View source for Plastic number

{{About|the irrational number|numbers stamped on [[plastic]]s for recycling identification|Resin identification code}}

{{Expand French|Nombre plastique|date=October 2017|topic=sci}}

{| class="infobox bordered" cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
| [[Binary numeral system|Binary]]
| {{gaps|1.0101|0011|0010|0000|1011|…}}
|-
| [[Decimal]]
| {{gaps|1.32471|79572|44746|02596|…}}
|-
| [[Hexadecimal]]
| {{gaps|1.5320|B74E|CA44|ADAC|1788|…}}
|-
| [[Continued fraction]]<ref>Sequence {{OEIS2C|A072117}} in the [[OEIS]]</ref>
| {{nowrap|[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80 ...]}}<br /><small>Note that this continued fraction is neither [[Finite continued fraction|finite]] nor [[Periodic continued fraction|periodic]].<br>(Shown in [[continued fraction#Notations|linear notation]])</small>
|-
| [[Algebraic number|Algebraic form]]
| <math>\sqrt[3]{\frac{9 + \sqrt{69}}{18}} + \sqrt[3]{\frac{9 - \sqrt{69}}{18}}</math>
|}
[[File:Triangles in ratio of the plastic number in a three armed counter clockwise spiral.svg|thumb|Triangles with sides in ratio of <math>\rho</math> form a closed spiral]]
[[File:Plastic number square spiral.svg|thumb|Squares with sides in ratio of <math>\rho</math> form a closed spiral]]

In [[mathematics]], the '''plastic number''' {{math|''ρ''}} (also known as the '''plastic constant''', the '''plastic ratio''', the '''minimal Pisot number''', the '''platin number''',<ref>{{Cite journal |last=Choulet |first=Richard |date=January–February 2010 |title=Alors argent ou pas ? Euh … je serais assez platine |url=http://www.apmep.fr/IMG/pdf/AAA10014.pdf |format=PDF |journal=Le Bulletin Vert |department=Pour chercher et approfondir |publisher=Association des Professeurs de Mathématiques de l'Enseignement Public (APMEP) Paris |issue=486 |pages=89–96 |issn=0240-5709 |oclc=477016293 |archive-url=https://web.archive.org/web/20171114033114/http://www.apmep.fr/IMG/pdf/AAA10014.pdf |archive-date=2017-11-14 |access-date=2017-11-14}}</ref> '''[[Carl Ludwig Siegel|Siegel]]'s number''' or, in French, '''{{lang|fr|le nombre radiant}}''') is a [[mathematical constant]] which is the unique real solution of the [[cubic function|cubic equation]]
: <math>x^3 = x + 1.</math>

It has the exact value<ref>{{MathWorld |title=Plastic Constant |id=PlasticConstant}}</ref>
: <math>\rho = \sqrt[3]{\frac{9+\sqrt{69}}{18}}+\sqrt[3]{\frac{9-\sqrt{69}}{18}}.</math>
Its decimal expansion begins with {{gaps|1.32471|79572|44746|02596|09088|54…}}.<ref>Sequence {{OEIS2C|id=A060006}} in the [[OEIS]].</ref>

== Properties ==
===Recurrences===
The powers of the plastic number {{math|''A''(''n'') {{=}} ''ρ''<sup>''n''</sup>}} satisfy the third-order linear recurrence relation {{math|''A''(''n'') {{=}} ''A''(''n'' − 2) + ''A''(''n'' − 3)}} for {{math|''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|Cordonnier numbers]] (better known as the Padovan sequence), the [[Perrin number]]s and the [[Van der Laan numbers]], and bears relationships to these sequences akin to the relationships of the [[golden ratio]] to the second-order [[Fibonacci number|Fibonacci]] and [[Lucas number|Lucas]] numbers, akin to the relationships between the [[silver ratio]] and the [[Pell number]]s.<ref>;{{harvtxt|Shannon|Anderson|Horadam|2006}}.</ref>

The plastic number satisfies the [[nested radical]] recurrence<ref>{{MathWorld|urlname=PlasticConstant|title=Plastic Constant|author=Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W.}}</ref>

: <math>\rho = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}}.</math>

===Number theory===
Because the plastic number has the [[Minimal polynomial (field theory)|minimal polynomial]] {{math|''x''<sup>3</sup> − ''x'' − 1 {{=}} 0,}} it is also a solution of the polynomial equation {{math|''p''(''x'') {{=}} 0}} for every polynomial {{math|''p''}} that is a multiple of {{math|''x''<sup>3</sup> − ''x'' − 1,}} but not for any other polynomials with integer coefficients. Since the [[discriminant]] of its minimal polynomial is −23, its [[splitting field]] over rationals is {{math|ℚ({{sqrt|−23}}, ''ρ''}}). This field is also a [[Hilbert class field]] of {{math|ℚ({{sqrt|−23}})}}.

The plastic number is the smallest [[Pisot–Vijayaraghavan number]]. Its [[algebraic conjugate]]s are
: <math>\left(-\tfrac12 \pm \tfrac{\sqrt3}{2}i\right) \sqrt[3]{\tfrac{1}{2} + \tfrac{1}{6} \sqrt{\tfrac{23}{3}}} + \left(-\tfrac12 \mp \tfrac{\sqrt3}{2}i\right) \sqrt[3]{\tfrac{1}{2} - \tfrac{1}{6} \sqrt{\tfrac{23}{3}}} \approx -0.662359 \pm 0.56228i,</math>
of [[absolute value]]&nbsp;≈&nbsp;0.868837 {{OEIS|A191909}}. This value is also {{math|1/{{sqrt|''ρ''}}}} because the product of the three roots of the minimal polynomial is 1.

===Trigonometry===
The plastic number can be written using the [[hyperbolic cosine]] ({{math|cosh}}) and its inverse:
:<math>\rho = \tfrac{1}{c} \cosh\left(\tfrac{1}{3} \cosh^{-1} (3 c)\right), \qquad
c = \cos\left(\tfrac{2 \pi}{12}\right) = \sin\left(\tfrac{2 \pi}{6}\right) = \tfrac{\sqrt{3}}{2} \, .</math>
(See [[Cubic function#Trigonometric (and hyperbolic) method]].)

===Geometry===
[[File:Plastic square partitions.svg|thumb|upright=1.5|Three partitions of a square into similar rectangles]]
There are precisely three ways of partitioning a square into three similar rectangles:<ref>Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
</ref><ref>{{citation |last1=de Spinadel|first1=Vera W. 
|author1-link=Vera W. de Spinadel |last2=Antonia|first2=Redondo Buitrago |title=Towards van der Laan's plastic number in the plane |journal=Journal for Geometry and Graphics |volume=13 |issue=2 |year=2009 |pages=163–175 |url=http://www.heldermann-verlag.de/jgg/jgg13/j13h2spin.pdf}}.</ref>
#The trivial solution given by three congruent rectangles with aspect ratio 3:1.
#The solution in which two of the three rectangles are congruent with the third one of twice the linear dimension of the congruent pair and where the rectangles have aspect ratio 3:2.
#The solution in which the three rectangles are mutually non congruent (all of different sizes) and where they have aspect ratio ''ρ''<sup>2</sup>. The ratios of the linear sizes of the three rectangles are: ''ρ'' (large:medium); ''ρ''<sup>2</sup> (medium:small); and ''ρ''<sup>3</sup> (large:small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio ''ρ.'' The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio ''ρ''<sup>4</sup>.
The fact that a rectangle of aspect ratio ''ρ''<sup>2</sup> can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ''ρ''<sup>2</sup> related to the [[Routh–Hurwitz theorem]]: all of its conjugates have positive real part.<ref>{{citation
 | last1 = Freiling | first1 = C.
 | last2 = Rinne | first2 = D.
 | doi = 10.4310/MRL.1994.v1.n5.a3
 | issue = 5
 | journal = Mathematical Research Letters
 | mr = 1295549
 | pages = 547–558
 | title = Tiling a square with similar rectangles
 | volume = 1
 | year = 1994}}</ref><ref>{{citation
 | last1 = Laczkovich | first1 = M.
 | last2 = Szekeres | first2 = G. | author2-link = George Szekeres
 | doi = 10.1007/BF02574063
 | issue = 3-4
 | journal = [[Discrete and Computational Geometry]]
 | mr = 1318796
 | pages = 569–572
 | title = Tilings of the square with similar rectangles
 | volume = 13
 | year = 1995}}</ref>

== History ==
[[File:Interieur bovenkerk, zicht op de middenbeuk met koorbanken voor de monniken - Mamelis - 20536587 - RCE.jpg|thumb|The 1967 [[St. Benedictusberg Abbey]] church by [[Hans van der Laan]] has plastic-number proportions.]]
{{further|Mathematics and art}}

=== Name ===

Dutch architect & [[Benedictine monk]] Dom [[Hans van der Laan]] gave the name ''plastic number'' ({{lang-nl|het plastische getal}}) to this number in 1928. In 1924, four years prior to van der Laan's christening of the number's name, French engineer {{ill|Gérard Cordonnier|fr}} had already discovered the number and referred to it as ''the radiant number'' ({{lang-fr|le nombre radiant}}). Unlike the names of the [[golden ratio]] and [[silver ratio]], the word plastic was not intended by van der Laan to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.<ref>{{harvtxt|Padovan|2002}}; {{harvtxt|Shannon|Anderson|Horadam|2006}}.</ref> This, according to [[Richard Padovan]], is because the characteristic ratios of the number, 3/4 and 1/7, relate to the limits of human perception in relating one physical size to another. Van der Laan designed the 1967 [[St. Benedictusberg Abbey]] church to these plastic number proportions.<ref>{{harvtxt|Padovan|2002}}.</ref>

The plastic number is also sometimes called the '''silver number''', a name given to it by [[Midhat J. Gazalé]]<ref>{{Cite book |url=https://www.worldcat.org/oclc/40298400 |title=Gnomon: From Pharaohs to Fractals |last=Gazalé |first=Midhat J. |date=April 19, 1999 |publisher=Princeton University Press |isbn=9780691005140 |location=Princeton, N.J. |pages=135–150 |chapter=Chapter VII: The Silver Number |oclc=40298400 |author-link=Midhat J. Gazalé}}</ref> and subsequently used by [[Martin Gardner]]{{Citation needed|reason=It's in Gardner's Workout, Chapter 16, pp. 121–128, just need to add a proper ref.|date=November 2017}}, but that name is more commonly used for the [[silver ratio]] {{math|1 + {{sqrt|2}}}}, one of the ratios from the family of [[metallic means]] first described by [[Vera W. de Spinadel]] in 1998.<ref>{{Cite journal |first=Vera W. |last=de Spinadel |title=The Metallic Means and Design |pages=141–157 |url=https://www.nexusjournal.com/the-nexus-conferences/nexus-1998/119-n1998-spinadel.html |journal=Nexus II: Architecture and Mathematics |editor-first=Kim |editor-last=Williams |location=Fucecchio (Florence) |publisher=Edizioni dell'Erba |date=1998}}</ref>

[[Donald E. Knuth]] has suggested referring to <math display="inline">\rho^2</math> as "high phi", along with suggesting a special typographic mark for it.{{Citation needed|reason=Also in Gardner's Workout, Chapter 16, pp. 121–128, but better to cite the 1994 Quantum version at this point in the text, which lacks the postscript from Gardner's book.|date=November 2017}}

== See also ==
*[[Snub icosidodecadodecahedron]]
*[[Supergolden ratio]]

==Notes==
{{reflist|30em}}

==References==
{{refbegin}}
*{{citation
 | last1 = Aarts | first1 = J.
 | last2 = Fokkink | first2 = R.
 | last3 = Kruijtzer | first3 = G.
 | title = Morphic numbers
 | journal = Nieuw Arch. Wiskd.
 | series = 5
 | volume = 2 | issue = 1 | year = 2001 | pages = 56–58
 | url = http://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-056.pdf}}.
*{{citation|first=Midhat J.|last=Gazalé|title=Gnomon|year=1999|publisher=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}}.
{{refend}}

==External links==
*[https://web.archive.org/web/20120320051231/http://members.fortunecity.com/templarser/padovan.html Tales of a Neglected Number] by [[Ian Stewart (mathematician)|Ian Stewart]]
*[http://www.maecla.it/tartapelago/museo/oro/rettangoli/en%20plasticrectangle.htm  Plastic rectangle and Padovan sequence] at Tartapelago by Giorgio Pietrocola
*{{cite web |last1=Harris |first1=Ed |title=The Plastic Ratio |url=https://www.youtube.com/watch?v=PsGUEj4w9Cc&t=488s |website=youtube |publisher=[[Brady Haran]] |accessdate=15 March 2019 |format=video}}

{{Irrational number}}
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