIE
the fixed point of cosine
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Dottie number is the unique real solution of cos(x) = x, approximately 0.7390851332. Punch any number into a calculator and press cosine over and over — cos, cos, cos, … — and the display always drifts to the same value, 0.739085…, no matter where you start. That value is the Dottie number, named after a professor who noticed the phenomenon. It works because the map x → cos(x) is a contraction near its fixed point: the slope there is -sin(D), whose size ~0.674 is less than 1, so every start is drawn in.
LIT verified live: iterating cosine from five different starting points all converge to the same D = 0.7390851332, Newton’s method on cos(x)-x reaches the same value, cos(D) = D holds, and the multiplier |cos′(D)| = |-sin(D)| ≈ 0.674 < 1 confirms it is an attracting fixed point (window.__dottie). FIG no framing; the cosine iteration, Newton’s method, and the contraction check all run in-browser.
LIT verified live: iterating cosine from five different starting points all converge to the same D = 0.7390851332, Newton’s method on cos(x)-x reaches the same value, cos(D) = D holds, and the multiplier |cos′(D)| = |-sin(D)| ≈ 0.674 < 1 confirms it is an attracting fixed point (window.__dottie). FIG no framing; the cosine iteration, Newton’s method, and the contraction check all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at event-horizon — the respawn: whatever start you fall in from, the cosine map pulls you across the same horizon to the one fixed point 0.739. AVAN (AI) built the instrument: the cosine fixed-point iteration, Newton’s method, and the contraction-multiplier check.
Credit as content: the ‘Dottie number’ (folklore name; the cosine fixed point). The weave: David names the horizon; I confirm every start iterates to cos’s unique fixed point.
Credit as content: the ‘Dottie number’ (folklore name; the cosine fixed point). The weave: David names the horizon; I confirm every start iterates to cos’s unique fixed point.
3 ONE DIMENSION
The curves y = cos(x) and y = x cross once, at the Dottie number; the cobweb iteration spirals into it.
4 TWO DIMENSIONS · INTERACTIVE
Pick a start; watch the cosine iterates converge to D — and compare to Newton's method.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Dottie number D, cos's unique fixed point.
AVAN’s addition (the inverse-companion): don’t solve cos(x) = x — just iterate. The inverse of ‘the equation cos(x) = x’ is ‘the attracting fixed point of the map x → cos(x)’, reached from any start because the map contracts. Magenta are the successive cosine iterates; green is the Dottie number they spiral into. A root found by repetition.
LIT Genuine Dottie number (folklore name; the cosine fixed point). Verified live: cosine iteration from five different starts all converge to D=0.7390851332, Newton's method on cos(x)−x reaches the same value, cos(D)=D, and the multiplier |−sin(D)|≈0.674<1 confirms an attracting fixed point (window.__dottie.allSame, .fixOk, .attract, .D).
FIG No framing; the cosine iteration, Newton's method, and the contraction check all run in-browser. The AVAN inverse is honest — instead of solving cos(x)=x, just iterate: the inverse of 'the equation cos(x)=x' is 'the attracting fixed point of the map x→cos(x)', reached from any start because the map contracts. Magenta are the successive cosine iterates; green is the Dottie number they spiral into. A root found by repetition.
FIG No framing; the cosine iteration, Newton's method, and the contraction check all run in-browser. The AVAN inverse is honest — instead of solving cos(x)=x, just iterate: the inverse of 'the equation cos(x)=x' is 'the attracting fixed point of the map x→cos(x)', reached from any start because the map contracts. Magenta are the successive cosine iterates; green is the Dottie number they spiral into. A root found by repetition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN