Plastic ratio

A spiral of rectangles in the plastic ratio.
Rationality	irrational algebraic
Symbol	ρ
Representations
Decimal	1.32471795724474602596...
Algebraic form	real root ofx3 =x + 1
Continued fraction (linear)	[1;3,12,1,1,3,2,3,2,4,2,141,80,...] [1] not periodic infinite

In mathematics, theplastic ratio is a geometricalproportion, given by the unique realsolution of the equationx³ =x + 1. Its decimal expansion begins with1.324717957244746... (sequenceA060006 in theOEIS).

The adjectiveplastic does not refer tothe artificial material, but to the formative and sculptural qualities of this ratio, as inplastic arts.

Three quantitiesa > b > c > 0 are in the plastic ratio if$${\displaystyle \frac{b}{c}=\frac{a}{b}=\frac{b+c}{a}}$$This ratio is commonly denoted⁠${\displaystyle \rho .}$⁠

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

Solving withCardano's formula,or, using thehyperbolic cosine,^[2]$${\displaystyle \rho =\frac{2}{\sqrt{3}}\cosh \left(\frac{1}{3}\mathrm{arcosh}\left(\frac{3\sqrt{3}}{2}\right)\right).}$$

⁠${\displaystyle \rho }$⁠ is the superstablefixed point of the iteration${\displaystyle x\leftarrow (2x^3+1)/(3x^2-1),}$ which is the update step ofNewton's method applied to⁠${\displaystyle x^3-x-1=0.}$⁠

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

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

The plastic ratio⁠${\displaystyle \rho }$⁠ andgolden ratio⁠${\displaystyle \phi }$⁠ are the only morphic numbers: real numbersx > 1 for which there exist natural numbers m and n such that⁠${\displaystyle x+1=x^m}$⁠ and⁠${\displaystyle x-1=x^{-n}.}$⁠ Morphic numbers can serve as basis for a system of measure.^[3]

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

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

in comparison to the golden ratio identity$${\displaystyle \phi =\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-2n}\text{ and }viceversa.}$$Additionally,${\displaystyle 1+{\phi }^{-1}+{\phi }^{-2}=2,}$ while${\displaystyle \sum _{n=0}^{13}{\rho }^{-n}=4.}$

For every integer⁠${\displaystyle n}$⁠ one hasfrom 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 theBring radical. If${\displaystyle y=x^5+x}$ then${\displaystyle x=BR(y).}$ Since${\displaystyle {\rho }^{-5}+{\rho }^{-1}=1,\rho =1/BR(1).}$

ARauzy fractal associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratiosρ⁵ : ρ² : ρ : 1.

A Rauzy fractal associated with Ⴔ, the plastic ratio-squared; with areas as above.

Continued fraction pattern of a few low powers

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

The plastic ratio is the smallestPisot number.^[4] By definition of these numbers, theabsolute value${\displaystyle 1/\sqrt{\rho }}$ of the algebraic conjugates is smaller than 1, thus powers of⁠${\displaystyle \rho }$⁠ generatealmost integers.For example:${\displaystyle {\rho }^{29}=\mathrm{3480.0002874...}\approx 3480+1/3479.}$ After 29 rotation steps thephases of the inward spiraling conjugate pair – initially close to⁠${\displaystyle \pm 45\pi /58}$⁠ – nearly align with the imaginary axis.

Theminimal polynomial of the plastic ratio${\displaystyle m(x)=x^3-x-1}$ hasdiscriminant${\displaystyle \mathrm{\Delta }=-23.}$ TheHilbert class field of imaginaryquadratic field${\displaystyle K=\mathbb{Q}(\sqrt{\mathrm{\Delta }})}$ can be formed by adjoining⁠${\displaystyle \rho }$⁠. With argument${\displaystyle \tau =(1+\sqrt{\mathrm{\Delta }})/2}$ a generator for thering of integers of⁠${\displaystyle K}$⁠, one has the special value ofDedekind eta quotient ^[5]$${\displaystyle \rho =\frac{e^{\pi i/24}\eta (\tau )}{\sqrt{2}\eta (2\tau )}.}$$

Expressed in terms of theWeber-Ramanujan class invariant G_n ^[a]$${\displaystyle \rho =\frac{\mathfrak{f}(\sqrt{\mathrm{\Delta }})}{\sqrt{2}}=\frac{G_{23}}{\sqrt[4]{2}}.}$$

Properties of the relatedKlein j-invariant⁠${\displaystyle j(\tau )}$⁠ result in near identity${\displaystyle e^{\pi \sqrt{-\mathrm{\Delta }}}\approx {\left(\sqrt{2}\rho \right)}^{24}-24.}$ The difference is< 1/12659.

Theelliptic integral singular value ^[6]${\displaystyle k_r={\lambda }^{\ast }(r)}$ for⁠${\displaystyle r=23}$⁠ has closed form expression$${\displaystyle {\lambda }^{\ast }(23)=\sin (\arcsin \left((\sqrt[4]{2}\rho {)}^{-12}\right)/2)}$$(which is less than 1/3 theeccentricity of the orbit of Venus).

In his quest for perceptible clarity, the DutchBenedictine monk and architect DomHans 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 are1/4 and7/1, spanning a singleorder of size.^[7] Requiring proportional continuity, he constructed ageometric series ofeight measures (types of size) with common ratio2 / (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 thePerrin andPadovan sequences. In combinatorics, the number ofcompositions of n into parts 2 and 3 is counted by thenth Van der Laan number.

The Van der Laan sequence is defined by the third-orderrecurrence relation$${\displaystyle V_n=V_{n-2}+V_{n-3}\text{ for }n>2,}$$with initial values$${\displaystyle V_1=0,V_0=V_2=1.}$$

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,... (sequenceA182097 in theOEIS).The limit ratio between consecutive terms is the plastic ratio:${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{V}_{n+1}/V_n=\rho .}$

Table of the eight Van der Laan measures

k	n − m	⁠${\displaystyle V_n/V_m}$⁠	err⁠${\displaystyle ({\rho }^k)}$⁠	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${\displaystyle V_n}$⁠ is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 (sequenceA112882 in theOEIS).^[b] The last number has 154 decimal digits.

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

Thegenerating function of the Van der Laan sequence is given by^[8]

The sequence is related to sums ofbinomial coefficients by^[9]$${\displaystyle V_n=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{k}{n-2k})}$$

Thecharacteristic equation of the recurrence is${\displaystyle x^3-x-1=0.}$ If the three solutions are real root⁠${\displaystyle \alpha }$⁠ and conjugate pair⁠${\displaystyle \beta }$⁠ and⁠${\displaystyle \gamma }$⁠, the Van der Laan numbers can be computed with theBinet formula^[9]$${\displaystyle V_{n-1}=a{\alpha }^n+b{\beta }^n+c{\gamma }^n,}$$with real⁠${\displaystyle a}$⁠ and conjugates⁠${\displaystyle b}$⁠ and⁠${\displaystyle c}$⁠ the roots of${\displaystyle 23x^3+x-1=0.}$

Since and${\displaystyle \alpha =\rho ,}$ the number⁠${\displaystyle V_n}$⁠ is the nearest integer to${\displaystyle a{\rho }^{n+1},}$ withn > 1 and${\displaystyle a=\rho /(3{\rho }^2-1)=}$0.3106288296404670777619027...

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

The first few terms are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119,... (sequenceA001608 in theOEIS).

ThisPerrin sequence has theFermat property: if p is prime,${\displaystyle P_p\equiv P_{1}modp.}$ The converse does not hold, but the small number ofpseudoprimes${\displaystyle n\mid P_n}$ makes the sequence special.^[10] The only 7 composite numbers below10⁸ to pass the test are n = 521², 904631, 16532714, 24658561, 27422714, 27664033, 46672291.^[11]

The Van der Laan numbers are obtained as integral powersn > 2 of amatrix with realeigenvalue⁠${\displaystyle \rho }$⁠^[8]

Thetrace of⁠${\displaystyle Q^n}$⁠ gives the Perrin numbers.

Alternatively,⁠${\displaystyle Q}$⁠ can be interpreted asincidence matrix for aD0LLindenmayer system on the alphabet⁠${\displaystyle \{a,b,c\}}$⁠ with correspondingsubstitution rule$${\displaystyle \{\begin{array}{l}a\mapsto b\\ b\mapsto ac\\ c\mapsto a\end{array}}$$and initiator⁠${\displaystyle w_0=c}$⁠. The series of words⁠${\displaystyle w_n}$⁠ produced by iterating the substitution have the property that the number ofc's, b's anda's are equal to successive Van der Laan numbers. Their lengths are${\displaystyle l(w_n)=V_{n+2}.}$

Associated to this string rewriting process is a set composed of three overlappingself-similar tiles called theRauzy fractal, that visualizes thecombinatorial information contained in a multiple-generation letter sequence.^[12]

There are precisely three ways of partitioning a square into three similar rectangles:^[13][14]

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 ratioρ². The ratios of the linear sizes of the three rectangles are:ρ (large:medium);ρ² (medium:small); andρ³ (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 ratioρ. 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 ratioρ⁴.

The fact that a rectangle of aspect ratioρ² can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the numberρ² related to theRouth–Hurwitz theorem: all of its conjugates have positive real part.^[15][16]

Thecircumradius of thesnub icosidodecadodecahedron for unit edge length is ^[17]$${\displaystyle \frac{1}{2}\sqrt{\frac{2\rho -1}{\rho -1}}.}$$

The unique positive node⁠${\displaystyle t}$⁠ that optimizes cubicLagrange interpolation on the interval[−1,1] is equal to0.41779130... The square of⁠${\displaystyle t}$⁠ is the single real root of polynomial${\displaystyle P(x)=25x^3+17x^2+2x-1}$ with discriminant⁠${\displaystyle D=-{23}^3.}$⁠^[18] Expressed in terms of the plastic ratio,${\displaystyle t=\sqrt{\rho }/({\rho }^2+1),}$ which is verified byinsertion into⁠${\displaystyle P.}$⁠

With optimal node set${\displaystyle T=\{-1,-t,t,1\},}$ theLebesgue function⁠${\displaystyle {\lambda }_3(x)}$⁠ evaluates to the minimal cubic Lebesgue constant${\displaystyle {\mathrm{\Lambda }}_3(T)=\frac{1+t^2}{1-t^2}}$ atcritical point${\displaystyle x_c={\rho }^{2}t.}$^[19][c]

The constants are related through${\displaystyle x_c+t=\sqrt{\rho }}$ and can be expressed as infinitegeometric seriesEach term of the series corresponds to the diagonal length of a rectangle with edges in ratio⁠${\displaystyle {\rho }^2,}$⁠ which results from the relation${\displaystyle {\rho }^n={\rho }^{n-1}+{\rho }^{n-5},}$ with⁠${\displaystyle n}$⁠ odd. The diagram shows the sequences of rectangles with common shrink rate⁠${\displaystyle {\rho }^{-4}}$⁠ converge at a single point on the diagonal of a rho-squared rectangle with length${\displaystyle \sqrt{\rho \phantom{/}}=\sqrt{1+{\rho }^{-4}}.}$

A spiral ofequilateral triangles with edges in ratio⁠${\displaystyle \rho }$⁠ tiles aplasticpentagon with four angles of 120 and one of 60 degrees.^[21] The initial triangle is positioned at the left-hand side of aparallelogram with base to side ratio⁠${\displaystyle \rho }$⁠ 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${\displaystyle \frac{1}{\rho },}$⁠ accompanied with a clockwise rotation by 60 degrees, the horizontal base is mapped onto the third edge of the triangle. Thecentre of rotation is on the short (falling) diagonal, dividing it in ratio⁠${\displaystyle {\rho }^3}$⁠, the expansion rate for a half-turn. Iteration of the process traces an infinite, closed sequence of equilateral triangles with pentagonal boundary.

Thelogarithmic spiral through the vertices of all triangles has polar slope${\displaystyle k=\frac{3}{\pi }\ln (\rho ).}$ For parallelogram base⁠${\displaystyle \rho }$⁠, the length of the short diagonal is${\displaystyle \sqrt{{\rho }^2-\rho +1}}$ with angle${\displaystyle \arctan (\frac{\sqrt{3}}{1-2\rho }).}$ The length of the discrete spiral is${\displaystyle {\rho }^5=\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-n};}$ the pentagon has area${\displaystyle \frac{\sqrt{3}}{4}{\rho }^3=\frac{\sqrt{3}}{4}\sum _{n=0}^{\mathrm{\infty }}{\rho }^{-2n}.}$

In the vector image, the construction is repeated on each side of a⁠${\displaystyle \rho }$⁠ triangle.John Rutherford Boyd discovered a related figure, build on the sides of the⁠${\displaystyle {\rho }^{-4}}$⁠ triangle.^[22]

Two plastic spirals with different initial radii.

Chambered nautilus shell and plastic spiral.

Aplastic spiral is alogarithmic spiral that gets wider by a factor of⁠${\displaystyle \rho }$⁠ for every quarter turn. It is described by thepolar equation${\displaystyle r(\theta )=a\exp (k\theta ),}$ with initial radius⁠${\displaystyle a}$⁠ and parameter${\displaystyle k=\frac{2}{\pi }\ln (\rho ).}$ If drawn on a rectangle with sides in ratio⁠${\displaystyle \rho }$⁠, 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${\displaystyle {\rho }^2}$⁠ which are perpendicularly aligned and successively scaled by a factor⁠${\displaystyle {\rho }^{-1}.}$⁠

In 1838Henry Moseley noticed that whorls of a shell of thechambered nautilus are in geometrical progression: "It will be found that the distance of any two of its whorls measured upon a radius vector isone-third that of the next two whorls measured upon the same radius vector ... The curve is therefore a logarithmic spiral."^[23] Moseley thus gave the expansion rate${\displaystyle \sqrt[4]{3}\approx \rho -1/116}$ for a quarter turn.^[d]Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.^[e]

ρ was first studied byAxel Thue in 1912 and byG. H. Hardy in 1919.^[25] French high school studentGérard Cordonnier [fr] discovered the ratio for himself in 1924. In his correspon­dence withHans van der Laan a few years later, he called it the radiant number (French:le nombre radiant). Van der Laan initially referred to it as the fundamental ratio (Dutch:de grondverhouding), using the plastic number (Dutch:het plastische getal) from the 1950s onward.^[26] In 1944Carl Siegel showed thatρ is the smallest possiblePisot–Vijayaraghavan number and suggested naming it in honour of Thue.

Unlike the names of thegolden andsilver 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.^[27] This, according toRichard Padovan, is because the characteristic ratios of the number,⁠3/4⁠ and⁠1/7⁠, relate to the limits of human perception in relating one physical size to another. Van der Laan designed the 1967St. Benedictusberg Abbey church to these plastic number proportions.^[28]

The plastic number is also sometimes called the silver number, a name given to it byMidhat J. Gazalé ^[29] and subsequently used byMartin Gardner,^[30] but that name is more commonly used for thesilver ratio1 +√2, one of the ratios from the family ofmetallic means first described byVera W. de Spinadel. Gardner suggested referring toρ² as "high phi", andDonald 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 letterpari ("Ⴔ").

• Solutions of equations similar to${\displaystyle x^3=x+1}$:
  • Golden ratio – the positive solution of the equation${\displaystyle x^2=x+1}$
  • Supergolden ratio – the real solution of the equation${\displaystyle x^3=x^2+1}$

• 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.
• Harriss, Edmund (15 March 2019),"The Plastic Ratio"(video),youtube,Brady Haran,archived from the original on 2021-12-21, retrieved15 March 2019.

• Cubic irrational numbers
• Mathematical constants
• History of geometry
• Integer sequences
• Composition in visual art

• CS1 interwiki-linked names
• Articles with short description
• Short description is different from Wikidata
• Pages using multiple image with auto scaled images
• Articles containing French-language text
• Articles containing Dutch-language text