Plastic number

The Plastic Number (also plastic number ) is a mathematical constant . It is the unique real solution of the cubic equation ${\displaystyle \psi }$

${\displaystyle x^{3}-x-1=0}$

It applies

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

As a decimal number, the plastic number begins with 1.324 717 957 244 746 025 960 908 854 ... (sequence A060006 in OEIS ). The definition of the plastic number goes back to the Dutch architect Hans van der Laan . The designation plastic number is misleading and does not correspond to van der Laan's intention, because not the material plastic , but the spatial extension (in architecture) was decisive for the name “plastic” .

properties

The two conjugate complex solutions of

${\displaystyle x^{3}-x-1=0}$

are

${\displaystyle \left(-{\tfrac {1}{2}}\pm {\tfrac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{{\tfrac {1}{2}}+{\tfrac {1}{6}}{\sqrt {\tfrac {23}{3}}}}}+\left(-{\tfrac {1}{2}}\mp {\tfrac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{{\tfrac {1}{2}}-{\tfrac {1}{6}}{\sqrt {\tfrac {23}{3}}}}}\approx -0{,}6626\pm 0{,}5623i}$

and can also be expressed by the plastic number : ${\displaystyle \psi }$

${\displaystyle -{\frac {\psi }{2}}\pm i{\sqrt {\frac {3-\psi }{4\psi }}}}$ .

Since the product of the three solutions of the cubic equation is equal to 1, the absolute value of the complex solutions is the same    (sequence A191909 in OEIS ). ${\displaystyle \psi ^{-1/2}\approx 0{,}86883696183}$

The plastic number is the limit value of the quotients of successive elements of the Padovan sequence :

${\displaystyle \psi =\lim _{n\to \infty }{\frac {P_{n}}{P_{n-1}}}}$

Individual evidence

1. ↑ ^{a b} Eric W. Weisstein : Plastic Constant , In: MathWorld
2. ^ Richard Padovan presents the plastic number , Nexus Network Journal
3. ^ Dom H. van der Laan: The architectural space. Leiden 1992.

Web links

Commons : Plastic number  - collection of images