Plastic number

List of numbers Irrational and suspected irrational numbers γ ζ(3) √2 √3 √5 φ ρ δS e π δ
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	$\frac{\sqrt[3]{108 + 12\sqrt{69}} + \sqrt[3]{108 - 12\sqrt{69}}}{6}$

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

$x^3=x+1\, .$

It has the exact value^[2]

$\rho = \frac{\sqrt[3]{108 + 12\sqrt{69}} + \sqrt[3]{108 - 12\sqrt{69}}}{6}\, .$

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.

Properties

Recurrences

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]

$\rho = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}} \, .$

Number theory

Because the plastic number has minimal polynomial x³ − x − 1 = 0, it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x³ − x − 1, but not for any other polynomials with integer coefficients. Since the discriminant of 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

$\left(-\tfrac12\pm\tfrac{\sqrt3}2i\right)\sqrt[3]{\tfrac{1}{2}+\tfrac{1}{6}\sqrt{\tfrac{23}{3}}}+\left(-\tfrac12\mp\tfrac{\sqrt3}2i\right)\sqrt[3]{\tfrac{1}{2}-\tfrac{1}{6}\sqrt{\tfrac{23}{3}}}\approx -0.662359 \pm 0.56228i ,$

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

Trigonometry

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

$\rho = \tfrac{1}{c} \cosh\left(\tfrac{1}{3} \cosh^{-1} (3 c)\right), \qquad c = \cos\left(\tfrac{2 \pi}{12}\right) = \sin\left(\tfrac{2 \pi}{6}\right) = \tfrac{\sqrt{3}}{2} \, .$

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

Geometry

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 the square of the plastic number as their aspect ratio.^[6]

History

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.

Notes

1. Sequence  A072117 in the OEIS
2. Weisstein, Eric W., "Plastic Constant", MathWorld.
3. Sequence  A060006 in the OEIS.
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).

References

• 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

• Tales of a Neglected Number by Ian Stewart