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 a special Pisot number (Smyth 1999, p. 1).

It is also given by

(4)

where

(5)

where is the j-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 a morphic number, a real number for which there exist positive integers and such that and . It was proven by Aarts et al. (2001, p. 58) that and the golden ratio are in fact the only such numbers.

The identity leads to the beautiful nested radical identity

(8)

The unit circle and the semicubical parabola intersect at

(9)

Indeed, gives (M. Schofield, pers. comm., Sep. 1, 2022).

The plastic constant is also connected with the ring of integers of the number field through a special value of the Weber modular function . This number field has discriminant and class number 3. Setting gives

			(10)
			(11)
			(12)
			(13)

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

The plastic constant is also the smallest Pisot number.

The plastic constant satisfies the near-identity

(14)

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 (E. Pegg, Jr., pers. comm., Mar. 21, 2025).

The limiting self-similar fractal is illustrated above, with its two first-level images colored separately. It is the attractor of the two similarity transformations underlying the quadrilateral substitution. These maps have scale factors and , obtained from the absolute values of the two complex multipliers. The Hokkaido fractal is another construction involving the nonreal roots of the same cubic equation.

A further construction uses a pentagon with vertices in the complex plane given by , , , , and , where is the root with positive imaginary part (Pegg 2026b). In this boundary order, its side lengths are proportional to , , , , and . The corresponding interior angles , , , , and satisfy , , and (Pegg 2026b). Starting with this pentagon, a fractal is obtained by iterating the two similarity transformations

			(15)
			(16)

The substitution system , , , with dominant eigenvalue , gives a Rauzy fractal with three subtiles (Coulbois and Minervino 2017, pp. 6 and 12-13). In complex eigenvector coordinates, the limiting attractor of the pentagonal construction is similar to one of these subtiles. The figure shows the seed at left and a finite approximation to the limiting attractor at right, with its two first-level images colored separately. The scale factors are and , whose squares sum to 1. The children extend outside the seed pentagon, so the construction is not a dissection of that pentagon. After three steps of replacing all largest pieces, the polygonal patch contains a triangular gap with side lengths in the ratio . These patches establish neither a tiling of the plane by these pentagons nor a nonperiodic tiling by them. The limiting fractal admits a tiling of the plane using three sizes of similar copies, with relative linear sizes , , and . A proof uses the algebraic conjugate obtained by replacing with . The inverse maps then partition the interval at . Together with the field norm, this gives uniform separation of the normalized translations in compositions of a fixed total exponent and implies positive area of the attractor. With similarity dimension 2, positive area implies nonempty interior (Schief 1994; Ye 2002, p. 44). Expanding a smaller copy lying wholly in the interior and subdividing at comparable scales produces the three-size tiling.

The numbers of copies in the pentagonal construction follow the Padovan sequence (Pegg 2026c). Replacing every largest copy produces children with linear scale factors and relative to their parent. For size levels with linear sizes , , and and counts , advancing the size threshold changes the counts to . A level may have zero copies, so advancing the threshold need not perform an actual replacement. Writing and , the total after replacements of all largest copies is . The figure shows stages with 4, 12, and 37 copies, colored by size. At stage 23, the largest, middle, and smallest copies number , , and , respectively, totaling . These are counts of substitution copies, including their polygonal approximations, and do not remove the triangular gaps described above.

The plastic constant also defines a tent tile using the maps and , with as above. For this attractor, Scheicher et al. (2026) leave the existence of a point lattice tiling by translates, with or without half-turned copies, unresolved. This is a different question from the three-size tiling of the pentagonal construction's limiting fractal.

Hinsley (2016) illustrates a two-piece tile under the name Ammonite tile. The illustration above shows a finite approximation of this tile, up to a similarity transformation, with its two first-level pieces colored separately. With as above, it is the attractor of the two similarity transformations

			(17)
			(18)

Its scale factors are and . The similarity transformation carries this attractor to the bean-shaped attractor of Pegg's pentagonal construction above. Hinsley's printed map list instead gives in place of , which does not reproduce his pictured tile.

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

(19)

With integer indices, the point set is

(20)

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