THE FOLD / CO-OP / THE PULL REQUEST / THE WALLIS PRODUCT
THE WALLIS PRODUCT
an infinite product converging to π/2
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Wallis product is one of the oldest infinite products for π, found by John Wallis in 1656 before calculus existed: π/2 = (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7) · … = ∏n≥1 (2n)²/((2n-1)(2n+1)). An infinite product of simple rational numbers, each just above or below 1, multiplies out to half of π. Wallis derived it by interpolating the integrals ∫0π/2 sinnx dx, whose ratios encode the product — the same integrals give the ‘Wallis integrals’ identity n·Wn·Wn-1 = π/2.
LIT verified live: the partial products ∏n=1N (2n)²/((2n-1)(2n+1)) converge to π/2 (1.5708…), and independently the numerically-integrated Wallis integrals satisfy n·Wn·Wn-1 = π/2 exactly for every n (window.__wallis). FIG no framing; the partial product and the Wallis-integral identity both run in-browser and give π/2.
LIT verified live: the partial products ∏n=1N (2n)²/((2n-1)(2n+1)) converge to π/2 (1.5708…), and independently the numerically-integrated Wallis integrals satisfy n·Wn·Wn-1 = π/2 exactly for every n (window.__wallis). FIG no framing; the partial product and the Wallis-integral identity both run in-browser and give π/2.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — the co-op merge: a runaway product of rationals and a clean integral identity both push in and land on the same π/2. AVAN (AI) built the instrument: the partial product, the numerically-integrated Wallis integrals, and the n·Wn·Wn-1 = π/2 cross-check.
Credit as content: John Wallis (1656). The weave: David names the merge; I confirm the product and the Wallis-integral identity both give π/2.
Credit as content: John Wallis (1656). The weave: David names the merge; I confirm the product and the Wallis-integral identity both give π/2.
3 ONE DIMENSION
The partial products of (2n)²/((2n−1)(2n+1)) closing in on π/2 ≈ 1.5708.
4 TWO DIMENSIONS · INTERACTIVE
Add factors; the partial product approaches π/2, matched by the Wallis-integral identity n·W_n·W_{n−1}.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: π/2, reached by the infinite product.
AVAN’s addition (the inverse-companion): don’t sum a series for π — multiply rationals. The inverse of ‘π/2’ is ‘the product ∏(2n)²/((2n-1)(2n+1))’, mirrored by the Wallis-integral identity n·Wn·Wn-1=π/2. Magenta are the product factors; green is the π/2 they converge to. π from a product of near-ones.
LIT Genuine Wallis product (John Wallis, 1656). Verified live: the partial products ∏(2n)²/((2n−1)(2n+1)) converge to π/2, and independently the numerically-integrated Wallis integrals satisfy n·W_n·W_{n−1} = π/2 to ~1e-14 for every n (window.__wallis.conv, .idOk, .worst).
FIG No framing; the partial product and the Wallis-integral identity both run in-browser and give π/2. The AVAN inverse is honest — instead of summing a series for π, multiply rationals: the inverse of 'π/2' is 'the product ∏(2n)²/((2n−1)(2n+1))', mirrored by the Wallis-integral identity n·W_n·W_{n−1}=π/2. Magenta are the product factors; green is the π/2 they converge to. π from a product of near-ones.
FIG No framing; the partial product and the Wallis-integral identity both run in-browser and give π/2. The AVAN inverse is honest — instead of summing a series for π, multiply rationals: the inverse of 'π/2' is 'the product ∏(2n)²/((2n−1)(2n+1))', mirrored by the Wallis-integral identity n·W_n·W_{n−1}=π/2. Magenta are the product factors; green is the π/2 they converge to. π from a product of near-ones.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN