Plastic ratio

Template:Short description Template:Infobox non-integer number In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation Template:Math Its decimal expansion begins with Template:Math Template:OEIS. It is the smallest Pisot number.

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

ρ = Template:Sfrac = Template:Sfrac = Template:Sfrac. For b = 1 the boxes have volumes Template:Math

Three quantities Template:Math are in the plastic ratio if $$\frac{b}{c}=\frac{a}{b}=\frac{b+c}{a}$$ This ratio is commonly denoted Template:Tmath (rho).

Substituting $b=\rho c$ and $a=\rho b={\rho }^{2}c$ in the last fraction, $$\rho =\frac{c(\rho +1)}{{\rho }^{2}c}.$$ It follows that the plastic ratio is the unique real solution of the cubic equation ${\rho }^3-\rho -1=0.$

Solving with Cardano's formula, $$\begin{array}{cc}{w}_{1,2}& =\frac{1}{2}(1\pm \frac{1}{3}\sqrt{\frac{23}{3}})\\ \rho & =\sqrt[3]{w_1}+\sqrt[3]{w_2}\end{array}$$ or, using the hyperbolic cosine, $$\rho =\frac{2}{\sqrt{3}}\cosh (\frac{1}{3}\mathrm{arcosh}\left(\frac{3\sqrt{3}}{2}\right)).$$

Newton's method for Template:Math Template:Nowrap and its complex conjugates at the nuclei of their basins of attraction. Julia set of the Newton map in orange, with unit circle and real curve for reference.

Template:Tmath is the superstable fixed point of the iteration $x\leftarrow (2x^3+1)/(3x^2-1),$ which is the update step of Newton's method applied to Template:Tmath

The iteration $x\leftarrow \sqrt{1+\frac{1}{x}}$ results in the continued reciprocal square root $$\rho =\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\ddots }}}}}}$$

Dividing the defining trinomial $x^3-x-1$ by Template:Tmath one obtains $x^2+\rho x+1/\rho ,$ and the conjugate elements of Template:Tmath are $$x_{1,2}=\frac{1}{2}(-\rho \pm i\sqrt{3{\rho }^2-4}),$$ with $x_1+x_2=-\rho$ and $x_1{x}_2=1/\rho .$

Properties

Rectangles in aspect ratios Template:Math (top) and Template:Math (bottom row) tile the square.

The plastic ratio Template:Tmath and golden ratio Template:Tmath are the only morphic numbers: real numbers Template:Math for which there exist natural numbers m and n such that Template:Tmath and Template:Tmath Morphic numbers can serve as basis for a system of measure.

Properties of Template:Tmath (m=3 and n=4) are related to those of Template:Tmath (m=2 and n=1). For example, The plastic ratio satisfies the continued radical $$\rho =\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{\phantom{|}1+\cdots }}},$$ while the golden ratio satisfies the analogous $$\phi =\sqrt{1+\sqrt{1+\sqrt{\phantom{|}1+\cdots }}}$$

The plastic ratio can be expressed in terms of itself as the infinite geometric series $$\begin{array}{cc}{\rho }^5& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-n}\\ {\rho }^3& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-2n}\\ {\rho }^2& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-3n}\\ \frac{{\rho }^5}{{\rho }^2+1}& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-4n}\\ \rho & =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-5n}\\ \frac{{\rho }^5}{\rho +2}& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-6n}\\ \frac{{\rho }^3}{2}& =\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-7n},\end{array}$$ in comparison to the series for the golden ratio $$\begin{array}{cc}{\phi }^2& =\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-n}\\ \phi & =\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-2n}\\ \frac{{\phi }^2}{2}& =\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-3n}.\end{array}$$ Additionally, $\sum _{n=0}^{13}{\rho }^{-n}=4,$ while $\sum _{n=0}^2{\phi }^{-n}=2.$

For every integer Template:Tmath one has $$\begin{array}{cc}{\rho }^n& ={\rho }^{n-2}+{\rho }^{n-3}\\ & ={\rho }^{n-1}+{\rho }^{n-5}\\ & ={\rho }^{n-3}+{\rho }^{n-4}+{\rho }^{n-5}\end{array}$$ 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 ${\rho }^{-5}+{\rho }^{-1}=1,\rho =1/BR(1).$

Continued fraction pattern of a few low powers $$\begin{array}{cc}{\rho }^{-1}& =[0;1,3,12,1,1,3,2,3,2,\mathrm{...}]\approx 0.7549(\frac{25}{33})\\ {\rho }^0& =[1]\\ {\rho }^1& =[1;3,12,1,1,3,2,3,2,4,\mathrm{...}]\approx 1.3247(\frac{45}{34})\\ {\rho }^2& =[1;1,3,12,1,1,3,2,3,2,\mathrm{...}]\approx 1.7549(\frac{58}{33})\\ {\rho }^3& =[2;3,12,1,1,3,2,3,2,4,\mathrm{...}]\approx 2.3247(\frac{79}{34})\\ {\rho }^4& =[3;12,1,1,3,2,3,2,4,2,\mathrm{...}]\approx 3.0796(\frac{40}{13})\\ {\rho }^5& =[4;12,1,1,3,2,3,2,4,2,\mathrm{...}]\approx 4.0796(\frac{53}{13})\mathrm{...}\\ {\rho }^7& =[7;6,3,1,1,4,1,1,2,1,1,\mathrm{...}]\approx 7.1592(\frac{93}{13})\mathrm{...}\\ {\rho }^9& =[12;1,1,3,2,3,2,4,2,141,\mathrm{...}]\approx 12.5635(\frac{88}{7})\end{array}$$

The convergents of the continued fraction expansion of the plastic ratio are good rational approximations: $$\frac{4}{3},\frac{49}{37},\frac{53}{40},\frac{102}{77},\frac{257}{194},\frac{359}{271},\frac{820}{619},\frac{2819}{2128},\frac{6458}{4875},\frac{28651}{21628},\frac{63760}{48131},\dots$$

The plastic ratio is the smallest Pisot number. By definition of these numbers, the absolute value $1/\sqrt{\rho }$ of the algebraic conjugates is smaller than 1, thus powers of Template:Tmath generate almost integers. For example: ${\rho }^{29}=\mathrm{3480.0002874...}$ Template:Tmath. After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to Template:Tmath – nearly align with the imaginary axis.

The infinite radicals Template:Efn $$\begin{array}{cc}\rho & ={}_{}{}^{\frac{1}{3}}(1+{}_{}{}^{\frac{1}{3}}(1+{}_{}{}^{\frac{1}{3}}(1+\cdots ))),\\ 1/\rho & ={}_{}{}^{-\frac{1}{2}}(1+{}_{}{}^{-\frac{1}{2}}(1+{}_{}{}^{-\frac{1}{2}}(1+\cdots )))\end{array}$$ are analogous to $$\begin{array}{cc}\xi & ={}_{}{}^{\frac{1}{4}}(1+{}_{}{}^{\frac{1}{4}}(1+{}_{}{}^{\frac{1}{4}}(1+\cdots ))),\\ 1/\xi & ={}_{}{}^{-\frac{1}{3}}(1+{}_{}{}^{-\frac{1}{3}}(1+{}_{}{}^{-\frac{1}{3}}(1+\cdots ))),\end{array}$$ with Perron number Template:Tmath the positive real root of $x^4=x+1.$ The other real root is the negative reciprocal of the second Pisot number Template:Tmath.

The three zeros of the cubic $z^3+z^2-\rho z-(\rho +1)$ lie on a circle with radius Template:Tmath. Similarly, the zeros of $z^3+\frac{1}{\xi }{z}^2-z-(\frac{1}{\xi }+1)$ lie on a circle with radius Template:Tmath. In both cases, the complex conjugate zeros have approximate argument Template:Tmath.

The minimal polynomial of the plastic ratio $m(x)=x^3-x-1$ has discriminant $\mathrm{\Delta }=-23.$ The Hilbert class field of imaginary quadratic field $K=\mathbb{Q}(\sqrt{\mathrm{\Delta }})$ can be formed by adjoining Template:Tmath. With argument $\tau =(1+\sqrt{\mathrm{\Delta }})/2$ a generator for the ring of integers of Template:Tmath, 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 G_n Template:Efn $$\rho =\frac{𝔣(\sqrt{\mathrm{\Delta }})}{\sqrt{2}}=\frac{G_{23}}{\sqrt[4]{2}}.$$

Properties of the related Klein j-invariant Template:Tmath result in near identity $e^{\pi \sqrt{-\mathrm{\Delta }}}\approx {\left(\sqrt{2}\rho \right)}^{24}-24.$ The difference is Template:Math.

The elliptic integral singular value  $k_r={\lambda }^{\ast }(r)$ for Template:Tmath has closed form expression $${\lambda }^{\ast }(23)=\sin (\arcsin ((\sqrt[4]{2}\rho {)}^{-12})/2)$$ (which is less than 1/3 the eccentricity of the orbit of Venus).

Van der Laan sequence

A fan of plastic Rauzy tiles with areas in ratio Ⴔ. The fractal boundary has box-counting 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 Template:Math and Template:Math, spanning a single order of size. Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio Template:Math 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,... Template:OEIS. The limit ratio between consecutive terms is the plastic ratio:$\underset{n\to \mathrm{\infty }}{lim}{V}_{n+1}/V_n=\rho .$

The 1924 Cordonnier cut. With Template:Math, the harmonic mean of Template:Math is Template:Math

Table of the eight Van der Laan measures

k	n − m	Template:Tmath	errTemplate:Tmath	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 Template:Tmath is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 Template:OEIS.Template:Efn The last number has 154 decimal digits.

The sequence can be extended to negative indices using $$V_n=V_{n+3}-V_{n+1}.$$

Powers of the plastic ratio can be written with Van der Laan numbers as quadratic coefficients $${\rho }^n={\rho }^2{V}_{n-2}+\rho V_{n-1}+V_{n-3},$$ which is proved by mathematical induction on Template:Tmath This relation also holds for Template:Tmath The order of the coefficients corresponds to the bottom row of matrix Template:Tmath below.

The generating function of the Van der Laan sequence is given by $$\frac{1}{1-x^2-x^3}=\sum _{n=0}^{\mathrm{\infty }}{V}_n{x}^n\text{ for }x<\frac{1}{\rho }$$

The sequence is related to sums of binomial coefficients by $$V_n=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{k}{n-2k})$$

The characteristic equation of the recurrence is $x^3-x-1=0.$ If the three solutions are real root Template:Tmath and conjugate pair Template:Tmath and Template:Tmath, the Van der Laan numbers are given by the Binet formula $$V_{n-1}=a{\rho }^n+b{\beta }^n+c{\gamma }^n,$$ with real Template:Tmath and conjugates Template:Tmath and Template:Tmath the roots of $23x^3+x-1=0.$

Since $|b{\beta }^n+c{\gamma }^n|<{\rho }^{-n/2}$, the number Template:Tmath is the nearest integer to Template:Tmath, with Template:Tmath and coefficient $a={\rho }^2/({\rho }^3+\rho +2)=$ Template:Gaps

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,... Template:OEIS.

This Perrin sequence has the Fermat property: if p is prime, $P_p\equiv P_{1}modp.$ 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 Template:Math to pass the test are n = 521Template:Sup, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.

A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios Template:Math

The Van der Laan numbers are obtained as integral powers Template:Math of a matrix with real eigenvalue Template:Tmath $$Q=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),$$

$$Q^n=\left(\begin{array}{ccc}{V}_n& V_{n+1}& V_{n-1}\\ V_{n-1}& V_n& V_{n-2}\\ V_{n-2}& V_{n-1}& V_{n-3}\end{array}\right)$$

The trace of Template:Tmath gives the Perrin numbers.

Alternatively, Template:Tmath can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet Template:Tmath with corresponding substitution rule $$\{\begin{array}{c}a\mapsto b\\ b\mapsto ac\\ c\mapsto a\end{array}$$ and initiator Template:Tmath. The series of words Template:Tmath produced by iterating the substitution have the property that the number of Template:Math and Template:Math 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

Partitioning the square

Three partitions of a square into similar rectangles, Template:Math.

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

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 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.

Cubic Lagrange interpolation

The relation Template:Math in a rho-squared rectangle.

The unique positive node Template:Tmath that optimizes cubic Lagrange interpolation on the interval Template:Math is equal to Template:Math The square of Template:Tmath is the single real root of polynomial $P(x)=25x^3+17x^2+2x-1$ with discriminant Template:Tmath Expressed in terms of the plastic ratio, $t=\sqrt{\rho }/({\rho }^2+1),$ which is verified by insertion into Template:Tmath

With optimal node set $T=\{-1,-t,t,1\},$ the Lebesgue function Template:Tmath evaluates to the minimal cubic Lebesgue constant ${\mathrm{\Lambda }}_3(T)=\frac{1+t^2}{1-t^2}$ at critical point $x_c={\rho }^{2}t.$Template:Efn

The constants are related through $x_c+t=\sqrt{\rho }$ and can be expressed as infinite geometric series $$\begin{array}{cc}{x}_c& =\sum _{n=0}^{\mathrm{\infty }}\sqrt{{\rho }^{-(8n+5)}}\\ t& =\sum _{n=0}^{\mathrm{\infty }}\sqrt{{\rho }^{-(8n+9)}}.\end{array}$$ Each term of the series corresponds to the diagonal length of a rectangle with edges in ratio Template:Tmath which results from the relation ${\rho }^n={\rho }^{n-1}+{\rho }^{n-5},$ with Template:Tmath odd. The diagram shows the sequences of rectangles with common shrink rate Template:Tmath converge at a single point on the diagonal of a rho-squared rectangle with length $\sqrt{\rho \phantom{/}}=\sqrt{1+{\rho }^{-4}}.$

Plastic pentagon

A triskeles of plastic pentagons.

A spiral of equilateral triangles with edges in ratio Template:Tmath tiles a plastic pentagon with four angles of 120 and one of 60 degrees. The initial triangle is positioned at the left-hand side of a parallelogram with base to side ratio Template:Tmath 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 Template:Tmath accompanied with a clockwise rotation by 60 degrees, the horizontal base is mapped onto the third edge of the triangle. The centre of rotation is on the short (falling) diagonal, dividing it in ratio Template:Tmath, the expansion rate for a 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 $k=\frac{3}{\pi }\ln (\rho ).$ For parallelogram base Template:Tmath, the length of the short diagonal is $\sqrt{{\rho }^2-\rho +1}$ with angle $\arctan (\frac{\sqrt{3}}{1-2\rho }).$ The length of the discrete spiral is ${\rho }^5=\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-n};$ the pentagon has area $\frac{\sqrt{3}}{4}{\rho }^3=\frac{\sqrt{3}}{4}\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-2n}.$

In the vector image, the construction is repeated on each side of a Template:Tmath triangle. John Rutherford Boyd discovered a related figure, build on the sides of the Template:Tmath triangle.

Plastic spiral

A plastic spiral is a logarithmic spiral that gets wider by a factor of Template:Tmath for every quarter turn. It is described by the polar equation $r(\theta )=a\exp (k\theta ),$ with initial radius Template:Tmath and parameter $k=\frac{2}{\pi }\ln (\rho ).$ If drawn on a rectangle with sides in ratio Template:Tmath, 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 Template:Tmath which are perpendicularly aligned and successively scaled by a factor Template:Tmath

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." Moseley thus gave the expansion rate $\sqrt[4]{3}\approx \rho -1/116$ for a quarter turn.Template:Efn Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.Template:Efn

Polyhedra

The plastic ratio is important in the study of the snub icosidodecadodecahedron.

History and names

Template:Math was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919. French high school student Template:Ill 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 (Template:Langx). Van der Laan initially referred to it as the fundamental ratio (Template:Langx), using the plastic number (Template:Langx) from the 1950s onward.Template:Sfn In 1944 Carl Siegel showed that Template:Math is the smallest possible Pisot–Vijayaraghavan number and suggested naming it in honour of Thue.

The 1967 St. Benedictusberg Abbey church designed by Hans van der Laan.

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, Template:Sfrac and Template:Sfrac, 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 Template:Math, one of the ratios from the family of metallic means first described by Vera W. de Spinadel. Gardner suggested referring to Template:Math 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").

See also

Solutions of equations similar to $x^3=x+1$:

• Golden ratio – the positive solution of the equation $x^2=x+1$
• Supergolden ratio – the real solution of the equation $x^3=x^2+1$

Further reading

• Template:Citation.
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External links

• Plastic rectangle and Padovan sequence at Tartapelago by Giorgio Pietrocola.
• The digital study room of Dom Hans van der Laan at The Van der Laan Archives.
• Template:CitationTemplate:Cbignore.