Plastic ratio

{{infobox non-integer number

| image=Triangles in ratio of the plastic number in a three armed counter clockwise spiral.svg

| image_caption=Triangles with sides in ratio {{math|ρ}} form a closed spiral

| 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,...]}} {{cite OEIS|A072117}}

| 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 proportion equal to {{math|{{val|1.32471795724474602596|end=...}}}};{{cite OEIS|A060006|Decimal expansion of real root of x^3 - x - 1 (the plastic constant)}} it is the unique real solution of the equation {{math|1=x{{isup|3}} = x + 1.}}

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

File:Plastic number square spiral.svg

Definition

Three quantities {{math|a > b > c > 0}} are in the plastic ratio {{tmath|\rho}} if

$\frac{b+c}{a} = \frac{a}{b} = \frac{b}{c} = \rho\,.$

Writing {{tmath|1=b=\rho c}} and {{tmath|1=a=\rho b = \rho^2c}}, the value of {{tmath|c}} cancels out and we get

$\frac{\rho + 1}{\rho^2} = \frac{\rho^2}{\rho} = \frac{\rho}{1}\,.$

It follows that the plastic ratio is the unique real solution of the cubic equation $\rho^3 -\rho -1 =0.$

Solving the equation with Cardano's formula,

$\begin{align}$

w_{1,2} &=\frac12 \left( 1 \pm \frac13 \sqrt{\frac{23}{3}} \right) \\

\rho &=\sqrt[3]{w_1} +\sqrt[3]{w_2} \end{align}

or, using the hyperbolic cosine,{{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}}

: $\rho =\frac{2}{ \sqrt{3}} \cosh \left( \frac{1}{3} \operatorname{arcosh} \left( \frac{3 \sqrt{3}}{2} \right) \right).$

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

The iteration $x \gets \sqrt{1 +\tfrac{1}{x}}$ results in the continued reciprocal square root

: $\rho =\sqrt{1 +\cfrac{1}{\sqrt{1 +\cfrac{1}{\sqrt{1 +\cfrac{1}{\ddots}}}}}}$

Dividing the defining trinomial $x^{3} -x -1$ by {{tmath|x -\rho}} one obtains $x^{2} +\rho x +1 /\rho$ , and the conjugate elements of {{tmath|\rho}} are

$x_{1,2} = \frac12 \left( -\rho \pm i \sqrt{3 \rho^2 - 4} \right),$

with $x_1 +x_2 =-\rho \;$ and $\; x_1x_2 =1 /\rho.$

Approximation as a fraction

Good approximations for the plastic ratio come from its continued fraction expansion, {{math|[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, ...]}}.{{cite oeis|A072117|2=Continued fraction expansion of smallest Pisot-Vijayaraghavan number (positive root of x^3 = x + 1 )}} The first few are:

:{{math|{{sfrac|4|3}}}}, {{math|{{sfrac|49|37}}}}, {{math|{{sfrac|53|40}}}}, {{math|{{sfrac|102|77}}}}, {{math|{{sfrac|359|271}}}}, {{math|{{sfrac|820|619}}}}, {{math|{{sfrac|2819|2128}}}}, {{math|{{sfrac|6458|4875}}}}, {{math|{{sfrac|28651|21628}}}}, {{math|{{sfrac|63760|48131}}}}, ....

Also see the #Van der Laan sequence below.

Properties

File:PlasticSquare_6.png

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

: $x +1 =x^{m}$ and $x -1 =x^{-n}$ .{{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}}

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 continued radical

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

while the golden ratio satisfies the analogous

: $\varphi =\sqrt{1 +\sqrt{1 +\sqrt{1 +\cdots}}}$

The plastic ratio can be expressed in terms of itself as the infinite geometric series

: $\rho = \sum_{n=0}^{\infty} \rho^{-5n}$ and $\,\rho^2 = \sum_{n=0}^{\infty} \rho^{-3n},$

in comparison to the golden ratio identity

: $\varphi = \sum_{n=0}^{\infty} \varphi^{-2n}$ and vice versa.

Additionally, $1 +\varphi^{-1} +\varphi^{-2} =2$ , while $\sum_{n=0}^{13} \rho^{-n} =4.$

For every integer {{tmath|n}} one has

$\begin{align}$

\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}

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 $y =x^{5} +x$ then $x = BR(y)$ . Since $\rho^{-5} +\rho^{-1} =1, \quad \rho =1 /BR(1).$

class="floatright"

File:Plastic_Rauzy_cub.png associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratios {{math|ρ{{sup|5}} : ρ{{sup|2}} : ρ : 1.}}]]

File:Plastic_Rauzy_sqr.png

Continued fraction pattern of a few low powers

$\begin{align}$

\rho^{-1} &= [0;1,3,12,1,1,3,2,3,2,...] \approx 0.7549 \;(25/33) \\

\rho^0 &= [1] \\

\rho^1 &= [1;3,12,1,1,3,2,3,2,4,...] \approx 1.3247 \;(45/34) \\

\rho^2 &= [1;1,3,12,1,1,3,2,3,2,...] \approx 1.7549 \;(58/33) \\

\rho^3 &= [2;3,12,1,1,3,2,3,2,4,...] \approx 2.3247 \;(79/34) \\

\rho^4 &= [3;12,1,1,3,2,3,2,4,2,...] \approx 3.0796 \;(40/13) \\

\rho^5 &= [4;12,1,1,3,2,3,2,4,2,...] \approx 4.0796 \;(53/13)\,... \\

\rho^7 &= [7;6,3,1,1,4,1,1,2,1,1,...] \approx 7.1592 \;(93/13)\,... \\

\rho^9 &= [12;1,1,3,2,3,2,4,2,141,...] \approx 12.5635 \;(88/7)

\end{align}

The plastic ratio is the smallest Pisot number.{{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}} Because the absolute value $1 /\sqrt{\rho}$ of the algebraic conjugates is smaller than 1, powers of {{tmath|\rho}} generate almost integers. For example: $\rho^{29} =3480.0002874... \approx 3480 +1/3479.$ After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to {{tmath| \pm 45 \pi/58}} – nearly align with the imaginary axis.

The minimal polynomial of the plastic ratio $m(x) = x^{3}-x-1$ has discriminant $\Delta=-23$ . The Hilbert class field of imaginary quadratic field $K = \mathbb{Q}( \sqrt{\Delta})$ can be formed by adjoining {{tmath|\rho}}. With argument $\tau=(1 +\sqrt{\Delta})/2\,$ a generator for the ring of integers of {{tmath|K}}, one has the special value of Dedekind eta quotient

: $\rho = \frac{ e^{\pi i/24}\,\eta(\tau)}{ \sqrt{2}\,\eta(2\tau)}$ .{{MathWorld |id=PlasticConstant |title=Plastic constant}}

Expressed in terms of the Weber-Ramanujan class invariant G_n

: $\rho = \frac{ \mathfrak{f} ( \sqrt{ \Delta} ) }{ \sqrt{2} } = \frac{ G_{23} }{ \sqrt[4]{2} }.$ {{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 Klein j-invariant {{tmath|j(\tau)}} result in near identity $e^{\pi \sqrt{- \Delta}} \approx \left( \sqrt{2}\,\rho \right)^{24} - 24$ . The difference is {{math|< 1/12659}}.

The elliptic integral singular value{{MathWorld |id=EllipticIntegralSingularValue |title=Elliptic integral singular value}} $k_{r} =\lambda^{*}(r)$ for {{tmath|r{{=}}23}} has closed form expression

: $\lambda^{*}(23) =\sin ( \arcsin \left( ( \sqrt[4]{2}\,\rho)^{-12} \right) /2)$

(which is less than 1/3 the eccentricity of the orbit of Venus).

Van der Laan sequence

File:Plastic5_Rauzy_sqr.png 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.{{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}} Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio {{math|2 / (3/4 + 1/7^1/7) ≈ 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 and Padovan sequences. In combinatorics, the number of compositions of n into parts 2 and 3 is counted by the nth Van der Laan number.

The Van der Laan sequence is defined by the third-order recurrence relation

$V_{n} =V_{n-2} +V_{n-3} \text{ for } n > 2,$

with initial values

$V_{1} =0, V_{0} =V_{2} =1.$

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: {{tmath|1=\lim_{n\rightarrow\infty} V_{n+1} / V_n = \rho}}.

File:Nombre plastique2.svg . 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

$V_{n} =V_{n+3} -V_{n+1}.$

The generating function of the Van der Laan sequence is given by

: $\frac{1}{1 -x^{2} -x^{3}} = \sum_{n=0}^{\infty} V_{n}x^{n} \text{ for } x <1 /\rho \;.$ {{OEIS|A182097}}

The sequence is related to sums of binomial coefficients by

: $V_{n} = \sum_{k =\lfloor (n +2)/3 \rfloor}^{\lfloor n /2 \rfloor}{k \choose n -2k}$ .{{OEIS|A000931}}

The characteristic equation of the recurrence is $x^{3} -x -1=0$ . 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 Binet formula

: $V_{n-1} =a \alpha^{n} +b \beta^{n} +c \gamma^{n}$ , with real {{tmath|a}} and conjugates {{tmath|b}} and {{tmath|c}} the roots of $23x^{3} +x -1 = 0$ .

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

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

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 property: if p is prime, $P_{p} \equiv P_{1} \bmod p$ . The converse does not hold, but the small number of pseudoprimes $\,n \mid P_{n}$ makes the sequence special.{{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 }} The only 7 composite numbers below {{math|10{{sup|8}} }} to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.{{OEIS|A013998}}

File:Plastic_Rauzy_ac.png

The Van der Laan numbers are obtained as integral powers {{math|n > 2}} of a matrix with real eigenvalue {{tmath|\rho}}

$Q = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} ,$

$Q^{n} = \begin{pmatrix} V_{n} & V_{n+1} & V_{n-1} \\ V_{n-1} & V_{n} & V_{n-2} \\ V_{n-2} & V_{n-1} & V_{n-3} \end{pmatrix}$

The trace of {{tmath|Q^{n} }} gives the Perrin numbers.

Alternatively, {{tmath|Q}} can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet {{tmath|\{a,b,c\} }} with corresponding substitution rule

$\begin{cases}$

a \;\mapsto \;b \\

b \;\mapsto \;ac \\

c \;\mapsto \;a \end{cases}

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 $l(w_n) =V_{n+2}.$

Associated to this string rewriting process is a set composed of three overlapping self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation letter sequence.{{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}}

Geometry

=Partitioning the square=

File:Plastic square partitions.svg

There are precisely three ways of partitioning a square into three similar rectangles:{{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 }}{{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}}

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 ρ². The ratios of the linear sizes of the three rectangles are: ρ (large:medium); ρ² (medium:small); and ρ³ (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 ρ⁴.

The fact that a rectangle of aspect ratio ρ² can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ² related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.{{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

}}{{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

}}

The circumradius of the snub icosidodecadodecahedron for unit edge length is

: $\frac{1}{2} \sqrt{ \frac{2 \rho -1}{\rho -1}}$ .{{MathWorld |id=SnubIcosidodecadodecahedron |title=Snub icosidodecadodecahedron}}

=Cubic Lagrange interpolation=

File:Rho-Lagrange.svg

The unique positive node {{tmath|t}} that optimizes cubic 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 $P(x) =25x^3 +17x^2 +2x -1$ with discriminant {{tmath|1=D =-23^3.}}{{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}} Expressed in terms of the plastic ratio, $t =\sqrt{\rho} /(\rho^2 +1),$ which is verified by insertion into {{tmath|P.}}

With optimal node set $T =\{-1,-t, t, 1\},$ the Lebesgue function {{tmath|\lambda_3(x)}} evaluates to the minimal cubic Lebesgue constant $\Lambda_3(T) = \frac{1 +t^2}{1 -t^2}\,$ at critical point $x_c =\rho^{2} t.$ {{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}}{{efn|The square of {{math|x{{sub|c}} }} is the single real root of polynomial {{math|1=R(x) = 25x{{sup|3}} − 23x{{sup|2}} + 7x − 1}} with discriminant {{math|1=D = −2{{sup|6}} 23.}} The equality {{math|1=x{{sub|c}} = ρ{{sup|2}}t}} is verified by insertion into {{math|R}}.}}

The constants are related through $x_c +t =\sqrt{\rho}$ and can be expressed as infinite geometric series

$\begin{align}$

x_c &=\sum_{n=0}^{\infty} \sqrt{\rho^{-(8n +5)}} \\

t &=\sum_{n=0}^{\infty} \sqrt{\rho^{-(8n +9)}}.\end{align}

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 $\rho^{n} =\rho^{n-1} +\rho^{n-5},$ 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 $\sqrt{\rho \vphantom{/}} =\sqrt{1 +\rho^{-4}}.$

=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 $r( \theta) =a \exp(k \theta),$ with initial radius {{tmath|a}} and parameter $k =\frac{2\ln( \rho)}{ \pi}.$ 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 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."{{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 }} Moseley thus gave the expansion rate $\sqrt[4]{3} \approx \rho -1/116$ 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".{{MathWorld |id=OmegaConstant |title=Omega constant}} The interval {{math|3{{sup|1/4}} < ρ < Ω{{sup|−1/2}} }} is smaller than {{math|0.012.}} }}

History and names

{{math|ρ}} was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919. French high school student {{ill|Gérard Cordonnier|fr|Gérard Cordonnier}} discovered the ratio for himself in 1924. In his correspondence 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 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

Unlike the names of the golden and silver ratios, 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.{{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 }} 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.{{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}}.

The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé{{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}} and subsequently used by Martin Gardner,{{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.) but that name is more commonly used for the silver ratio {{math|1 + {{radic|2}}}}, one of the ratios from the family of metallic means 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 ("Ⴔ").

Notes

References

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}}.

Category:Cubic irrational numbers

Category:Mathematical constants

Category:History of geometry

Category:Integer sequences

Category:Composition in visual art