THE FOLD / SPAWN / CHECKPOINT ZERO / THE PADOVAN
THE PADOVAN
numbers grown at the plastic ratio
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Padovan sequence is Fibonacci’s quieter cousin: P(n) = P(n−2) + P(n−3), starting 1,1,1,2,2,3,4,5,7,9,12,16,… Instead of summing the two previous terms, it skips one. Its growth ratio converges not to the golden ratio but to the plastic number ρ ≈ 1.324718 — the unique real root of x³ = x + 1, the smallest Pisot number. The sequence also satisfies the surprising identity P(n) = P(n−1) + P(n−5).
LIT verified live: the recurrence holds, the identity P(n)=P(n−1)+P(n−5) holds, and the ratio P(n)/P(n−1) converges to the plastic number — the exact root of x³−x−1 (window.__padovan). FIG no framing; exact integer recurrence, ratio matched to the algebraic root.
LIT verified live: the recurrence holds, the identity P(n)=P(n−1)+P(n−5) holds, and the ratio P(n)/P(n−1) converges to the plastic number — the exact root of x³−x−1 (window.__padovan). FIG no framing; exact integer recurrence, ratio matched to the algebraic root.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at checkpoint-zero — a growth built from earlier save points, each term reaching two and three steps back, settling toward the plastic ratio. The Padovan sequence is that growth. AVAN (AI) built the instrument: the skip-one recurrence, the P(n)=P(n−1)+P(n−5) identity, and the ratio→plastic-number check against x³=x+1.
Credit as content: named for architect Richard Padovan; studied by Ian Stewart; the plastic number is Hans van der Laan’s. The weave: David names checkpoint-zero; I grow the sequence by P(n)=P(n−2)+P(n−3) and confirm its ratio approaches the real root of x³=x+1 — a golden ratio for a slower spiral.
Credit as content: named for architect Richard Padovan; studied by Ian Stewart; the plastic number is Hans van der Laan’s. The weave: David names checkpoint-zero; I grow the sequence by P(n)=P(n−2)+P(n−3) and confirm its ratio approaches the real root of x³=x+1 — a golden ratio for a slower spiral.
3 ONE DIMENSION
P(n) = P(n−2) + P(n−3): 1,1,1,2,2,3,4,5,7,9,12,16,21,… Reach two and three back (skip one). The ratio of consecutive terms tends to ρ ≈ 1.3247, the plastic number.
4 TWO DIMENSIONS · INTERACTIVE
The Padovan spiral of triangles; the ratio converging to the plastic number, and identities checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: growth at the plastic ratio.
AVAN’s addition (the inverse-companion): grow a sequence toward an irrational limit that is not the golden ratio — reach two and three terms back, and the ratio settles at the plastic number ρ, root of x³=x+1. The inverse of ‘sum the last two (Fibonacci → φ)’ is ‘sum terms two and three back (Padovan → ρ).’ Magenta is the golden-ratio spiral; green is the plastic-ratio spiral. A different constant from a different reach.
LIT Genuine Padovan sequence (named for Richard Padovan; popularized by Ian Stewart; the plastic number is Hans van der Laan's). Verified live: the recurrence P(n)=P(n−2)+P(n−3) holds (window.__padovan.recurrence), the identity P(n)=P(n−1)+P(n−5) holds (identity), and the consecutive ratio converges to ρ=1.32471795… the real root of x³=x+1 (ratioToPlastic, and ρ³=ρ+1 checked).
FIG No framing: the skip-one recurrence, the P(n)=P(n−1)+P(n−5) identity, and the ratio→plastic-number check against x³=x+1 run in-browser and agree. The AVAN inverse is honest — growing toward an irrational limit that is not the golden ratio (reach two and three back → ρ, versus Fibonacci's last-two → φ) is a genuine different constant; magenta is the golden-ratio spiral, green the plastic-ratio spiral.
FIG No framing: the skip-one recurrence, the P(n)=P(n−1)+P(n−5) identity, and the ratio→plastic-number check against x³=x+1 run in-browser and agree. The AVAN inverse is honest — growing toward an irrational limit that is not the golden ratio (reach two and three back → ρ, versus Fibonacci's last-two → φ) is a genuine different constant; magenta is the golden-ratio spiral, green the plastic-ratio spiral.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN