Plastic ratio

In mathematics, the plastic ratio is a geometrical proportion close to $ρ$. Its true value is the real solution of the equation $x3 = x + 1$

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



Definition
Three quantities $53/40$ are in the plastic ratio if
 * $$ \frac{a}{b} = \frac{b+c}{a} = \frac{b}{c} $$.

The ratio $$ \frac{a}{b} $$ is commonly denoted $\rho.$

Let $$ and $\,b=1$, then

$$ \rho^{2} =1 +c \,\land \,\rho =1 /c $$

$$ \implies\rho^{2} -1 =\rho^{-1} $$.

It follows that the plastic ratio is found as the unique real solution of the cubic equation $$\rho^{3} -\rho -1 =0.$$ The decimal expansion of the root begins as $$1.324\,717\,957\,244\,746...$$.

Solving the equation with Cardano's formula,
 * $$ w_{1,2} = \left( 1 \pm \frac{1}{3} \sqrt{\frac{23}{3}} \right) /2 $$
 * $$ \rho =\sqrt[3]{w_1} +\sqrt[3]{w_2} $$

or, using the hyperbolic cosine,
 * $$\rho =\frac{2}{ \sqrt{3}} \cosh \left( \frac{1}{3} \operatorname{arcosh} \left( \frac{3 \sqrt{3}}{2} \right) \right).$$

$\rho$ is the superstable fixed point of the iteration $$ x \gets (2x^{3}+1) /(3x^{2}-1) $$.

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 $x -\rho$ one obtains $$ x^{2} +\rho x +1 /\rho $$, and the conjugate elements of $\rho$ are
 * $$ x_{1,2} = \left( -\rho \pm i \sqrt{3 \rho^2 - 4} \right) /2,$$

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

Properties


The plastic ratio $\rho$ and golden ratio $\varphi$ are the only morphic numbers: real numbers $x3 = x + 1.$ for which there exist natural numbers m and n such that
 * $$ x +1 =x^{m} $$ and $$ x -1 =x^{-n} $$.

Morphic numbers can serve as basis for a system of measure.

Properties of $\rho$ (m=3 and n=4) are related to those of $\varphi$ (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 $$n$$ 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}$$

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

Continued fraction pattern of a few low powers
 * $$ \rho^{-1} = [0;1,3,12,1,1,3,2,3,2,...] \approx 0.7549 $$ ($ρ$)
 * $$\ \rho^{0} = [1] $$
 * $$\ \rho^{1} = [1;3,12,1,1,3,2,3,2,4,...] \approx 1.3247 $$ ($a > b > c > 0$)
 * $$\ \rho^{2} = [1;1,3,12,1,1,3,2,3,2,...] \approx 1.7549 $$ ($ρ, ρ2, ρ3$)
 * $$\ \rho^{3} = [2;3,12,1,1,3,2,3,2,4,...] \approx 2.3247 $$ ($ρ2, ρ, ρ3$)
 * $$\ \rho^{4} = [3;12,1,1,3,2,3,2,4,2,...] \approx 3.0796 $$ ($x > 1$)
 * $$\ \rho^{5} = [4;12,1,1,3,2,3,2,4,2,...] \approx 4.0796 $$ ($ρ5 : ρ2 : ρ : 1.$) ...
 * $$\ \rho^{7} = [7;6,3,1,1,4,1,1,2,1,1,...] \approx 7.1592 $$ ($25/33$) ...
 * $$\ \rho^{9} = [12;1,1,3,2,3,2,4,2,141,...] \approx 12.5635 $$ ($45/34$)

The plastic ratio is the smallest Pisot number. Because the absolute value $$1 /\sqrt{\rho}$$ of the algebraic conjugates is smaller than 1, powers of $\rho$ 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 $\pm 45 \pi/58$ – 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 $\rho$. With argument $$ \tau=(1 +\sqrt{\Delta})/2\, $$ a generator for the ring of integers of $K$, one has the special value of Dedekind eta quotient
 * $$ \rho = \frac{ e^{\pi i/24}\,\eta(\tau)}{ \sqrt{2}\,\eta(2\tau)} $$.

Expressed in terms of the Weber-Ramanujan class invariant Gn
 * $$ \rho = \frac{ \mathfrak{f} ( \sqrt{ \Delta} ) }{ \sqrt{2} } = \frac{ G_{23} }{ \sqrt[4]{2} } $$.

Properties of the related Klein j-invariant $j(\tau)$ result in near identity $$ e^{\pi \sqrt{- \Delta}} \approx \left( \sqrt{2}\,\rho \right)^{24} - 24 $$. The difference is $58/33$.

The elliptic integral singular value $$ k_{r} =\lambda^{*}(r) $$ for $$ has closed form expression
 * $$ \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


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 $79/34$, spanning a single order of size. Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio $40/13$. 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
 * $$ V_{n} =V_{n-2} +V_{n-3} $$ for $53/13$,

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,... . The limit ratio between consecutive terms is the plastic ratio.



The first 14 indices n for which $V_{n} /V_{m}$ is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264. 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
 * $$ \frac{1}{1 -x^{2} -x^{3}} = \sum_{n=0}^{\infty} V_{n}x^{n} $$ for $$x < 1 /\rho \;.$$

The sequence is related to sums of binomial coefficients by
 * $$ V_{n} = \sum_{k =\lfloor (n +2)/3 \rfloor}^{\lfloor n /2 \rfloor}{k \choose n -2k} $$.

The characteristic equation of the recurrence is $$x^{3} -x -1=0$$. If the three solutions are real root $(\rho^{k})$ and conjugate pair $V_{n}$ and $\alpha$, 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 $\beta$ and conjugates $\gamma$ and $a$ the roots of $$ 23x^{3} +x -1 = 0 $$.

Since $$ \left\vert b \beta^{n} +c \gamma^{n} \right\vert < 1 /\sqrt{ \alpha^{n}} $$ and $$ \alpha = \rho $$, the number $b$ is the nearest integer to $$ a\,\rho^{n+1} $$, with $93/13$ and $$ a =\rho /(3 \rho^{2} -1) =$$ 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,... .

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. The only 7 composite numbers below $88/7$ to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.



The Van der Laan numbers are obtained as integral powers $< 1/12659$ of a matrix with real eigenvalue $c$
 * $$ Q = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} ,$$



The trace of $V_{n}$ gives the Perrin numbers.

Alternatively, $\rho$ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet $Q^{n}$ with corresponding substitution rule
 * $$\begin{cases}

a \;\mapsto \;b\\ b \;\mapsto \;ac\\ c \;\mapsto \;a \end{cases}$$ and initiator $Q$. The series of words $\{a,b,c\}$ produced by iterating the substitution have the property that the number of $1/4 and 7/1$ and $2 / (3/4 + 1/7^{1/7}) ≈ ρ$ are equal to successive Van der Laan numbers. Their lengths are $$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.

Geometry


There are precisely three ways of partitioning a square into three similar rectangles: 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.
 * 1) The trivial solution given by three congruent rectangles with aspect ratio 3:1.
 * 2) 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.
 * 3) 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 circumradius of the snub icosidodecadodecahedron for unit edge length is
 * $$ \frac{1}{2} \sqrt{ \frac{2 \rho -1}{\rho -1}} $$.

History and names


$$ was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919. French high school student 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 (Le nombre radiant). Van der Laan initially referred to it as The fundamental ratio (De grondverhouding), using The plastic number (Het plastische getal) from the 1950's onward. In 1944 Carl Siegel showed that ρ is the smallest possible Pisot–Vijayaraghavan number and suggested naming it in honour of Thue.

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. This, according to Richard Padovan, is because the characteristic ratios of the number, $w_n$ and $\rho$, 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.

The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé and subsequently used by Martin Gardner, but that name is more commonly used for the silver ratio $n > 2$, one of the ratios from the family of metallic means first described by Vera W. de Spinadel. Gardner suggested referring to $S1 = 3, S2 = 4, S3 = 5$ 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").