The golden ratio φ satisfies φ² = φ + 1. The plastic number ρ satisfies the analogous cubic ρ³ = ρ + 1. Its only real solution is ρ ≈ 1.32471. Dutch architect Hans van der Laan named it "the plastic number" in the 1920s while studying three-dimensional proportions that feel harmonious to the human eye and hand.
Padovan: 1,1,1,2,2,3,4,5,7,9,12,16,21... each term = sum two and three steps back. Ratios converge to rho.
ρ is the smallest Pisot-Vijayaraghavan number: an algebraic integer greater than 1 whose conjugate roots all lie strictly inside the unit circle. Pisot numbers have special properties in harmonic analysis, tiling theory, and the structure of quasicrystals. The next Pisot number after ρ is the golden ratio φ.
Van der Laan designed the Saint Benedict Abbey in Vaals, Netherlands using proportions derived from ρ. He argued that only ratios between 1:1 and 1:7 are perceptible as "different but related", and that ρ divides this range in the most harmonious way. Full value: 1.32471795724474602596090885447809734…
The Padovan sequence 1,1,1,2,2,3,4,5,7,9,12… each term = term two ago + term three ago. The bars grow asymptotically at rate ρ ≈ 1.3247 per step. The golden ratio governs 2-step Fibonacci; the plastic number governs this 3-step variant.
The plastic number rho ≈ 1.32471 is the real root of x^3 = x + 1. Named by Dutch architect Hans van der Laan in the 1920s for its role in three-dimensional proportion. Rho is the smallest Pisot-Vijayaraghavan number: an algebraic integer greater than 1 with all conjugate roots inside the unit circle. The Padovan sequence 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16... has ratios converging to rho. Van der Laan used rho proportions in the Saint Benedict Abbey in Vaals, Netherlands.