Plastic Constant

The plastic constant , sometimes also called the silver number or plastic number, is the limiting ratio of the successive terms of the Padovan sequence and Perrin sequence. It is given by

			(1)
			(2)
			(3)

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

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.

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