Plastic number

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Binary 1.01010011001000001011
Decimal 1.32471795724474602596
Hexadecimal 1.5320B74ECA44ADAC1788
Continued fraction[1] [1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80 ...]
Note that this continued fraction is neither finite nor periodic.
(Shown in linear notation)
Algebraic form

In mathematics, the plastic number ρ (also known as the plastic constant or the minimal Pisot number) is a mathematical constant which is the unique real solution of the cubic equation

It has the exact value[2]

Its decimal expansion begins with 1.324717957244746025960908854….[3] and at least 10,000,000,000 decimal digits have been computed.[4]

The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio 1 + 2.



The powers of the plastic number A(n) = ρn satisfy the recurrence relation A(n) = A(n − 2) + A(n − 3) for n > 2. Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the Padovan sequence and the Perrin sequence, and bears the same relationship to these sequences as the golden ratio does to the Fibonacci sequence and the silver ratio does to the Pell numbers.

The plastic number satisfies the nested radical recurrence:[5]

Number theory[edit]

Because the plastic number has minimal polynomial x3x − 1 = 0, it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x3x − 1, but not for any other polynomials with integer coefficients. Since the discriminant of its minimal polynomial is −23, its splitting field over rationals is ℚ(−23, ρ). This field is also Hilbert class field of ℚ(−23).

The plastic number is the smallest Pisot–Vijayaraghavan number. Its algebraic conjugates are

of absolute value ≈ 0.868837 (sequence A191909 in the OEIS). This value is also 1/ρ because the product of the three roots of the minimal polynomial is 1.


The plastic number can be written using the hyperbolic cosine (cosh) and its inverse:

(See Cubic function#Trigonometric (and hyperbolic) method.)


There are two ways of partitioning a square into three similar rectangles: the trivial solution given by three equal rectangles with aspect ratio 1:3, and another solution in which the three rectangles all have different sizes, but the same shape, with their aspect ratio equal to ρ2; in this case the ratios of the linear sizes of the rectangles are: ρ (large:medium) and ρ2 (medium:small).[6]


The name plastic number (het plastische getal in Dutch) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio and silver ratio, the word plastic was not intended to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.[7] This is because, according to Padovan, the characteristic ratios of the number, 3/4 and 1/7, relate to the limits of human perception in relating one physical size to another.


  1. ^ Sequence OEISA072117 in the OEIS
  2. ^ Weisstein, Eric W. "Plastic Constant". MathWorld. 
  3. ^ Sequence OEISA060006 in the OEIS.
  4. ^ [1]
  5. ^ Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W. "Plastic Constant". MathWorld. 
  6. ^ de Spinadel, Vera W.; Antonia, Redondo Buitrago (2009), "Towards van der Laan’s plastic number in the plane" (PDF), Journal for Geometry and Graphics, 13 (2): 163–175 .
  7. ^ Padovan (2002);Shannon, Anderson & Horadam (2006).


  • Gazalé, Midhat J. (1999), Gnomon, Princeton University Press .
  • Padovan, Richard (2002), "Dom Hans Van Der Laan And The Plastic Number", Nexus IV: Architecture and Mathematics, Kim Williams Books, pp. 181–193 .
  • Shannon, A. G.; Anderson, P. G.; Horadam, A. F. (2006), "Properties of Cordonnier, Perrin and Van der Laan numbers", International Journal of Mathematical Education in Science and Technology, 37 (7): 825–831, doi:10.1080/00207390600712554 .
  • Aarts, J.; Fokkink, R.; Kruijtzer, G. (2001), "Morphic numbers" (PDF), Nieuw Arch. Wiskd., 5, 2 (1): 56–58 .

External links[edit]