Plastic number

In mathematics, a plastic number (also known as a plastic constant ) is the only real root of the equation

x^{3}=x+1.\;

{\ displaystyle x ^ {3} = x + 1. \;}

x ^ {3} = x + 1. \;

Its numerical value

\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}}}}},

{\ 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}}}}, }

{\ 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}}}}, }

approximately 1.32471795724474602596090885447809734073440405690173336453401505030282785124554759405469934798178728032991 ... (the numbers form the sequence A060006 in OEIS ).

A plastic number is sometimes also called a silver number , but more often this name is used for the silver section $1+{\sqrt {2}}$ ${\ displaystyle 1 + {\ sqrt {2}}}$ $1 + {\ sqrt 2}$ .

The name plastic number (originally in the Dutch plastische getal ) was given in 1928 by Hans van der Laan. Unlike the names of the golden and silver sections, the word plastic had nothing to do with any substance, but rather referred to the fact that it can be given a three-dimensional shape (Padovan 2002; Shannon, Anderson, and Horadam 2006).

Properties

A plastic number is the limit of the ratio of consecutive terms of the Padovan and Perrin sequences and has the same meaning for them as the golden ratio for the Fibonacci sequence and the silver ratio for Pell numbers .

The plastic number is also the root of the equations:

x^{5}=x^{4}+1

{\ displaystyle x ^ {5} = x ^ {4} +1}

x^{5}=x^{2}+x+1

{\ displaystyle x ^ {5} = x ^ {2} + x + 1}

x^{5}=x^{4}+x^{3}-x

{\ displaystyle x ^ {5} = x ^ {4} + x ^ {3} -x}

x^{6}=x^{2}+2x+1

{\ displaystyle x ^ {6} = x ^ {2} + 2x + 1}

etc.

The plastic number is represented as infinitely nested radicals :

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

{\ displaystyle \ rho = {\ sqrt [{3}] {1 + {\ sqrt [{3}] {1 + {\ sqrt [{3}] {1 + {\ sqrt [{3}] {\ cdots }}}}}}}}}}

.

The plastic number is the smallest Piso number .

Links

Midhat J. Gazalé. Gnomon. - Princeton University Press, 1999.
Padovan, Richard (2002), Dom Hans Van Der Laan And The Plastic Number , Nexus IV: Architecture and Mathematics, Kim Williams Books, pp. 181-193.
Shannon, AG; Anderson, PG; Horadam, AF Properties of Cordonnier, Perrin and Van der Laan numbers (neopr.) // International Journal of Mathematical Education in Science and Technology. - 2006. - T. 37 , No. 7 . - S. 825-831 . - DOI : 10.1080 / 00207390600712554 .
Ian Stewart , Tales of a Neglected Number
Piezas, Tito III; van Lamoen, Floor; Weisstein, Eric W. Plastic Constant on Wolfram MathWorld .

Plastic number

Properties

Links

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