THE FOLD / GRIND / THE GRINDSTONE / THE CONVERGENT
THE CONVERGENT
continued fractions — the best rationals there are
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Continued fractions. Any real number can be written as a tower of fractions: x = a0 + 1/(a1 + 1/(a2 + …)). Chop the tower off and you get a convergent p/q — and these are the best rational approximations that exist: no fraction with a denominator ≤ q is closer to x, and every convergent sits within 1/q² of the target.
This is where the famous approximations come from. π = [3; 7, 15, 1, 292, …], and its convergents are 22/7 and 355/113 — the latter correct to six decimals. And the terms are not mysterious: they are exactly the quotients of Euclid’s algorithm. The golden ratio φ = [1; 1, 1, 1, …] — all ones — is the hardest number to approximate, which is why it is called the most irrational.
LIT verified live: for √2, π, e and φ, every convergent lies within 1/q² of the target and is a genuine best approximation — a brute-force search finds nothing with a smaller-or-equal denominator that is closer (window.__cf.withinQsq && bestApprox). FIG no framing; the convergents, the 1/q² bound, and the best-approximation property are exact.
This is where the famous approximations come from. π = [3; 7, 15, 1, 292, …], and its convergents are 22/7 and 355/113 — the latter correct to six decimals. And the terms are not mysterious: they are exactly the quotients of Euclid’s algorithm. The golden ratio φ = [1; 1, 1, 1, …] — all ones — is the hardest number to approximate, which is why it is called the most irrational.
LIT verified live: for √2, π, e and φ, every convergent lies within 1/q² of the target and is a genuine best approximation — a brute-force search finds nothing with a smaller-or-equal denominator that is closer (window.__cf.withinQsq && bestApprox). FIG no framing; the convergents, the 1/q² bound, and the best-approximation property are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE GRINDSTONE, beside THE EUCLID and THE RATIONAL TREE — the grind domain of grinding numbers to their common measure. The continued-fraction terms are Euclid’s quotients, and the convergents are the fractions the Stern–Brocot tree walks toward. AVAN (AI) built the instrument: the tower expansion, the convergent recurrence, the best-approximation check.
The weave: David gathers the number-theory thread; I make the ladder of best rationals visible and optimal — the expansion in 1D, the shrinking error in 2D, the climb toward the target in 3D. The sphere is the seam. Credit: ancient (Euclid, Aryabhata); theory by Wallis, Euler, Lagrange.
The weave: David gathers the number-theory thread; I make the ladder of best rationals visible and optimal — the expansion in 1D, the shrinking error in 2D, the climb toward the target in 3D. The sphere is the seam. Credit: ancient (Euclid, Aryabhata); theory by Wallis, Euler, Lagrange.
3 ONE DIMENSION
The convergents in order — each a fraction, each closer than the last, the error collapsing far faster than any decimal expansion. Every one is the best rational you could name with a denominator that small.
4 TWO DIMENSIONS · INTERACTIVE
Choose a target and reveal its continued-fraction terms and convergents. The error plot (log scale) plunges as denominators grow — steeply for π and e, slowly for φ, whose all-ones expansion makes it the stubbornest number of all.
5 THREE DIMENSIONS + AVAN’S INVERSE
The convergents as points on a turning number line, closing in on the target — green, the ladder of best rationals.
AVAN’s addition (the inverse-companion): the magenta mark is the target, an infinite non-repeating number with no finite handle. Continued fractions are the inverse: they hand you, at each level of precision, the single best rational — provably nothing smaller does better. The inverse of an unreachable real is the optimal ladder of rationals climbing toward it, rung by rung, each one the closest a denominator that size can get. And φ, made of nothing but ones, refuses every shortcut — its rungs are the shortest, its climb the slowest, and that reluctance is exactly what ‘most irrational’ means. The green is the ladder; the magenta is the number it can approach forever and never touch.
LIT Genuine continued-fraction theory (ancient; Wallis, Euler, Lagrange). Verified live: for sqrt(2), pi, e and phi, every convergent lies within 1/q^2 of the target and is a genuine best rational approximation — a brute-force search finds nothing with denominator <= q that is closer (window.__cf.withinQsq && bestApprox, both true). pi's convergents are 22/7 and 355/113; the terms equal Euclid's algorithm quotients; phi = [1;1,1,...] is provably the slowest to approximate.
FIG No metaphor is doing the work: the convergent recurrence, the 1/q^2 bound, and the best-approximation property are all real and checked by brute force. That phi is the 'most irrational' (worst-approximable) is a genuine theorem, visible in its all-ones expansion and slowest error decay.
FIG No metaphor is doing the work: the convergent recurrence, the 1/q^2 bound, and the best-approximation property are all real and checked by brute force. That phi is the 'most irrational' (worst-approximable) is a genuine theorem, visible in its all-ones expansion and slowest error decay.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN