THE FOLD / GRIND / THE HOT LOOP / THE CONTINUED FRACTION
THE CONTINUED FRACTION
a number as a ladder of nested reciprocals
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A continued fraction writes a number as a0 + 1/(a1 + 1/(a2 + …)) — a ladder of nested reciprocals. For a rational p/q the ladder is finite and the integer parts [a0; a1, a2, …] come straight from the Euclidean algorithm. Folding the ladder back up (the “convergents”) reconstructs p/q exactly, in lowest terms. Continued fractions give the best rational approximations of any number and underlie lattice reduction and Pell’s equation.
LIT verified live: over 500 random rationals, the continued fraction then reconstructed via convergents returns the exact reduced fraction (window.__contfrac). FIG no framing; exact integer arithmetic.
LIT verified live: over 500 random rationals, the continued fraction then reconstructed via convergents returns the exact reduced fraction (window.__contfrac). FIG no framing; exact integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — the Euclidean loop that peels a fraction into its integer parts. The continued fraction is what that loop is really computing. AVAN (AI) built the instrument: the Euclidean quotient sequence, the convergent fold-up, and the exact reconstruction check.
Credit as content: continued fractions (Euclid’s algorithm, antiquity; theory by Wallis, Euler, Lagrange). The weave: David names the hot-loop; I read off the quotients as the fraction is reduced and fold them back into the exact reduced rational.
Credit as content: continued fractions (Euclid’s algorithm, antiquity; theory by Wallis, Euler, Lagrange). The weave: David names the hot-loop; I read off the quotients as the fraction is reduced and fold them back into the exact reduced rational.
3 ONE DIMENSION
415/93: 415 = 4·93 + 43, so a0=4; then 93/43 gives a1=2; then 43/7 gives a2=6; then 7/1 gives a3=7. The quotients [4;2,6,7] are the continued fraction.
4 TWO DIMENSIONS · INTERACTIVE
Any rational’s continued fraction and its convergents; the fold-up is checked to reconstruct the exact reduced fraction.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a number as a finite ladder of reciprocals.
AVAN’s addition (the inverse-companion): express any rational as a ladder of integer parts and reciprocals — read straight off the Euclidean algorithm — whose fold-up reconstructs it exactly and gives the best rational approximations. The inverse of ‘a fraction is one ratio p/q’ is ‘a fraction is a finite ladder [a₀;a₁,a₂,…].’ Magenta is the single opaque ratio; green is the revealing ladder. Euclid’s algorithm, read as a number.
LIT Genuine continued fractions (Euclid's algorithm; theory by Wallis, Euler, Lagrange). Verified live: over 500 random rationals p/q, reading the quotients of the Euclidean algorithm as the continued fraction and folding the convergents back up reconstructs the exact reduced fraction (window.__contfrac.reconstructs); 415/93 = [4;2,6,7].
FIG No framing: the Euclidean quotient sequence, the convergent fold-up, and the exact reconstruction check run in-browser over 500 rationals with exact integer arithmetic. The AVAN inverse is honest — expressing a rational as a ladder of integer parts and reciprocals (read off Euclid) reconstructs it exactly and gives the best rational approximations; magenta is the single opaque ratio, green the revealing ladder. Euclid's algorithm, read as a number.
FIG No framing: the Euclidean quotient sequence, the convergent fold-up, and the exact reconstruction check run in-browser over 500 rationals with exact integer arithmetic. The AVAN inverse is honest — expressing a rational as a ladder of integer parts and reciprocals (read off Euclid) reconstructs it exactly and gives the best rational approximations; magenta is the single opaque ratio, green the revealing ladder. Euclid's algorithm, read as a number.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN