THE FOLD / GRIND / THE GRINDSTONE / THE CONTINUED FRACTION OF E
THE CONTINUED FRACTION OF E
the number e written as a patterned fraction
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The continued fraction of e hides a beautiful regularity where you’d expect chaos. Unlike π (whose CF looks random), e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] follows a clean pattern: a 2, then repeating triples (1, 2m, 1) for m = 1, 2, 3, … The even numbers 2, 4, 6, 8, … march through, each flanked by ones. Truncating the fraction gives rational convergents that rush toward e faster than any decimal expansion reveals.
LIT verified live: the generated terms obey the 2;(1,2m,1) pattern, and the convergent p/q (exact BigInt) matches e to floating precision (window.__cfe). FIG no framing; exact pattern check and BigInt convergents.
LIT verified live: the generated terms obey the 2;(1,2m,1) pattern, and the convergent p/q (exact BigInt) matches e to floating precision (window.__cfe). FIG no framing; exact pattern check and BigInt convergents.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the pattern grinds out term by term, 1, 2, 1, 1, 4, 1, 1, 6, …, each turn adding structure that sharpens the approximation to e. AVAN (AI) built the instrument: the pattern generator, the BigInt convergent recurrence, and the pattern + convergence-to-e checks.
Credit as content: the regular continued fraction of e (Euler, 1737). The weave: David names the-grindstone; I generate e’s continued-fraction terms by the 2;(1,2m,1) rule, fold them into exact rational convergents, and confirm both that the pattern holds and that the convergents close in on e.
Credit as content: the regular continued fraction of e (Euler, 1737). The weave: David names the-grindstone; I generate e’s continued-fraction terms by the 2;(1,2m,1) rule, fold them into exact rational convergents, and confirm both that the pattern holds and that the convergents close in on e.
3 ONE DIMENSION
e = [2; 1,2,1, 1,4,1, 1,6,1, 1,8,1, …] — a 2, then (1, 2m, 1) for m=1,2,3,… The even terms 2,4,6,8 step upward, each between two ones.
4 TWO DIMENSIONS · INTERACTIVE
The CF terms and the convergents approaching e; the pattern and error to e checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: e as a patterned fraction.
AVAN’s addition (the inverse-companion): write the transcendental e not as an endless decimal but as a patterned continued fraction whose convergents are the best rational approximations — 2;(1,2m,1) marching the even numbers through. The inverse of ‘expand e = 2.71828… digit by digit’ is ‘read its continued fraction — structure, and fast convergence.’ Magenta is the featureless decimal; green is the patterned fraction. Order inside a transcendental.
LIT Genuine regular continued fraction of e (Euler 1737). Verified live: the terms generated by the 2;(1,2m,1) rule obey that exact pattern (window.__cfe.pattern), and the BigInt convergent p/q matches Math.E to floating precision — |p/q − e| < 1e-12 (window.__cfe.convergesToE).
FIG No framing: the pattern generator, the BigInt convergent recurrence, and the pattern + convergence-to-e checks run in-browser (BigInt so the convergents stay exact) and agree. The AVAN inverse is honest — writing e as a patterned continued fraction whose convergents are the best rational approximations genuinely differs from an endless decimal; magenta is the featureless 2.71828…, green the patterned fraction. Order inside a transcendental.
FIG No framing: the pattern generator, the BigInt convergent recurrence, and the pattern + convergence-to-e checks run in-browser (BigInt so the convergents stay exact) and agree. The AVAN inverse is honest — writing e as a patterned continued fraction whose convergents are the best rational approximations genuinely differs from an endless decimal; magenta is the featureless 2.71828…, green the patterned fraction. Order inside a transcendental.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN