TOPICS
Search

Plastic Constant


The plastic constant P, 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 given by

P=(x^3-x-1)_1
(1)
=((9-sqrt(69))^(1/3)+(9+sqrt(69))^(1/3))/(2^(1/3)3^(2/3))
(2)
=1.32471795...
(3)

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

It is also given by

 P=(11r+54)/(5r-61)
(4)

where

 r=-1/5[-j(tau_0)]^(1/3),
(5)

where j(tau) is the j-function and the half-period ratio is equal to tau_0=(1+isqrt(23))/2.

The plastic constant P 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).

P satisfies the algebraic identities

 P-1=P^(-4)
(6)

and

 P+1=P^3
(7)

and is therefore is one of the numbers x for which there exist natural numbers k and l such that x+1=x^k and x-1=x^(-l). It was proven by Aarts et al. (2001) that P and the golden ratio phi are in fact the only such numbers.

The identity P+1=P^3 leads to the beautiful nested radical identity

 P=RadicalBox[{1, +, RadicalBox[{1, +, RadicalBox[{1, +, ...}, 3]}, 3]}, 3].
(8)

The plastic constant is also connected with the ring of integers Z(tau=(1+isqrt(23))/2) of the number field Q(sqrt(-23)) since it the real root of the Weber function for the smallest negative discriminant with class number 3, namely -23. In particular,

Q=P^(24)
(9)
=-1/(f_2^(24)(tau))
(10)
=-[(eta(tau))/(sqrt(2)eta(2tau))]^(24)
(11)
=853.025791919196...
(12)

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

The plastic constant is also the smallest Pisot number.

The plastic constant satisfies the near-identity

 e^(pisqrt(23)) approx 2^(12)P^(24)-24,
(13)

where the difference is 7.9×10^(-5).

Surprisingly, the plastic constant is connected to the metric properties of the snub icosidodecadodecahedron.


See also

Class Number, Dedekind Eta Function, Discriminant, Golden Ratio, j-Function, Nested Radical, Padovan Sequence, Perrin Sequence, Pisot Number, Snub Icosidodecadodecahedron, Wallis's Constant, Weber Functions

Portions of this entry contributed by Tito Piezas III

Portions of this entry contributed by Floor van Lamoen

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Aarts, J.; Fokkink, R. J.; and Kruijtzer, G. "Morphic Numbers." Nieuw Arch. Wisk 5-2, 56-58, 2001. http://www.math.leidenuniv.nl/~naw/serie5/deel02/mrt2001/pdf/archi.pdf.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 9, 2003.Gazale, M. J. Ch. 7 in Gnomon: From Pharaohs to Fractals. Princeton, NJ: Princeton University Press, 1999.Piezas, T. "Ramanujan's Constant and Its Cousins." http://www.geocities.com/titus_piezas/Ramanujan_a.htm.Sloane, N. J. A. Sequences A060006 and A116397 in "The On-Line Encyclopedia of Integer Sequences."Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274, 102-103, Jun. 1996.van der Laan, H. Le Nombre Plastique: quinze Leçons sur l'Ordonnance architectonique. Leiden: Brill, 1960.Weng, A. "Class Polynomials of CM-Fields." http://www.exp-math.uni-essen.de/zahlentheorie/classpol/class.html.

Referenced on Wolfram|Alpha

Plastic Constant

Cite this as:

Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W. "Plastic Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlasticConstant.html

Subject classifications