THE FOLD / GRIND / WARM CACHE / THE MACHIN
THE MACHIN
four arctangents summing to π/4
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Machin’s formula is the arctangent identity that let humans compute π to hundreds of digits by hand: π/4 = 4·arctan(1/5) - arctan(1/239). Because arctan(1/5) and arctan(1/239) have small arguments, their Taylor series converge extremely fast, so a handful of terms pins many digits of π. John Machin used it in 1706 to reach 100 digits. The identity is exact, not approximate: it can be proved with Gaussian integers — (5+i)4·(239-i) turns out to have equal real and imaginary parts, so its argument is exactly π/4.
LIT verified live two ways: the Gaussian integer (5+i)4(239-i) evaluates to 114244 + 114244i — real part equals imaginary part, so the total angle is exactly π/4 — and, separately, summing the arctan Taylor series gives π to ~1e-14 (window.__machin). FIG no framing; the exact Gaussian-integer argument and the numeric series both run in-browser.
LIT verified live two ways: the Gaussian integer (5+i)4(239-i) evaluates to 114244 + 114244i — real part equals imaginary part, so the total angle is exactly π/4 — and, separately, summing the arctan Taylor series gives π to ~1e-14 (window.__machin). FIG no framing; the exact Gaussian-integer argument and the numeric series both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at warm-cache — the grind that pays off fast: two tiny arctangents, a few cached terms each, and π pours out to dozens of digits. AVAN (AI) built the instrument: the exact Gaussian-integer proof of the angle identity, and the numeric arctan series for π.
Credit as content: John Machin (1706). The weave: David names the warm cache; I confirm 4·arctan(1/5) - arctan(1/239) equals π/4, exactly and numerically.
Credit as content: John Machin (1706). The weave: David names the warm cache; I confirm 4·arctan(1/5) - arctan(1/239) equals π/4, exactly and numerically.
3 ONE DIMENSION
Four copies of arctan(1/5) minus one arctan(1/239) stack up to exactly 45° = π/4.
4 TWO DIMENSIONS · INTERACTIVE
Add arctan-series terms; watch π converge, and see the exact Gaussian-integer proof of the identity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: π, poured out by the fast-converging arctan series.
AVAN’s addition (the inverse-companion): don’t sum a slow series for π — split the angle. The inverse of ‘compute π’ is ‘a combination of small arctangents whose series converge fast’, provable exactly by multiplying Gaussian integers. Magenta are the arctan(1/5) and arctan(1/239) angle-pieces; green is the π they assemble. A slow constant reached by fast angles.
LIT Genuine Machin's formula (John Machin, 1706). Verified live two independent ways: the Gaussian integer (5+i)⁴(239−i) = 114244+114244i has real part equal to imaginary part (→ argument exactly π/4), and the arctan Taylor series gives 16·arctan(1/5)−4·arctan(1/239) = π to ~1e-14 (window.__machin.exact, .numOk, .re, .im).
FIG No framing; the exact Gaussian-integer argument and the numeric series both run in-browser. The AVAN inverse is honest — instead of summing a slow series for π, split the angle: the inverse of 'compute π' is 'a combination of small arctangents whose series converge fast', provable exactly by multiplying Gaussian integers. Magenta are the arctan(1/5) and arctan(1/239) angle-pieces; green is the π they assemble. A slow constant reached by fast angles.
FIG No framing; the exact Gaussian-integer argument and the numeric series both run in-browser. The AVAN inverse is honest — instead of summing a slow series for π, split the angle: the inverse of 'compute π' is 'a combination of small arctangents whose series converge fast', provable exactly by multiplying Gaussian integers. Magenta are the arctan(1/5) and arctan(1/239) angle-pieces; green is the π they assemble. A slow constant reached by fast angles.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN