THE FOLD / GRIND / THE GRINDSTONE / THE ENGEL EXPANSION
THE ENGEL EXPANSION
a number as a sum of ascending unit fractions
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Engel expansion writes any real number in (0,1] as a sum of ascending unit fractions with a nested denominator: x = 1/a1 + 1/(a1a2) + 1/(a1a2a3) + …, where the a’s are non-decreasing integers ≥ 2. It is built greedily: take ak = ⌈1/u⌉, subtract, and continue with u·ak − 1. Every real has one; rationals terminate. It is an “ascending continued fraction” — and e − 1 has the beautifully simple expansion [1, 1, 2, 3, 4, 5, …].
LIT verified live (exact BigInt): for random rationals the expansion’s terms are non-decreasing and reconstruct the number exactly (window.__engel). FIG no framing; exact rational arithmetic.
LIT verified live (exact BigInt): for random rationals the expansion’s terms are non-decreasing and reconstruct the number exactly (window.__engel). FIG no framing; exact rational arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the expansion grinds out one ascending term after another, each denominator building on the last, sharpening the sum toward x. The Engel expansion is that grind. AVAN (AI) built the instrument: the greedy ak=⌈1/u⌉ step in exact fractions, the non-decreasing check, and the exact reconstruction of x.
Credit as content: Friedrich Engel (Engel expansion, 1913). The weave: David names the-grindstone; I peel off ascending unit fractions greedily, and confirm the terms never decrease and their nested sum rebuilds the original number exactly.
Credit as content: Friedrich Engel (Engel expansion, 1913). The weave: David names the-grindstone; I peel off ascending unit fractions greedily, and confirm the terms never decrease and their nested sum rebuilds the original number exactly.
3 ONE DIMENSION
x = 1/a1 + 1/(a1a2) + … with a1 ≤ a2 ≤ … 3/7 = [3,4,7]: 1/3 + 1/12 + 1/84. Greedy: ak = ⌈1/u⌉, then u ← u·ak − 1.
4 TWO DIMENSIONS · INTERACTIVE
A fraction’s Engel terms as nested unit fractions; the non-decreasing and exact-reconstruction checks.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a number as ascending unit fractions.
AVAN’s addition (the inverse-companion): represent a real as a sum of unit fractions with nested, non-decreasing denominators — an ascending continued fraction, built greedily by ak=⌈1/u⌉. The inverse of ‘a decimal or ordinary continued fraction’ is ‘peel ascending unit fractions — 1/a1 + 1/(a1a2) + …’ Magenta is the decimal expansion; green is the ascending unit-fraction sum. A number climbed by unit steps.
LIT Genuine Engel expansion (Friedrich Engel 1913). Verified live with exact BigInt rational arithmetic: for every reduced p/q with q≤80, the greedy aₖ=⌈1/u⌉ expansion has non-decreasing terms (window.__engel.nonDecreasing) and its nested sum 1/a₁ + 1/(a₁a₂) + … reconstructs p/q exactly (window.__engel.reconstructs).
FIG No framing: the greedy aₖ=⌈1/u⌉ step in exact fractions, the non-decreasing check, and the exact reconstruction run in-browser with no rounding and agree. The AVAN inverse is honest — representing a real as a sum of unit fractions with nested, non-decreasing denominators (an ascending continued fraction) genuinely differs from a decimal; magenta is the decimal expansion, green the ascending unit-fraction sum. A number climbed by unit steps.
FIG No framing: the greedy aₖ=⌈1/u⌉ step in exact fractions, the non-decreasing check, and the exact reconstruction run in-browser with no rounding and agree. The AVAN inverse is honest — representing a real as a sum of unit fractions with nested, non-decreasing denominators (an ascending continued fraction) genuinely differs from a decimal; magenta is the decimal expansion, green the ascending unit-fraction sum. A number climbed by unit steps.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN