Plastic number

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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Plastic recycling — is the process of recovering scrap or waste plastics and reprocessing the material into useful products, sometimes completely different from their original state. For instance, this could mean melting down soft drink bottles then making model… …   Wikipedia

  • Plastic — is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products. Plastics are typically polymers of high molecular weight, and may contain other substances to… …   Wikipedia

  • Plastic surgery — Plastic surgeon Occupation Names Doctor, Medical Specialist Activity sectors Surgery Description Education required MD or MBBS or DO US[1] Plastic surgery is a …   Wikipedia

  • Plastic SCM — is a commercial SCM system developed by Codice Software. Plastic tries to focus on parallel development, branching and merging and security. Features Branching support To enable parallel development Plastic focuses on branching. The main… …   Wikipedia

  • Plastic armour — Plastic armour, called plastic protection in the United States, was a type of vehicle armour originally developed for merchant ships by the British Admiralty in 1940. The original composition was described as 50% clean granite of half inch size,… …   Wikipedia

  • plastic surgery — plastic surgeon. the branch of surgery dealing with the repair or replacement of malformed, injured, or lost organs or tissues of the body, chiefly by the transplant of living tissues. [1830 40] * * * Surgery to correct disfigurement, restore… …   Universalium

  • Plastic pants — Plastic pants, plastic panties, and diaper covers are all basically the same thing. They are devices worn over a diaper for the purpose of containing liquid or solid waste that may otherwise leak through the fabric. Today, plastic pants are… …   Wikipedia

  • Plastic Penny — was a UK 1960s pop band, formed in November 1967 before splitting up in August 1968. Their existence was notable only for their one hit single, early in 1968, with the song Everything I Am . It reached a high of number six in the UK Singles Chart …   Wikipedia

  • Plastic headlight restoration — is currently the cost effective solution for aging, cloudy, and foggy oxidized headlamps.It is the act of refinishing aged headlights that have succumbed to headlamp oxidation and other environmental hazards of travel, weather, and exposure to… …   Wikipedia

  • Number 1 (Goldfrapp song) — Number 1 Single by Goldfrapp from the album Supernature B side …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”