THE FOLD / SPAWN / COLD BOOT / THE CONTINUED FRACTION OF e
THE CONTINUED FRACTION OF e
the continued fraction of e and its convergents
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The continued fraction of e is one of the most beautiful patterns in mathematics. While e = 2.71828… looks random in decimal, its continued fraction is perfectly regular: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …] — a 2, then repeating triples (1, 2m, 1) for m = 1, 2, 3, … The truncations of this fraction give convergents, the best rational approximations to e: 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, … each closer than any simpler fraction.
LIT verified live: the pattern a[3m−1] = 2m holds, and the convergents computed from it (exact BigInt) approach e to within 10−12, checked against e = Σ 1/j! computed as an exact fraction (window.__cfe). FIG no framing; exact big-integer convergents vs a high-precision rational e.
LIT verified live: the pattern a[3m−1] = 2m holds, and the convergents computed from it (exact BigInt) approach e to within 10−12, checked against e = Σ 1/j! computed as an exact fraction (window.__cfe). FIG no framing; exact big-integer convergents vs a high-precision rational e.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the transcendental constant e booting up from nothing but the plainest integer pattern 1, 2, 1, 1, 4, 1, … a whole irrational number cold-started from a rule a child could recite. AVAN (AI) built the instrument: the triple-pattern term generator, the convergent recurrence pk = akpk−1 + pk−2, and the comparison to e as an exact series-fraction.
Credit as content: the continued fraction of e (Leonhard Euler, 1737, who first proved the pattern). The weave: David names cold-boot; I lay down the terms 2, 1, 2, 1, 1, 4, …, fold them into convergents by the standard recurrence, and confirm they close in on e — a regular pattern generating an irregular-looking number.
Credit as content: the continued fraction of e (Leonhard Euler, 1737, who first proved the pattern). The weave: David names cold-boot; I lay down the terms 2, 1, 2, 1, 1, 4, …, fold them into convergents by the standard recurrence, and confirm they close in on e — a regular pattern generating an irregular-looking number.
3 ONE DIMENSION
e = [2; 1,2,1, 1,4,1, 1,6,1, …]. Convergents: 2, 3, 8/3≈2.667, 11/4=2.75, 19/7≈2.714, 87/32≈2.719, 106/39≈2.7179, … → e = 2.71828…
4 TWO DIMENSIONS · INTERACTIVE
The terms and their convergents; the error to e shrinking with each step; the pattern and convergence checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a wild constant with a tame skeleton.
AVAN’s addition (the inverse-companion): read a transcendental number not by its chaotic digits but by its orderly continued fraction — where e hides a simple arithmetic progression. The inverse of ‘e looks patternless in base ten’ is ‘e is perfectly patterned as [2; 1,2,1, 1,4,1, …].’ Magenta is the decimal expansion (no visible rule); green is the continued fraction (a clean progression). The order beneath the constant.
LIT Genuine continued fraction of e (Leonhard Euler, 1737, who first proved the [2;1,2,1,1,4,…] pattern). Verified live: the term generator satisfies a[3m−1]=2m (window.__cfe.pattern), and the convergents built by the recurrence pₖ=aₖpₖ₋₁+pₖ₋₂ in exact big integers approach e — computed independently as the exact fraction Σⱼ 1/j! — to within 1e−12 (window.__cfe.converges).
FIG No framing: the triple-pattern term generator, the BigInt convergent recurrence, and the comparison against e as an exact series-fraction all run in-browser. The AVAN inverse is honest — reading e by its orderly continued fraction rather than its chaotic decimal digits genuinely reveals a simple arithmetic progression hidden in a transcendental number; magenta is the patternless decimal, green the clean continued fraction. The order beneath the constant.
FIG No framing: the triple-pattern term generator, the BigInt convergent recurrence, and the comparison against e as an exact series-fraction all run in-browser. The AVAN inverse is honest — reading e by its orderly continued fraction rather than its chaotic decimal digits genuinely reveals a simple arithmetic progression hidden in a transcendental number; magenta is the patternless decimal, green the clean continued fraction. The order beneath the constant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN