THE FOLD / SPAWN / FIRST LIGHT / THE VIETE
THE VIETE
π from an endless nested radical
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Viète’s formula (1593) is the very first time in history that a constant was written as an infinite product — the dawn of analysis. It expresses 2/π as an endless product of nested square roots of two: 2/π = (√2/2)·(√(2+√2)/2)·(√(2+√(2+√2))/2)·… Each factor ak/2 is built from the last by ak+1 = √(2 + ak), a value that creeps toward 2. Geometrically it is Archimedes’ doubling of a polygon’s sides made algebraic: each nested radical is the cosine of an angle halved again and again.
LIT verified live: the partial product converges to 2/π — after 30 nested factors it matches to machine precision, giving π to twelve digits (window.__viete). FIG no framing; the nested-radical recurrence and the convergence to 2/π run in-browser.
LIT verified live: the partial product converges to 2/π — after 30 nested factors it matches to machine precision, giving π to twelve digits (window.__viete). FIG no framing; the nested-radical recurrence and the convergence to 2/π run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — the first light of analysis itself: the earliest infinite product ever written, π emerging from an endless tower of nested square roots of two. AVAN (AI) built the instrument: the nested-radical recurrence, the running product, and the convergence to 2/π.
Credit as content: François Viète (1593). The weave: David names first-light; I confirm the infinite product of nested radicals converges to 2/π.
Credit as content: François Viète (1593). The weave: David names first-light; I confirm the infinite product of nested radicals converges to 2/π.
3 ONE DIMENSION
The nested radicals a₁=√2, a₂=√(2+√2), … each creeping toward 2; the running product of aₖ/2 approaches 2/π.
4 TWO DIMENSIONS · INTERACTIVE
Add nested factors; the partial product locks onto 2/π and the π estimate gains digits fast.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the value 2/π the infinite product converges to.
AVAN’s addition (the inverse-companion): don’t sum a series — multiply nested roots. The inverse of ‘compute π’ is ‘the endless product of √(2+√(2+…))/2’, each factor a halved-angle cosine. Magenta is the tower of nested radicals; green is the 2/π they multiply to. π as an infinite descent of square roots.
LIT Genuine Viète's formula (François Viète, 1593 — the first infinite product). Verified live: the partial product ∏ aₖ/2 with a₁=√2, aₖ₊₁=√(2+aₖ) converges to 2/π; after 30 nested factors the error is ~2e-16, giving π=3.14159265359 (window.__viete.converges, .e30, .piEst).
FIG No framing; the nested-radical recurrence and the convergence to 2/π run in-browser. The AVAN inverse is honest — instead of summing a series, multiply nested roots: the endless product of √(2+√(2+…))/2, each factor a halved-angle cosine. Magenta is the tower of nested radicals; green is the 2/π they multiply to. π as an infinite descent of square roots.
FIG No framing; the nested-radical recurrence and the convergence to 2/π run in-browser. The AVAN inverse is honest — instead of summing a series, multiply nested roots: the endless product of √(2+√(2+…))/2, each factor a halved-angle cosine. Magenta is the tower of nested radicals; green is the 2/π they multiply to. π as an infinite descent of square roots.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN