THE FOLD / CHEAT / THE SPEEDRUN / THE AGM
THE AGM
two means racing to one limit
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The arithmetic-geometric mean is one of the fastest-converging processes in mathematics. Start with two positive numbers a and b and replace them, over and over, by their arithmetic mean (a+b)/2 and their geometric mean √(ab). The two sequences rush toward each other and meet at a common limit M(a, b) — and they do so quadratically: the gap between them squares each step, so the number of correct digits doubles every iteration. Gauss discovered it links to elliptic integrals, and it is the engine of the Gauss–Legendre algorithm that computes π to millions of digits in a handful of steps.
LIT verified live: for thousands of random starting pairs the two means converge to a single limit, the gap shrinking quadratically (gap ≈ previous-gap²/8M), and the AGM-driven Gauss–Legendre iteration reaches π to ~1e-15 in just four steps (window.__agm). FIG no framing; the AGM iteration, the quadratic-rate check, and the Gauss–Legendre π computation all run in-browser.
LIT verified live: for thousands of random starting pairs the two means converge to a single limit, the gap shrinking quadratically (gap ≈ previous-gap²/8M), and the AGM-driven Gauss–Legendre iteration reaches π to ~1e-15 in just four steps (window.__agm). FIG no framing; the AGM iteration, the quadratic-rate check, and the Gauss–Legendre π computation all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — a genuine speedrun of convergence: the two means double their agreeing digits every single step, blazing to the limit (and to π) in a few iterations. AVAN (AI) built the instrument: the arithmetic/geometric mean iteration, the quadratic-convergence check, and the Gauss–Legendre π algorithm it drives.
Credit as content: Carl Friedrich Gauss (1799); the π algorithm by Salamin & Brent (1976). The weave: David names the speedrun; I confirm the means meet quadratically and drive π to machine precision in four steps.
Credit as content: Carl Friedrich Gauss (1799); the π algorithm by Salamin & Brent (1976). The weave: David names the speedrun; I confirm the means meet quadratically and drive π to machine precision in four steps.
3 ONE DIMENSION
The arithmetic mean (falling) and geometric mean (rising) rush together to their common limit M(a,b).
4 TWO DIMENSIONS · INTERACTIVE
Step the AGM; the gap squares each iteration, and the Gauss–Legendre π estimate gains digits just as fast.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the common limit M(a,b) the two means race to.
AVAN’s addition (the inverse-companion): don’t average once — average both ways forever. The inverse of ‘two different means of a and b’ is ‘iterate the arithmetic and geometric means together until they coincide’, converging quadratically to one number. Magenta are the two racing means; green is the single limit they meet at. Two means folded into one.
LIT Genuine arithmetic-geometric mean (Carl Friedrich Gauss, 1799; π algorithm by Salamin & Brent, 1976). Verified live: for 2000 random starting pairs the arithmetic and geometric means converge to a single limit with the gap shrinking quadratically (gap≈previous²/8M), and the AGM-driven Gauss-Legendre iteration reaches π to ~1e-15 in four steps (window.__agm.convOk, .quadOk, .piOk).
FIG No framing; the AGM iteration, the quadratic-rate check, and the Gauss-Legendre π computation all run in-browser. The AVAN inverse is honest — instead of averaging once, average both ways forever: iterate the arithmetic and geometric means together until they coincide, converging quadratically to one number. Magenta are the two racing means; green is the single limit they meet at. Two means folded into one.
FIG No framing; the AGM iteration, the quadratic-rate check, and the Gauss-Legendre π computation all run in-browser. The AVAN inverse is honest — instead of averaging once, average both ways forever: iterate the arithmetic and geometric means together until they coincide, converging quadratically to one number. Magenta are the two racing means; green is the single limit they meet at. Two means folded into one.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN