Plastic Constant

The plastic constant, sometimes also called le nombre radiant, the minimal Pisot number, plastic number, plastic ratio, platin number, Siegel's number, or silver number, is the limiting ratio of the successive terms of the Padovan sequence or Perrin sequence. It is denoted using or and given by

			(1)
			(2)
			(3)

(OEIS A060006), where denotes a polynomial root. It is therefore an algebraic number of degree 3.

The plastic constant is the self-summing ratio.

It is also given by

(4)

where

(5)

where is the -function and the half-period ratio is equal to .

The plastic constant was originally studied in 1924 by Gérard Cordonnier when he was 17. In his later correspondence with Dom Hans van der Laan, he described applications to architecture, using the name "radiant number." In 1958, Cordonnier gave a lecture tour that illustrated the use of the constant in many existing buildings and monuments (C. Mannu, pers comm., Mar. 11, 2006).

satisfies the algebraic identities

(6)

and

(7)

and is therefore is one of the numbers for which there exist natural numbers and such that and . It was proven by Aarts et al. (2001) that and the golden ratio are in fact the only such numbers.

The identity leads to the beautiful nested radical identity

(8)

The plastic constant is also connected with the ring of integers of the number field since it the real root of the Weber function for the smallest negative discriminant with class number 3, namely . In particular,

			(9)
			(10)
			(11)
			(12)

(OEIS A116397), where is the Dedekind eta function.

The plastic constant is also the smallest Pisot number.

The plastic constant satisfies the near-identity

(13)

where the difference is .

Surprisingly, the plastic constant is connected to the metric properties of the snub icosidodecadodecahedron. It is also involved in the definition of maverick graphs.

A single Beloch fold gives a geometric construction of the plastic constant (Pegg 2025).

The plastic constant satisfies , where is the golden ratio, so has a reciprocal proportion triangle.

A substitution tiling of quadrilaterals can be obtained using quadrilaterals having sides in the ratio and vertices in the complex plane given by powers of the complex root of , as noticed by Pegg (2019).

The first twelve iterations of this substitution are illustrated above (pers. comm., E. Pegg Jr., Mar. 21, 2025).

There is also a configuration of 19 points in three-dimensional Euclidean space whose 171 pairwise distances are all integer powers of (Pegg 2026). The illustration joins every pair of points, giving a geometric realization of the complete graph . Set and , and define

(14)

With integer indices, the point set is

(15)

The squares of its distances are for integers between and 6. This distance property does not establish a closed substitution system.