Plastic ratio
| Binary | 1.0101001100100000101… |
| Decimal | 1.32471795724474602596… |
| Hexadecimal | 1.5320B74ECA44ADAC1788… |
| Continued fraction | |
| Algebraic form | Note that this continuing fraction is not periodic. |
![{\displaystyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/632f36cfd2cc1e85c932d625257359ebf7ed3330)
In mathematics, the plastic number ρ (also known as the plastic constant) is a mathematical constant which is the unique real solution of the cubic equation
It has the value
![{\displaystyle \rho ={\sqrt[{3}]{{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {\frac {23}{3}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8340072b5bc6fe3953fc60e5c81ffb38878297a2)
which is approximately 1.324717957244746025960908854 (sequence A060006 in the OEIS). The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio . 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 name plastic number (originally in Dutch plastische getal) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio and silver number, 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 (Padovan 2002; Shannon, Anderson, and Horadam 2006).
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. This is a property of all minimal polynomials.
The plastic number is the smallest Pisot-Vijayaraghavan number.
References
- Midhat J. Gazalé, Gnomon, 1999 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.
External links
- Tales of a Neglected Number by Ian Stewart
- Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W. "Plastic Constant". MathWorld.
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