View source for Plastic ratio

{{short description|Number, approximately 1.3247}}
{{infobox non-integer number
| image=Plastic squares.svg
| image_caption=A spiral of squares in the plastic ratio.
| rationality=irrational algebraic
| symbol={{mvar|ρ}}
| decimal={{val|1.32471795724474602596|end=...}}
| continued_fraction_linear={{math|[1;3,12,1,1,3,2,3,2,4,2,141,80,...]}}&nbsp;<ref>{{cite OEIS|A072117}}</ref>
| continued_fraction_periodic=not periodic
| continued_fraction_finite=infinite
| algebraic=real root of {{math|1=''x''{{isup|3}} = ''x'' + 1}}
}}
In mathematics, the '''plastic ratio''' is a  geometrical [[aspect ratio|proportion]], given by the unique real [[polynomial root|solution]] of the equation {{math|1=''x''{{isup|3}} = ''x'' + 1.}} Its decimal expansion begins with {{math|{{val|1.324717957244746|end=...}} }} {{OEIS|A060006}}.<!-- PLEASE DO NOT add additional digits to the value of ρ in this equation; there is a long-standing consensus that additional digits do not add to understanding. Thank you.-->

The adjective ''plastic'' does not refer to [[Plastic|the artificial material]], but to the formative and sculptural qualities of this ratio, as in ''plastic arts''.

==Definition==
[[File:Plastic cuboid.svg |thumb|upright=1.25 |ρ = {{sfrac|a|b}} = {{sfrac|b+c|a}} = {{sfrac|b|c}}. For b = 1 the boxes have volumes {{math|1=ρ{{isup|3}} = ρ (red) + 1 (green).}}]]

Three quantities {{math|a > b > c > 0}} are in the plastic ratio if
<math display="block"> \frac{a}{b} =\frac{b+c}{a} =\frac{b}{c}</math>
The ratio {{tmath|\frac{a}{b} }} is commonly denoted {{tmath|\rho.}}

Substituting <math> b=\rho c \,</math> and <math> a=\rho b =\rho^2 c \,</math> in the middle fraction,
<math display="block"> \rho =\frac{c(\rho+1)}{\rho^2 c}.</math> It follows that the plastic ratio is the unique real solution of the [[cubic equation]] <math>\rho^3 -\rho -1 =0.</math>

Solving with [[Cubic equation#Cardano's formula|Cardano's formula]],
<math display="block"> \begin{align}
 w_{1,2} &=\frac12 \left( 1 \pm \frac13 \sqrt{\frac{23}{3}} \right) \\
 \rho &=\sqrt[3]{w_1} +\sqrt[3]{w_2} \end{align} </math>
or, using the [[Cubic equation#Trigonometric and hyperbolic solutions|hyperbolic cosine]],<ref>{{cite web|url=https://www.youtube.com/watch?v=CpBeRv-P5RM |title=What is the plastic ratio? |last=Tabrizian|first=Peyam|date=2022|website=YouTube|access-date=26 November 2023|language=en}}</ref>
:<math>\rho =\frac{2}{ \sqrt{3}} \cosh \left( \frac{1}{3} \operatorname{arcosh} \left( \frac{3 \sqrt{3}}{2} \right) \right).</math>

[[File:Plastic Newton map.svg |thumb|upright=1.25 |Newton's method for {{math|1=p(z) = z{{isup|3}} − z − 1:}} {{nowrap|{{math|ρ}} (right)}} and its complex conjugates at the nuclei of their [[Attractor#Basins_of_attraction|basins of attraction]]. [[Julia set]] of the Newton map in orange, with unit circle and real curve for reference.]]

{{tmath|\rho}} is the superstable [[Fixed-point iteration|fixed point]] of the iteration <math> x \gets (2x^{3}+1) /(3x^{2}-1) </math>, which is the update step of [[Newton's method]] applied to {{tmath|1=x^3 - x - 1 = 0}}.

The iteration <math> x \gets \sqrt{1 +\tfrac{1}{x}} </math> results in the continued reciprocal square root
:<math> \rho =\sqrt{1 +\cfrac{1}{\sqrt{1 +\cfrac{1}{\sqrt{1 +\cfrac{1}{\ddots}}}}}} </math>

Dividing the defining trinomial <math>x^{3} -x -1</math> by {{tmath|x -\rho}} one obtains <math> x^{2} +\rho x +1 /\rho </math>, and the [[Conjugate element (field theory)|conjugate elements]] of {{tmath|\rho}} are
<math display="block"> x_{1,2} = \frac12 \left( -\rho \pm i \sqrt{3 \rho^2 - 4} \right),</math>
with <math>x_1 +x_2 =-\rho \;</math> and <math>\; x_1x_2 =1 /\rho.</math>

==Properties==
[[File:PlasticSquare_6.png |thumb|upright=1.25 |Rectangles in aspect ratios {{math|ρ, ρ{{sup|2}}, ρ{{sup|3}} }} (top) and {{math|ρ{{sup|2}}, ρ, ρ{{sup|3}} }} (bottom row) tile the square.]]

The plastic ratio {{tmath|\rho}} and [[golden ratio]] {{tmath|\varphi}} are the only morphic numbers: real numbers {{math|''x'' > 1}} for which there exist natural numbers m and n such that
:<math> x +1 =x^{m} </math> and <math> x -1 =x^{-n}.</math><ref>{{cite journal|last1=Aarts|first1=Jan|last2=Fokkink|first2=Robbert|last3=Kruijtzer|first3=Godfried|date=2001|title=Morphic numbers|url=https://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-056.pdf |journal=[[Nieuw Archief voor Wiskunde]]|series=5|volume=2|issue=1|pages=56–58|access-date=26 November 2023|language=en}}</ref>
Morphic numbers can serve as basis for a system of measure.

Properties of {{tmath|\rho}} (m=3 and n=4) are related to those of {{tmath|\varphi}} (m=2 and n=1). For example, The plastic ratio satisfies the [[Nested radical#Infinitely nested radicals|continued radical]]
:<math> \rho =\sqrt[3]{1 +\sqrt[3]{1 +\sqrt[3]{1 +\cdots}}} </math>,
while the golden ratio satisfies the analogous
:<math> \varphi =\sqrt{1 +\sqrt{1 +\sqrt{1 +\cdots}}} </math>

The plastic ratio can be expressed in terms of itself as the infinite [[geometric series]]
:<math> \rho = \sum_{n=0}^{\infty} \rho^{-5n}</math> and <math> \,\rho^2 = \sum_{n=0}^{\infty} \rho^{-3n},</math>
in comparison to the golden ratio identity
:<math> \varphi = \sum_{n=0}^{\infty} \varphi^{-2n} </math> and ''vice versa''.
Additionally, <math> 1 +\varphi^{-1} +\varphi^{-2} =2 </math>, while <math> \sum_{n=0}^{13} \rho^{-n} =4.</math>

For every integer {{tmath|n}} one has
<math display="block"> \begin{align}
 \rho^{n} &=\rho^{n-2} +\rho^{n-3}\\
 &=\rho^{n-1} +\rho^{n-5}\\
 &=\rho^{n-3} +\rho^{n-4} +\rho^{n-5}
\end{align}</math>
From this an infinite number of further relations can be found.

The algebraic solution of a reduced quintic equation can be written in terms of square roots, cube roots and the [[Bring radical]]. If <math> y =x^{5} +x </math> then <math> x = BR(y) </math>. Since <math> \rho^{-5} +\rho^{-1} =1, \quad \rho =1 /BR(1).</math>

{{multiple image
| total_width=320
| direction=vertical
| image_gap=6
| image1=Plastic_Rauzy_cub.png
| caption1=A [[#matrix|Rauzy fractal]] associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratios {{math|ρ{{isup|5}} : ρ{{isup|2}} : ρ : 1.}}
| image2=Plastic_Rauzy_sqr.png
| caption2=A Rauzy fractal associated with Ⴔ, the plastic ratio-squared; with areas as above.
}}

[[Simple continued fraction|Continued fraction]] pattern of a few low powers
<math display="block"> \begin{align}
 \rho^{-1} &= [0;1,3,12,1,1,3,2,3,2,...] \approx 0.7549 \;(\tfrac{25}{33}) \\
 \rho^0 &= [1] \\
 \rho^1 &= [1;3,12,1,1,3,2,3,2,4,...] \approx 1.3247 \;(\tfrac{45}{34}) \\
 \rho^2 &= [1;1,3,12,1,1,3,2,3,2,...] \approx 1.7549 \;(\tfrac{58}{33}) \\
 \rho^3 &= [2;3,12,1,1,3,2,3,2,4,...] \approx 2.3247 \;(\tfrac{79}{34}) \\
 \rho^4 &= [3;12,1,1,3,2,3,2,4,2,...] \approx 3.0796 \;(\tfrac{40}{13}) \\
 \rho^5 &= [4;12,1,1,3,2,3,2,4,2,...] \approx 4.0796 \;(\tfrac{53}{13})\,... \\
 \rho^7 &= [7;6,3,1,1,4,1,1,2,1,1,...] \approx 7.1592 \;(\tfrac{93}{13})\,... \\
 \rho^9 &= [12;1,1,3,2,3,2,4,2,141,...] \approx 12.5635 \;(\tfrac{88}{7})
\end{align}</math>

The simplest rational approximations of {{tmath|\rho}} are:<math display="block"> \tfrac{4}{3},\tfrac{49}{37},\tfrac{53}{40},\tfrac{102}{77},\tfrac{257}{194},\tfrac{359}{271},\tfrac{820}{619},\tfrac{2819}{2128},\tfrac{6458}{4875},\tfrac{28651}{21628},\tfrac{63760}{48131},... </math>

The plastic ratio is the smallest [[Pisot number]].<ref>{{cite journal |last=Panju|first=Maysum|date=2011|title=A systematic construction of almost integers |url=https://mathreview.uwaterloo.ca/archive/voli/2/panju.pdf |journal=The Waterloo Mathematics Review|volume=1|issue=2|pages=35–43|access-date=29 November 2023|language=en}}</ref> Because the [[Absolute value#Complex numbers|absolute value]] <math>1 /\sqrt{\rho}</math> of the algebraic conjugates is smaller than 1, powers of {{tmath|\rho}} generate [[almost integer]]s. For example: <math>\rho^{29} =3480.0002874... \approx 3480 +1/3479.</math> After 29 rotation steps the [[Phase (waves)|phase]]s of the inward spiraling conjugate pair &ndash; initially close to {{tmath| \pm 45 \pi/58}} &ndash; nearly align with the imaginary axis.

The [[Minimal polynomial (field theory)|minimal polynomial]] of the plastic ratio <math> m(x) = x^{3}-x-1 </math> has [[discriminant]] <math>\Delta=-23</math>. The [[Hilbert class field]] of imaginary [[quadratic field]] <math> K = \mathbb{Q}( \sqrt{\Delta}) </math> can be formed by adjoining {{tmath|\rho}}. With argument <math> \tau=(1 +\sqrt{\Delta})/2\, </math> a generator for the [[ring of integers]] of {{tmath|K}}, one has the special value of [[Dedekind eta function|Dedekind eta]] quotient
:<math> \rho = \frac{ e^{\pi i/24}\,\eta(\tau)}{ \sqrt{2}\,\eta(2\tau)} </math>.<ref>{{MathWorld |id=PlasticConstant |title=Plastic constant}}</ref>

Expressed in terms of the [[Weber modular function|Weber-Ramanujan class invariant G<sub>n</sub>]]
:<math> \rho = \frac{ \mathfrak{f} ( \sqrt{ \Delta} ) }{ \sqrt{2} } = \frac{ G_{23} }{ \sqrt[4]{2} }.</math>{{efn|German Wikipedia has a table of analytical values of the {{ill|Ramanujan G-function|de|Ramanujansche g-Funktion und G-Funktion#Spezielle Werte}} for odd arguments below 47.}}

Properties of the related [[J-invariant#The q-expansion and moonshine|Klein j-invariant]] {{tmath|j(\tau)}} result in near identity <math> e^{\pi \sqrt{- \Delta}} \approx \left( \sqrt{2}\,\rho \right)^{24} - 24 </math>. The difference is {{math|< 1/12659}}.

The [[Modular lambda function#Lambda-star|elliptic integral singular value]]<ref>{{MathWorld |id=EllipticIntegralSingularValue |title=Elliptic integral singular value}}</ref> <math> k_{r} =\lambda^{*}(r) </math> for {{tmath|r{{=}}23}} has closed form expression
:<math> \lambda^{*}(23) =\sin ( \arcsin \left( ( \sqrt[4]{2}\,\rho)^{-12} \right) /2) </math>
(which is less than 1/3 the [[Eccentricity (mathematics)#Ellipses|eccentricity]] of the orbit of Venus).

==Van der Laan sequence==
[[File:Plastic5_Rauzy_sqr.png |thumb|upright=1.25 |A fan of plastic Rauzy tiles with areas in ratio Ⴔ. The fractal boundary has [[Minkowski–Bouligand dimension|box-counting]] [[Fractal dimension|dimension]] 1.11]]

In his quest for perceptible clarity, the Dutch [[Benedictine monk]] and architect Dom [[Hans van der Laan]] (1904-1991) asked for the minimum difference between two sizes, so that we will clearly perceive them as distinct. Also, what is the maximum ratio of two sizes, so that we can still relate them and perceive nearness. According to his observations, the answers are {{math|1/4 and 7/1}}, spanning a single ''order of size''.<ref>{{cite web|url=https://domhansvanderlaan.nl/theory-practice/theory/the-plastic-number-series-8/ |title=1:7 and a series of 8|last=Voet|first=Caroline|author-link=:nl:Caroline Voet|date=2019|website=The digital study room of Dom Hans van der Laan|publisher=Van der Laan Foundation|access-date=28 November 2023|language=en}}</ref> Requiring proportional continuity, he constructed a [[geometric series]] of [[#types|eight measures]] (''types of size'') with common ratio {{math|2 / (3/4 + 1/7<sup>1/7</sup>) ≈ ρ.}} Put in rational form, this architectonic system of measure is constructed from a subset of the numbers that bear his name.

The Van der Laan numbers have a close connection to the [[Perrin number|Perrin]] and [[Padovan sequence]]s. In combinatorics, the number of [[Composition (combinatorics)|compositions]] of n into parts 2 and 3 is counted by the ''n''th Van der Laan number.

The Van der Laan sequence is defined by the third-order [[recurrence relation]]
<math display="block"> V_{n} =V_{n-2} +V_{n-3} \text{ for } n > 2,</math>
with initial values
<math display="block"> V_{1} =0, V_{0} =V_{2} =1.</math>

The first few terms are 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86,... {{OEIS|A182097}}.
The limit ratio between consecutive terms is the plastic ratio:<math> \lim_{n\rightarrow\infty} V_{n+1}/V_n =\rho.</math>

[[File:Nombre plastique2.svg |thumb|upright=1.5 |The 1924 [[#History and names|Cordonnier cut]]. With {{math |1= S{{sub|1}} = 3, S{{sub|2}} = 4, S{{sub|3}} = 5}}, the [[harmonic mean]] of {{math |{{sfrac |S{{sub|2}} |S{{sub|1}} }}, {{sfrac |S{{sub|1}} + S{{sub|2}} |S{{sub|3}} }} and {{sfrac |S{{sub|3}} |S{{sub|2}} }} }} is {{math|3 / {{pars |s=133% |{{sfrac|3|4}} + {{sfrac|5|7}} + {{sfrac|4|5}} }} ≈ ρ + 1/4922.}}]]

{| class="wikitable" id="types"
|+ Table of the eight Van der Laan measures
! k !! n - m !! {{tmath|V_{n} /V_{m} }} !! err{{tmath|(\rho^{k}) }} !! interval
|-
| 0 || 3 - 3 || 1 /1 || 0 || minor element
|-
| 1 || 8 - 7 || 4 /3 || 1/116 || major element
|-
| 2 || 10 - 8 || 7 /4 ||-1/205 || minor piece
|-
| 3 || 10 - 7 || 7 /3 || 1/116 || major piece
|-
| 4 || 7 - 3 || 3 /1 || -1/12 || minor part
|-
| 5 || 8 - 3 || 4 /1 || -1/12 || major part
|-
| 6 || 13 - 7 || 16 /3 || -1/14 || minor whole
|-
| 7 || 10 - 3 || 7 /1 || -1/6 || major whole
|}

The first 14 indices n for which {{tmath|V_n }} is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 {{OEIS|A112882}}.{{efn |{{math|1=V{{sub|n}} = Pa{{sub| n+3}} }} }} The last number has 154 decimal digits.

The sequence can be extended to negative indices using
<math display="block"> V_{n} =V_{n+3} -V_{n+1}.</math>

The [[generating function]] of the Van der Laan sequence is given by
:<math> \frac{1}{1 -x^{2} -x^{3}} = \sum_{n=0}^{\infty} V_{n}x^{n} \text{ for } x <1 /\rho \;.</math><ref name="oeisGF">{{OEIS|A182097}}</ref>

The sequence is related to sums of [[binomial coefficient]]s by
:<math> V_{n} = \sum_{k =\lfloor (n +2)/3 \rfloor}^{\lfloor n /2 \rfloor}{k \choose n -2k} </math>.<ref name="oeispado">{{OEIS|A000931}}</ref>

The [[Characteristic equation (calculus)|characteristic equation]] of the recurrence is <math>x^{3} -x -1=0</math>. If the three solutions are real root {{tmath|\alpha}} and conjugate pair {{tmath|\beta}} and {{tmath|\gamma}}, the Van der Laan numbers can be computed with the [[Fibonacci sequence#Binet's formula|Binet formula]]  <ref name="oeispado"/>
<math display="block"> V_{n-1} =a \alpha^n +b \beta^n +c \gamma^n ,</math>
with real {{tmath|a}} and conjugates {{tmath|b}} and {{tmath|c}} the roots of <math> 23x^3 +x -1 =0.</math>

Since <math> \left\vert b \beta^{n} +c \gamma^{n} \right\vert < 1 /\alpha^{n/2} </math> and <math> \alpha =\rho </math>, the number {{tmath|V_{n} }} is the nearest integer to <math> a\,\rho^{n+1} </math>, with {{math|''n'' > 1}} and <math> a =\rho /(3 \rho^{2} -1) =</math> {{gaps|0.31062|88296|40467|07776|19027...}}

Coefficients <math> a =b =c =1 </math> result in the Binet formula for the related sequence <math> P_{n} =2V_{n} +V_{n-3} </math>.

The first few terms are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119,... {{OEIS|A001608}}.

This [[Perrin sequence]] has the [[Fermat's little theorem|Fermat property]]: if p is prime, <math> P_{p} \equiv P_{1} \bmod p </math>. The converse does not hold, but the small number of [[pseudoprime]]s <math>\,n \mid P_{n} </math> makes the sequence special.<ref>{{cite journal |last1=Adams |first1=William |last2=Shanks |first2=Daniel |author-link2=Daniel Shanks |title=Strong primality tests that are not sufficient |journal=Math. Comp. |publisher=AMS |date=1982 |volume=39 |issue=159 |pages=255–300 |doi=10.2307/2007637 |doi-access=free |jstor=2007637 }}</ref> The only 7 composite numbers below {{math|10{{sup|8}} }} to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.<ref>{{OEIS|A013998}}</ref>

[[File:Plastic_Rauzy_ac.png |thumb|upright=1.33 |A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios {{math|ρ{{sup|5}} : ρ{{sup|2}} : ρ : 1.}}]]

The Van der Laan numbers are obtained as integral powers {{math|''n'' > 2}} of a [[Matrix (mathematics)|matrix]] with real [[Eigenvalues and eigenvectors#Eigenvalues and eigenvectors of matrices|eigenvalue]] {{tmath|\rho}} <ref name="oeisGF"/>
<math display="block"> Q = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} ,</math>

<math display="block" id="matrix"> Q^{n} = \begin{pmatrix} V_{n} & V_{n+1} & V_{n-1} \\ V_{n-1} & V_{n} & V_{n-2} \\ V_{n-2} & V_{n-1} & V_{n-3} \end{pmatrix} </math>

The [[Trace (linear algebra)#Relationship to eigenvalues|trace]] of {{tmath|Q^{n} }} gives the Perrin numbers.

Alternatively, {{tmath|Q}} can be interpreted as [[incidence matrix]] for a [[Morphic word#D0L system|D0L]] [[L-system|Lindenmayer system]] on the alphabet {{tmath|\{a,b,c\} }} with corresponding  [[Semi-Thue system|substitution rule]]
<math display="block">\begin{cases}
a \;\mapsto \;b \\
b \;\mapsto \;ac \\
c \;\mapsto \;a \end{cases}</math>
and initiator {{tmath|w_0{{=}}c}}. The series of words {{tmath|w_n}} produced by iterating the substitution have the property that the number of {{math|c's, b's}} and {{math|a's}} are equal to successive Van der Laan numbers. Their lengths are <math>l(w_n) =V_{n+2}.</math>

Associated to this string rewriting process is a set composed of three overlapping [[Self-similarity|self-similar]] tiles called the [[Rauzy fractal]], that visualizes the [[Combinatorics on words|combinatorial]] information contained in a multiple-generation letter sequence.<ref>{{cite journal |last1=Siegel |first1=Anne |last2=Thuswaldner |first2=Jörg M. |date=2009 |title=Topological properties of Rauzy fractals |url=http://numdam.org/item/MSMF_2009_2_118__1_0/ |journal=Mémoires de la Société Mathématique de France |volume=118 |series=2 |pages=1–140 |doi=10.24033/msmf.430}}</ref>

==Geometry==
===Partitioning the square===
[[File:Plastic square partitions.svg |thumb|upright=1.5 |Three partitions of a square into similar rectangles, {{math|1= 1 = 3·{{sfrac|1|3}} = {{sfrac|2|3}} + 2·{{sfrac|1|6}} = {{sfrac|1|ρ{{sup|2}} }} + {{sfrac|1|ρ{{sup|4}} }} + {{sfrac|1|ρ{{sup|8}} }} }}.]]

There are precisely three ways of partitioning a square into three similar rectangles:<ref>{{cite journal |last=Stewart |first=Ian |date=1996 |title=Tales of a neglected number |url=http://members.fortunecity.com/templarser/padovan.html |journal=Scientific American |volume=274 |issue=6 |pages=102–103 |doi=10.1038/scientificamerican0696-102|bibcode=1996SciAm.274f.102S |archive-url=https://web.archive.org/web/20120320051231/http://members.fortunecity.com/templarser/padovan.html |archive-date=2012-03-20 }} Feedback in: {{cite journal |last=Stewart |first=Ian |author-link=Ian Stewart (mathematician)|date=1996 |title=A guide to computer dating |journal=Scientific American |volume=275 |issue=5 |page=118 |doi=10.1038/scientificamerican1196-116|bibcode=1996SciAm.275e.116S }}</ref><ref>{{citation |last1=Spinadel |first1=Vera W. de |author1-link=Vera W. de Spinadel |last2=Redondo Buitrago |first2=Antonia |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 and the third one has twice the side lengths of the other two, where the rectangles have aspect ratio 3:2.
#The solution in which the three rectangles are 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]]: [[:File:RhoSqr Newton map.svg |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| doi-access = free
 }}</ref><ref>{{citation
 | last1 = Laczkovich | first1 = M.
 | last2 = Szekeres | first2 = G. | author2-link = George Szekeres
 | doi = 10.1007/BF02574063
 | issue = 3–4
 | journal = [[Discrete & Computational Geometry]]
 | mr = 1318796
 | pages = 569–572
 | title = Tilings of the square with similar rectangles
 | volume = 13
 | year = 1995| doi-access = free
 }}</ref>

The [[circumradius]] of the [[snub icosidodecadodecahedron]] for unit edge length is
:<math> \frac{1}{2} \sqrt{ \frac{2 \rho -1}{\rho -1}}.</math><ref>{{MathWorld |id=SnubIcosidodecadodecahedron |title=Snub icosidodecadodecahedron}}</ref>

===Cubic Lagrange interpolation===
[[File:Rho-Lagrange.svg |thumb |upright=1.5 |The relation {{math|1={{font color |crimson |''x{{sub|c}}''}} + {{font color |green |''t''}} = ''√ρ''}} in a rho-squared rectangle.]]

The unique positive node {{tmath|t}} that optimizes cubic [[Lagrange polynomial|Lagrange interpolation]] on the interval {{math|[−1,1]}} is equal to {{math|0.41779130...}} The square of {{tmath|t}} is the single real root of polynomial <math>P(x) =25x^3 +17x^2 +2x -1 </math> with discriminant {{tmath|1=D =-23^3.}}<ref>{{cite book |last1=Rack |first1=Heinz-Joachim |chapter=An example of optimal nodes for interpolation revisited |editor-last1=Anastassiou |editor-first1=George A.|editor-last2=Duman |editor-first2=Oktay |title=Advances in applied Mathematics and Approximation Theory 2012 |series=Springer Proceedings in Mathematics and Statistics |volume=41 |year=2013 |pages=117–120 |doi=10.1007/978-1-4614-6393-1 |issn=2194-1009 |isbn=978-1-4614-6393-1}}</ref> Expressed in terms of the plastic ratio, <math>t =\sqrt{\rho} /(\rho^2 +1),</math> which is verified by [[Substitution (logic)#Algebra|insertion]] into {{tmath|P.}}

With optimal node set <math>T =\{-1,-t, t, 1\},</math> the [[Lebesgue constant|Lebesgue function]] {{tmath|\lambda_3(x)}} evaluates to the minimal cubic Lebesgue constant <math>\Lambda_3(T) = \frac{1 +t^2}{1 -t^2}\,</math> at [[Critical point (mathematics)|critical point]] <math>x_c =\rho^{2} t.</math><ref>{{cite journal |last1=Rack |first1=Heinz-Joachim |last2=Vajda |first2=Robert |year=2015 |title=Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant |url=https://www.cs.ubbcluj.ro/~studia-m//2015-2/01-Rack-Vajda-final.pdf |journal=Studia Universitatis Babeş-Bolyai Mathematica |volume=60 |issue=2 |publisher=Babeş-Bolyai University, Faculty of Mathematics and Computer Science |location=Cluj-Napoca (Romania) |pages=151–171  |issn=0252-1938 |access-date=January 9, 2025}}</ref>{{efn|The square of {{math|x{{sub|c}} }} is the single real root of polynomial {{math|1=R(''x'') = 25''x''{{sup|3}} − 23''x''{{sup|2}} + 7''x'' − 1}} with discriminant {{math|1=D = −2{{sup|6}} 23.}}<ref>{{harvtxt |Rack |2013 |p=119}}</ref> The equality {{math|1=x{{sub|c}} = ρ{{sup|2}}t}} is verified by insertion into {{math|R}}.}}

The constants are related through <math>x_c +t =\sqrt{\rho}</math> and can be expressed as infinite [[geometric series]]
<math display="block"> \begin{align}
 x_c &=\sum_{n=0}^{\infty} \sqrt{\rho^{-(8n +5)}} \\
 t &=\sum_{n=0}^{\infty} \sqrt{\rho^{-(8n +9)}}.\end{align} </math>
Each term of the series corresponds to the diagonal length of a rectangle with edges in ratio {{tmath|\rho^2,}} which results from the relation <math>\rho^{n} =\rho^{n-1} +\rho^{n-5},</math> with {{tmath|n}} odd. The diagram shows the sequences of rectangles with common shrink rate {{tmath|\rho^{-4} }} converge at a single point on the diagonal of a rho-squared rectangle with length <math>\sqrt{\rho \vphantom{/}} =\sqrt{1 +\rho^{-4}}.</math>

===Plastic pentagon===
[[File:Plastic pentagons.svg |thumb|upright=1.3 |A [[triskelion|triskeles]] of plastic pentagons.]]
A spiral of [[equilateral triangle]]s with edges in ratio {{tmath|\rho}} tiles a ''plastic [[pentagon]]'' with four angles of 120 and one of 60 degrees.<ref>{{harvtxt| Spinadel |Redondo Buitrago |2009 |p=169–172}}</ref> The initial triangle is positioned at the left-hand side of a [[parallelogram]] with base to side ratio {{tmath|\rho}} and left base angle 60 degrees, so that two edges of the triangle are collinear with sides of the parallelogram. Scaling the parallelogram in ratio {{tmath|\tfrac{1}{\rho},}} accompanied with a clockwise rotation by 60 degrees, the horizontal base is mapped onto the third edge of the triangle. The [[Rotation (mathematics)#Related definitions and terminology|centre of rotation]] is on the short (falling) diagonal, dividing it in ratio {{tmath|\rho^3}}, the expansion rate for one half-turn. Iteration of the process traces an infinite, closed sequence of equilateral triangles with pentagonal boundary.

The [[logarithmic spiral]] through the vertices of all triangles has polar slope <math> k =\frac{3}{\pi} \ln( \rho).</math> For parallelogram base {{tmath|\rho}}, the length of the short diagonal is <math> \sqrt{\rho^2 - \rho + 1}</math> with angle <math> \arctan( \tfrac{\sqrt{3}}{1 -2\rho}).</math> The length of the discrete spiral is <math> \rho^5 =\sum_{n=0}^{\infty} \rho^{-n};</math> the pentagon has area <math> \tfrac{\sqrt{3}}{4} \rho^3 =\tfrac{\sqrt{3}}{4} \sum_{n=0}^{\infty} \rho^{-2n}.</math>

In the vector image, the construction is repeated on each side of a {{tmath|\rho}} triangle. [[Rutherford Boyd|John Rutherford Boyd]] discovered a [[:File:Plastic pentagons_alt.svg |related figure]], build on the sides of the {{tmath|\rho^{-4} }} triangle.<ref>{{cite journal |last=Rutherford Boyd |first=John |title=In a laboratory of design |year=1949 |journal=Scripta Mathematica |volume=15 |publisher=Yeshiva College |location=New York, NY |pages=183–192 |issn=0036-9713 }}</ref>

===Plastic spiral===
{{multiple image
|total_width=340
|direction=vertical
|image1=Plastic_spiral.svg
|caption1=Two plastic spirals with different initial radii.
|image2=Chambered_nautilus_shell_and_plastic_spiral.svg
|caption2=Chambered nautilus shell and plastic spiral.
}}
A ''plastic spiral'' is a [[logarithmic spiral]] that gets wider by a factor of {{tmath|\rho}} for every quarter turn. It is described by the [[polar equation]] <math>r( \theta) =a \exp(k \theta),</math> with initial radius {{tmath|a}} and parameter <math> k =\frac{2}{\pi} \ln( \rho).</math> If drawn on a rectangle with sides in ratio {{tmath|\rho}}, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio {{tmath|\rho^2}} which are perpendicularly aligned and successively scaled by a factor {{tmath|1/ \rho.}}


In 1838 [[Henry_Moseley_(mathematician)|Henry Moseley]] noticed that whorls of a shell of the [[chambered nautilus]] are in geometrical progression: "It will be found that the distance of any two of its whorls measured upon a radius vector is ''one-third'' that of the next two whorls measured upon the same radius vector ... The curve is therefore a logarithmic spiral."<ref>{{cite journal |last=Moseley |first=Henry |title=On the Geometrical Forms of Turbinated and Discoid Shells  |journal=Philosophical Transactions of the Royal Society of London |year=1838 |volume=128 |pages=351–370 [355–356] |doi=10.1098/rstl.1838.0018 |doi-access=free|jstor=108202 }}</ref> Moseley thus gave the expansion rate <math> \sqrt[4]{3} \approx \rho -1/116</math> for a quarter turn.{{efn |1=For a typical 8" nautilus shell the difference in diameter between the apertures of perfect 3{{sup|1/4}} and {{math|ρ}}−sized specimens is about 1 mm. Allowing for [[phenotypic plasticity]], they may well be indistinguishable.}}
Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.{{efn |1=An alternative is the [[omega constant]] {{math|0.567143...}} which satisfies {{math|1=Ω⋅exp(Ω) = 1.}} Resembling {{math|1=φ (φ−1) = 1,}} ''Mathworld'' suggests it is like a "golden ratio for exponentials".<ref>{{MathWorld |id=OmegaConstant |title=Omega constant}}</ref> The interval {{math|3{{sup|1/4}} < ρ < Ω{{sup|−1/2}} }} is smaller than {{math|0.012.}} }}

{{br}}

==History and names==
{{math|ρ}} was first studied by [[Axel Thue]] in 1912 and by [[G. H. Hardy]] in 1919.<ref>{{harvtxt |Panju |2011 |p=37}}</ref> French high school student {{ill|Gérard Cordonnier|fr|Gérard Cordonnier}} discovered the ratio for himself in 1924. In his correspon&shy;dence with [[Hans van der Laan]] a few years later, he called it the radiant number ({{langx|fr|le nombre radiant}}). Van der Laan initially referred to it as the fundamental ratio ({{langx|nl|de grondverhouding}}), using the plastic number ({{langx|nl|het plastische getal}}) from the 1950s onward.{{sfn|Voet|2016|loc=note 12}} In 1944 [[Carl Ludwig Siegel|Carl Siegel]] showed that {{math|ρ}} is the smallest possible [[Pisot–Vijayaraghavan number]] and suggested naming it in honour of Thue.

[[File:Interieur bovenkerk, zicht op de middenbeuk met koorbanken voor de monniken - Mamelis - 20536587 - RCE.jpg |thumb|upright=1.33 |The 1967 St. Benedictusberg Abbey church designed by Hans van der Laan.]]
Unlike the names of the [[golden ratio|golden]] and [[silver ratio]]s, 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>{{cite journal |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|s2cid=119808971 }}</ref> This, according to [[Richard Padovan]], is because the characteristic ratios of the number, {{sfrac|3|4}} and {{sfrac|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>{{citation |url=https://www.nexusjournal.com/the-nexus-conferences/nexus-2002/148-n2002-padovan.html |last=Padovan |first=Richard |author-link=Richard Padovan |title=Dom Hans van der Laan and The plastic number |date=2002 |journal=Nexus IV: Architecture and Mathematics |location=Fucecchio (Florence) |publisher=Kim Williams Books |pages=181–193}}.</ref>

The plastic number is also sometimes called the silver number, a name given to it by [[Midhat J. Gazalé]]<ref>{{cite book |last=Gazalé |first=Midhat J. |author-link=Midhat J. Gazalé |date=1999 |title=Gnomon: From Pharaohs to Fractals |chapter=Chapter VII: The silver number |location=Princeton, NJ |publisher=Princeton University Press |pages=135–150}}</ref> and subsequently used by [[Martin Gardner]],<ref>{{cite book |last=Gardner |first=Martin |author-link=Martin Gardner |date=2001 |title=A Gardner's Workout |chapter=Six challenging dissection tasks |chapter-url=https://static.nsta.org/pdfs/QuantumV4N5.pdf |location=Natick, MA |publisher=A K Peters |pages=121–128}} (Link to the 1994 Quantum article without Gardner's Postscript.)</ref> but that name is more commonly used for the [[silver ratio]] {{math|1 + {{radic|2}}}}, one of the ratios from the family of [[metallic mean]]s first described by [[Vera W. de Spinadel]]. Gardner suggested referring to {{math|ρ{{sup|2}}}} as "high phi", and [[Donald Knuth]] created a special typographic mark for this name, a variant of the Greek letter phi ("φ") with its central circle raised, resembling the Georgian letter [[Pari (letter)|pari]] ("Ⴔ").

==See also==
* Solutions of equations similar to <math>x^{3}=x+1</math>:
** [[Golden ratio]] – the only positive solution of the equation <math>x^{2}=x+1</math>
** [[Supergolden ratio]] – the only real solution of the equation <math>x^{3}=x^{2}+1</math>

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Further reading==
{{Refbegin}}
*{{citation
 | last = Laan, van der | first = Hans | author-link = Hans van der Laan
 | title = Le nombre plastique, Quinze leçons sur l'ordonnance architectonique
 | publisher = Brill | publication-place = Leiden | year = 1960
 }}.
*{{citation
 | last1 = Padovan | first1 = Richard |author-link1 = Richard Padovan
 | last2 = Eck | first2 = Caroline van |author-link2 = Caroline van Eck
 | last3 = Scheepmaker | first3 = H.J.
 | title = Dom Hans van der Laan: Modern Primitive | year = 1994
 | publisher = Architectura & Natura | publication-place = Amsterdam
 }}.
*{{citation
 | last = Voet | first = Caroline | author-link =:nl:Caroline Voet
 | title = Between Looking and Making: Unravelling Dom Hans van der Laan's Plastic Number
 | journal = [[Architectural Histories]] | volume = 4 | issue = 1 | year = 2016
 | publisher = European Architectural History Network | location = London
 | url = https://journal.eahn.org/article/id/7510/
 }}.
{{Refend}}

==External links==
*[http://www.maecla.it/tartapelago/museo/oro/rettangoli/en%20plasticrectangle.htm Plastic rectangle and Padovan sequence] at Tartapelago by Giorgio Pietrocola.
*[https://domhansvanderlaan.nl/theory-practice/ The digital study room of Dom Hans van der Laan] at The Van der Laan Archives.
*{{citation |last1=Harriss |first1=Edmund |author1-link=Edmund Harriss |title=The Plastic Ratio |url=https://www.youtube.com/watch?v=PsGUEj4w9Cc&t=488s |archive-url=https://ghostarchive.org/varchive/youtube/20211221/PsGUEj4w9Cc |archive-date=2021-12-21 |url-status=live|website=youtube |date=15 March 2019 |publisher=[[Brady Haran]] |access-date=15 March 2019 |format=video}}{{cbignore}}.

{{Algebraic numbers}}

[[Category:Cubic irrational numbers]]
[[Category:Mathematical constants]]
[[Category:History of geometry]]
[[Category:Integer sequences]]
[[Category:Composition in visual art]]