Plastic number

In math, the plastic number (also known as the plastic constant) is the unique real solution of the equation

:$x^3=x+1,;$

and has the value

:$ho\; =\; sqrt\; [3]\; \{frac\{1\}\{2\}+frac\{1\}\{6\}sqrt\{frac\{23\}\{3\}+sqrt\; [3]\; \{frac\{1\}\{2\}-frac\{1\}\{6\}sqrt\{frac\{23\}\{3\}$

which is approximately 1.324717957244746025960908854 OEIS|id=A060006. The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio $1+sqrt\; 2$. The plastic number is the limiting ratio of successive terms of 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).

Additional properties

The plastic number is also a solution of the following equations:

:$x^5\; =\; x^4\; +\; 1;$

:$x^5\; =\; x^2\; +\; x\; +\; 1;$

:$x^5\; =\; x^4\; +\; x^3\; -\; x;$

:$x^6\; =\; x^2\; +\; 2x\; +\; 1;$

:$x^6\; =\; x^4\; +\; x\; +\; 1;$

:$x^7\; =\; 2x^5\; -\; 1;$

:$x^7\; =\; 2x^4\; +\; 1;$

:$x^8\; =\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1;$

:$x^9\; =\; x^6\; +\; x^4\; +\; x^2\; +\; x\; +\; 1;$

:$x^\{12\}\; =\; 2x^\{10\}\; -\; x^4\; -\; 1;$

:$x^\{14\}\; =\; 4x^9\; +\; 1;$

The plastic number is the lowest Pisot-Vijayaraghavan number.

References

* Midhat J. Gazalé, "Gnomon", 1999 Princeton University Press.

*citation
url = http://www.nexusjournal.com/conferences/N2002-Padovan.html
contribution = Dom Hans Van Der Laan And The Plastic Number
last = Padovan | first = Richard | authorlink = Richard Padovan
title = Nexus IV: Architecture and Mathematics
pages = 181–193 | publisher = Kim Williams Books | year = 2002.

*citation
last1 = Shannon | first1 = A. G.
last2 = Anderson | first2 = P. G.
last3 = Horadam | first3 = A. F.
title = Properties of Cordonnier, Perrin and Van der Laan numbers
journal = International Journal of Mathematical Education in Science and Technology
volume = 37 | issue = 7 | year = 2006 | pages = 825–831
doi = 10.1080/00207390600712554.

External links

* [http://members.fortunecity.com/templarser/padovan.html Tales of a Neglected Number] by Ian Stewart
*MathWorld|urlname=PlasticConstant|title=Plastic Constant|author=Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W.