Plastic ratio

In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation Its decimal expansion begins with .

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

thumb|upright=1.25 |ρ = = = . For b = 1 the boxes have volumes

Three quantities are in the plastic ratio if <math display=block>\frac{b}{c} = \frac{a}{b} = \frac{b+c}{a}</math>

This ratio is commonly denoted

Substituting <math> b=\rho c \,</math> and <math> a=\rho b =\rho^2 c \,</math> in the last 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 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 hyperbolic cosine,

<math display=block>\rho =\frac{2}{ \sqrt{3 \cosh \left( \frac{1}{3} \operatorname{arcosh} \left( \frac{3 \sqrt{3{2} \right) \right).</math>

thumb|upright=1.25 |Newton's method for and the complex roots (left) 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.]]

is the superstable 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

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

<math display=block> \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 one obtains <math> x^2 +\rho x +1 /\rho ,</math> and the conjugate elements of 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

thumb|upright=1.25 |Rectangles in aspect ratios (top) and (bottom row) tile the square.

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

Properties of (m=3 and n=4) are related to those of (m=2 and n=1). For example, The plastic ratio satisfies the continued radical

<math display=block> \rho =\sqrt[3]{1 +\sqrt[3]{1 +\sqrt[3]{1 +\cdots} ,</math>

while the golden ratio satisfies the analogous

<math display=block> \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 display=block> \begin{align}

\rho &=\sum_{n=0}^{\infty} \rho^{-5n} \\

\rho^2 &=\sum_{n=0}^{\infty} \rho^{-3n},\end{align}</math>

in comparison to the golden ratio identity

<math display=block> \varphi =\sum_{n=0}^{\infty} \varphi^{-2n} \text{ and } vice~versa.</math>

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

For every integer 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, \;\rho =1 /BR(1).</math>

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 convergents of the continued fraction expansion of the plastic ratio are good rational approximations:

<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}, \ldots</math>

The plastic ratio is the smallest Pisot number. By definition of these numbers, the absolute value <math>1 /\sqrt{\rho}</math> of the algebraic conjugates is smaller than 1, thus powers of generate almost integers.

For example: <math>\rho^{29} =3480.0002874... \approx 3480 +1/3479.</math> After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to – nearly align with the imaginary axis.

The 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 . With argument <math> \tau=(1 +\sqrt{\Delta})/2\, </math> a generator for the ring of integers of , one has the special value of Dedekind eta quotient

Expressed in terms of the Weber-Ramanujan class invariant Gn

<math display=block> \rho =\frac{ \mathfrak{f} ( \sqrt{ \Delta} ) }{ \sqrt{2} } = \frac{ G_{23} }{ \sqrt[4]{2} }.</math>

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

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

<math display=block> \lambda^{*}(23) =\sin ( \arcsin \left( ( \sqrt[4]{2}\,\rho)^{-12} \right) /2) </math>

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

Van der Laan sequence

thumb|upright=1.25 |A fan of plastic Rauzy tiles with areas in ratio Ⴔ. The fractal boundary has [[Minkowski–Bouligand dimension|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 and , spanning a single order of size. Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio 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

with initial values

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:<math> \lim_{n\rightarrow\infty} V_{n+1}/V_n =\rho.</math>

thumb|upright=1.5 |The 1924 [[#History and names|Cordonnier cut. With , the harmonic mean of is ]]

{| class="wikitable" id="types"

|+ Table of the eight Van der Laan measures

! k !! n − m !! !! err !! 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 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

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

<math display=block> \frac{1}{1 -x^2 -x^3} = \sum_{n=0}^{\infty} V_n x^n \text{ for } x <\tfrac{1}{\rho} </math>

The sequence is related to sums of binomial coefficients by

<math display=block> V_n =\sum_{k =\lfloor (n +2)/3 \rfloor}^{\lfloor n /2 \rfloor}{k \choose n -2k} </math>

The characteristic equation of the recurrence is <math>x^3 -x -1 =0.</math> If the three solutions are real root and conjugate pair and , the Van der Laan numbers can be computed with the Binet formula The only 7 composite numbers below to pass the test are n = 521, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.

thumb|upright=1.33 |A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios

The Van der Laan numbers are obtained as integral powers of a matrix with real eigenvalue

Geometry

Partitioning the square

thumb|upright=1.5 |Three partitions of a square into similar rectangles, .

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

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.

The circumradius of the snub icosidodecadodecahedron for unit edge length is 

Cubic Lagrange interpolation

thumb |upright=1.5 |The relation in a rho-squared rectangle.

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

With optimal node set <math>T =\{-1,-t, t, 1\},</math> the Lebesgue function evaluates to the minimal cubic Lebesgue constant <math>\Lambda_3(T) = \frac{1 +t^2}{1 -t^2}\,</math> at critical point <math>x_c =\rho^2 t.</math>

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 which results from the relation <math>\rho^n =\rho^{n-1} +\rho^{n-5},</math> with odd. The diagram shows the sequences of rectangles with common shrink rate 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

thumb|upright=1.3 |A [[triskelion|triskeles of plastic pentagons.]]

A spiral of equilateral triangles with edges in ratio 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 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 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 , 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 <math> k =\frac{3}{\pi} \ln( \rho).</math> For parallelogram base , 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 triangle. John Rutherford Boyd discovered a related figure, build on the sides of the triangle.

frameless |border |upright=.14 |Spiral design by Rutherford Boyd.

Plastic spiral

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

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 <math> \sqrt[4]{3} \approx \rho -1/116</math> for a quarter turn.

Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.

History and names

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

thumb|upright=1.33 |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, and , 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 , one of the ratios from the family of metallic means first described by Vera W. de Spinadel. Gardner suggested referring to 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

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.
.

Definition

Properties

Van der Laan sequence

Geometry

Partitioning the square

Cubic Lagrange interpolation

Plastic pentagon

Plastic spiral

History and names

See also

Notes

References

Further reading

External links