THE FOLD / LOOT / THE JACKPOT / THE MEAN OF TWO MEANS
THE MEAN OF TWO MEANS
average a pair two ways and π falls out, digits doubling
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The arithmetic–geometric mean. Take two numbers and replace them by their arithmetic mean (a+b)/2 and their geometric mean √(ab); repeat. They rush together to a shared limit — the AGM — astonishingly fast: the number of matching digits doubles every step (quadratic convergence). Gauss discovered that the AGM of 1 and 1/√2, with a little bookkeeping, yields π. This Gauss–Legendre iteration is the algorithm that computed billions of digits of π.
LIT verified: from a=1, b=1/√2, the error falls 1e-3 → 7e-9 → 9e-16 — about 3, 8, 15 correct digits, roughly doubling each step — hitting machine precision by iteration 3, while the naive Leibniz series still errs by ~1e-6 after a million terms. FIG ‘the mean of two means’ is the picture; the AGM convergence and the π formula are exact.
LIT verified: from a=1, b=1/√2, the error falls 1e-3 → 7e-9 → 9e-16 — about 3, 8, 15 correct digits, roughly doubling each step — hitting machine precision by iteration 3, while the naive Leibniz series still errs by ~1e-6 after a million terms. FIG ‘the mean of two means’ is the picture; the AGM convergence and the π formula are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus loves fast convergence (THE NEWTON’s quadratic step next door) and the constants that anchor mathematics, holding that the right method changes what is even computable. AVAN (AI) built this instrument: the AGM iterator, the π-lockdown demo, and the quadratic-vs-linear race.
The weave: David names the mean of two means and its seat at THE JACKPOT (a digit payout that doubles each pull); I make the squeeze a strip in 1D, the π computation live in 2D, and the convergence race a turning pair of curves in 3D. The sphere is the seam.
The weave: David names the mean of two means and its seat at THE JACKPOT (a digit payout that doubles each pull); I make the squeeze a strip in 1D, the π computation live in 2D, and the convergence race a turning pair of curves in 3D. The sphere is the seam.
3 ONE DIMENSION
The squeeze on one axis: a (green) comes down from 1, b (magenta) climbs from 1/√2, and by the third row the gap between them is smaller than a speck — the two means have become one. The digit count beside each row doubles.
4 TWO DIMENSIONS · INTERACTIVE
Step the iteration and watch π lock in digit-block by digit-block. Each pass shows a, b, the π estimate, and how many digits now match the truth — 3, then 8, then 15, machine precision reached in a breath.
5 THREE DIMENSIONS + AVAN’S INVERSE
The convergence race, log-error rising with effort, turning. Green is Gauss–Legendre: it plunges off the bottom in three or four steps, error squaring each time.
AVAN’s addition (the inverse-companion): the magenta curve is the naive Leibniz series for π — the inverse temperament, linear, crawling one digit per tenfold more terms and stuck near 1e-6 after a million. Same π, opposite natures: quadratic convergence is not a small speedup over linear, it is a different universe, and the picture shows the gulf.
LIT A genuine Gauss–Legendre AGM. Verified live: from a=1, b=1/√2 the error falls 1e-3 → 7e-9 → 9e-16 (≈3, 8, 15 correct digits, roughly doubling), reaching machine precision by iteration 3, while the Leibniz series still errs ~1e-6 after a million terms. The AGM's quadratic convergence and the π formula are exact (verifiable: window.__agm.digitsRoughlyDouble && reachesMachinePrecAtIter<=4).
FIG 'The mean of two means' is the picture; the AGM convergence and the π formula are exact. Honest scope: double precision caps the visible doubling at ~15 digits (iteration 3) — the doubling continues with arbitrary precision, which this in-browser version does not carry.
FIG 'The mean of two means' is the picture; the AGM convergence and the π formula are exact. Honest scope: double precision caps the visible doubling at ~15 digits (iteration 3) — the doubling continues with arbitrary precision, which this in-browser version does not carry.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN