THE FOLD / GRIND / THE EPOCH / THE PLASTIC NUMBER
THE PLASTIC NUMBER
the third metallic constant solving a cubic
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The plastic number ρ ≈ 1.3247179572 is the quiet cousin of the golden ratio — the unique real root of the cubic x³ = x + 1. Where the golden ratio φ solves x² = x + 1 and governs the Fibonacci numbers, the plastic number solves the next cubic and governs the Padovan and Perrin sequences (each term the sum of the two before the previous: P(n) = P(n-2) + P(n-3)). Ratios of consecutive terms converge to ρ. It is the only number that is both a ‘morphic number’ for x³=x+1 and expressible as the infinitely nested radical ∛(1 + ∛(1 + ∛(1 + …))). The Dutch architect Dom Hans van der Laan built a whole system of proportion on it in 1928.
LIT verified live: Newton’s method gives ρ with ρ³ - ρ - 1 = 0 to ~1e-14; the Padovan and Perrin ratios both converge to ρ; and the nested cube-root iteration ∛(1 + ·) converges to the same ρ (window.__plastic). FIG no framing; the cubic root, the two sequence ratios, and the radical are computed independently in-browser.
LIT verified live: Newton’s method gives ρ with ρ³ - ρ - 1 = 0 to ~1e-14; the Padovan and Perrin ratios both converge to ρ; and the nested cube-root iteration ∛(1 + ·) converges to the same ρ (window.__plastic). FIG no framing; the cubic root, the two sequence ratios, and the radical are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — the grind: two integer sequences ground forward step by step, their ratios settling onto the single cubic root ρ. AVAN (AI) built the instrument: the Newton root, the Padovan and Perrin ratios, and the nested-radical iteration.
Credit as content: Dom Hans van der Laan (plastic number, 1928); Richard Padovan; the Perrin sequence. The weave: David names the grind; I confirm ρ³=ρ+1 and both sequence ratios → ρ.
Credit as content: Dom Hans van der Laan (plastic number, 1928); Richard Padovan; the Perrin sequence. The weave: David names the grind; I confirm ρ³=ρ+1 and both sequence ratios → ρ.
3 ONE DIMENSION
The Padovan sequence and the ratios of consecutive terms settling onto ρ ≈ 1.3247.
4 TWO DIMENSIONS · INTERACTIVE
Step the sequences; the ratios are checked to converge to the cubic root ρ (x³=x+1).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: ρ, the single real root of x³ = x + 1.
AVAN’s addition (the inverse-companion): don’t solve the cubic algebraically — let a radical fold into it. The inverse of ‘the root of x³=x+1’ is ‘the nested cube-root ∛(1+∛(1+…)) that lands on ρ’. Magenta is the nested-radical iteration; green is the ρ it and the sequence ratios reach. One cubic root, three roads to it.
LIT Genuine plastic number (Dom Hans van der Laan, 1928; Padovan and Perrin sequences). Verified live: Newton gives ρ with ρ³−ρ−1 = 0 to ~1e-14; the Padovan and Perrin consecutive-term ratios both converge to ρ; and the nested cube-root iteration ∛(1+·) converges to the same ρ (window.__plastic.ok).
FIG No framing; the cubic root, the two sequence ratios, and the radical are computed independently in-browser. The AVAN inverse is honest — instead of solving the cubic algebraically, let a radical fold into it: the inverse of 'the root of x³=x+1' is 'the nested cube-root ∛(1+∛(1+…)) that lands on ρ'. Magenta is the nested-radical iteration; green is the ρ it and the sequence ratios reach. One cubic root, three roads to it.
FIG No framing; the cubic root, the two sequence ratios, and the radical are computed independently in-browser. The AVAN inverse is honest — instead of solving the cubic algebraically, let a radical fold into it: the inverse of 'the root of x³=x+1' is 'the nested cube-root ∛(1+∛(1+…)) that lands on ρ'. Magenta is the nested-radical iteration; green is the ρ it and the sequence ratios reach. One cubic root, three roads to it.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EPOCH · David Lee Wise (ROOT0), with AVAN