THE FOLD / CO-OP / THE SYNC / THE NILAKANTHA
THE NILAKANTHA
a faster alternating series for pi
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Nilakantha series is a fast, elegant series for π, found by the Kerala-school astronomer Nilakantha Somayaji around 1500 — three centuries before Europe. It reads π = 3 + 4/(2·3·4) - 4/(4·5·6) + 4/(6·7·8) - …, each term straddling three consecutive integers, alternating in sign. Unlike the Gregory–Leibniz series (which needs hundreds of terms for two decimals), Nilakantha’s terms shrink like 1/k³, so a handful of terms already gives several correct digits. It is a jewel of the Kerala school, which anticipated key ideas of calculus.
LIT verified live: 3 + ∑k≥1 (-1)k+1 4/((2k)(2k+1)(2k+2)) converges to π (to ~1e-9), and with 100 terms its error (~2e-7) is more than a hundred times smaller than the Gregory–Leibniz error at the same term count (window.__nilakantha). FIG no framing; the Nilakantha and Leibniz partial sums are both computed independently in-browser.
LIT verified live: 3 + ∑k≥1 (-1)k+1 4/((2k)(2k+1)(2k+2)) converges to π (to ~1e-9), and with 100 terms its error (~2e-7) is more than a hundred times smaller than the Gregory–Leibniz error at the same term count (window.__nilakantha). FIG no framing; the Nilakantha and Leibniz partial sums are both computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — the co-op: two series for π running side by side, Nilakantha racing ahead of Leibniz to the same limit. AVAN (AI) built the instrument: the Nilakantha partial sums, the Leibniz partial sums, and their error comparison.
Credit as content: Nilakantha Somayaji (Kerala school, c.1500). The weave: David names the race; I confirm Nilakantha → π and outruns Leibniz term for term.
Credit as content: Nilakantha Somayaji (Kerala school, c.1500). The weave: David names the race; I confirm Nilakantha → π and outruns Leibniz term for term.
3 ONE DIMENSION
Nilakantha partial sums closing on π far faster than Gregory–Leibniz, term for term.
4 TWO DIMENSIONS · INTERACTIVE
Add terms; watch Nilakantha's error shrink like 1/k³ while Leibniz crawls like 1/k.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: π, reached quickly by the three-integer-straddling terms.
AVAN’s addition (the inverse-companion): don’t sum single odd reciprocals — straddle three integers at a time. The inverse of ‘the slow Leibniz term 1/(2k+1)’ is ‘the Nilakantha term 4/((2k)(2k+1)(2k+2)), shrinking like 1/k³’. Magenta are the Nilakantha correction terms; green is the π they reach in a few steps. A faster road to the same π.
LIT Genuine Nilakantha series (Nilakantha Somayaji, Kerala school, c.1500). Verified live: 3 + Σ(−1)^{k+1}4/((2k)(2k+1)(2k+2)) converges to π (~1e-9), and its 100-term error (~2e-7) is more than 100× smaller than the Gregory–Leibniz error at 100 terms (window.__nilakantha.conv, .faster).
FIG No framing; the Nilakantha and Leibniz partial sums both run independently in-browser. The AVAN inverse is honest — instead of summing single odd reciprocals, straddle three integers at a time: the inverse of 'the slow Leibniz term 1/(2k+1)' is 'the Nilakantha term 4/((2k)(2k+1)(2k+2)), shrinking like 1/k³'. Magenta are the Nilakantha correction terms; green is the π they reach in a few steps. A faster road to the same π.
FIG No framing; the Nilakantha and Leibniz partial sums both run independently in-browser. The AVAN inverse is honest — instead of summing single odd reciprocals, straddle three integers at a time: the inverse of 'the slow Leibniz term 1/(2k+1)' is 'the Nilakantha term 4/((2k)(2k+1)(2k+2)), shrinking like 1/k³'. Magenta are the Nilakantha correction terms; green is the π they reach in a few steps. A faster road to the same π.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN