Plastic ratio

In mathematics, the plastic ratio is a geometrical proportion close to 53/40. Its true value is the real solution of the equation x³ = x + 1.

Plastic ratio

Triangles with sides in ratio ρ form a closed spiral
Rationality	irrational algebraic
Symbol	ρ
Representations
Decimal	1.3247179572447460259609088...
Algebraic form	real root of x3 = x + 1
Continued fraction (linear)	[1;3,12,1,1,3,2,3,2,4,2,141,80,...]  not periodic infinite

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

Squares with sides in ratio ρ form a closed spiral

Definition

Three quantities a > b > c > 0 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 ⁠$a=\rho \,$⁠ 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 $1.324\,717\,957\,244\,746...$ (sequence A060006 in the OEIS).

Solving the equation with Cardano's formula,

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

$\rho ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}$

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

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

with $x_{1}+x_{2}=-\rho \;$ and $\;x_{1}x_{2}=1/\rho .$

Properties

Rectangles in aspect ratios ρ, ρ², ρ³ (top) and ρ², ρ, ρ³ (bottom row) tile the square.

The plastic ratio ⁠$\rho$⁠ and golden ratio ⁠$\varphi$⁠ are the only morphic numbers: real numbers 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

${\begin{aligned}\rho ^{n}&=\rho ^{n-2}+\rho ^{n-3}\\&=\rho ^{n-1}+\rho ^{n-5}\\&=\rho ^{n-3}+\rho ^{n-4}+\rho ^{n-5}.\end{aligned}}$

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

A Rauzy fractal associated with the plastic ratio-cubed. The central tile and its three subtiles have areas in the ratios ρ5 : 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

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

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 G_n

$\rho ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{23}}{\sqrt[{4}]{2}}}$.

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 < 1/12659.

The elliptic integral singular value $k_{r}=\lambda ^{*}(r)$ for ⁠$r=23$⁠ 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

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 1/4 and 7/1, spanning a single order of size. Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio 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

$V_{n}=V_{n-2}+V_{n-3}$ 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,... (sequence A182097 in the OEIS). The limit ratio between consecutive terms is the plastic ratio.

The 1924 Cordonnier cut. With S₁ = 3, S₂ = 4, S₃ = 5, the harmonic mean of ⁠S₂/S₁ ⁠, ⁠S₁ + S₂/S₃ ⁠ and ⁠S₃/S₂ ⁠ is 3 / (⁠3/4⁠ + ⁠5/7⁠ + ⁠4/5⁠ ) ≈ ρ + 1/4922.

Table of the eight Van der Laan measures

k	n - m	⁠$V_{n}/V_{m}$⁠	err⁠$(\rho ^{k})$⁠	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 ⁠$V_{n}$⁠ is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 (sequence A112882 in the OEIS). 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 ⁠$\alpha$⁠ and conjugate pair ⁠$\beta$⁠ and ⁠$\gamma$⁠, 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 ⁠$a$⁠ and conjugates ⁠$b$⁠ and ⁠$c$⁠ 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 ⁠$V_{n}$⁠ is the nearest integer to $a\,\rho ^{n+1}$, with n > 1 and $a=\rho /(3\rho ^{2}-1)=$ 0.3106288296404670777619027...

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,... (sequence A001608 in the OEIS).

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 10⁸ to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.

A plastic Rauzy fractal: the combined surface and the three separate tiles have areas in the ratios ρ⁵ : ρ² : ρ : 1.

The Van der Laan numbers are obtained as integral powers n > 2 of a matrix with real eigenvalue ⁠$\rho$⁠

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

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

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

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

${\begin{cases}a\;\mapsto \;b\\b\;\mapsto \;ac\\c\;\mapsto \;a\end{cases}}$

and initiator ⁠$w_{0}=c$⁠. The series of words ⁠$w_{n}$⁠ produced by iterating the substitution have the property that the number of c's, b's and 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.

Geometry

Three partitions of a square into similar rectangles, 1 = 3·⁠1/3⁠ = ⁠2/3⁠ + 2·⁠1/6⁠ = ⁠1/ρ² ⁠ + ⁠1/ρ⁴ ⁠ + ⁠1/ρ⁸ ⁠.

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.

The circumradius of the snub icosidodecadodecahedron for unit edge length is

${\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}}$.

Rho-squared rectangle

Nested rho-squared rectangles with side lengths in powers of ρ.

Given a rectangle of height 1, length ⁠$\rho ^{2}$⁠ and diagonal length ${\sqrt {\rho ^{5}}}$ (according to $1+\rho ^{4}=\rho ^{5}$). The triangles on the diagonal have altitudes $1/{\sqrt {\rho }}\,;$ each perpendicular foot divides the diagonal in ratio ⁠$\rho ^{4}$⁠.

On the left-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio ⁠$\rho :1$⁠ (according to $\rho ^{2}-1=\rho ^{-1}$). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.

The parent rho-squared rectangle and the two scaled copies along the diagonal have linear sizes in the ratios $\rho ^{3}:\rho :1.$ The areas of the rectangles opposite the diagonal are both equal to ⁠$1/\rho ^{3}$⁠, with aspect ratios ⁠$\rho ^{3}$⁠ (below) and ⁠$\rho$⁠ (above).

If the diagram is further subdivided by perpendicular lines through the feet of the altitudes, the lengths of the diagonal and its (thus far) seven distinct subsections are in ratios $\rho ^{6}:\rho ^{5}:\rho ^{4}:\rho ^{2}+1:$ $\,\rho ^{3}:\rho ^{2}:\rho :1,$ where ⁠$\rho ^{2}+1$⁠ corresponds to the span between both feet.

Nested rho-squared rectangles with diagonal lengths in ratios ⁠$1/\rho ^{2}$⁠ converge at distance $t={\sqrt {\rho }}/(\rho ^{2}+1)=0.41779130...$ from the intersection point. This is equal to the unique positive node that optimizes cubic Lagrange interpolation on the interval [−1,1]. With optimal node set T = {−1,−t, t, 1}, the Lebesgue function ⁠$\lambda _{3}(x)$⁠ evaluates to the minimal cubic Lebesgue constant ⁠$\Lambda _{3}(T)$⁠ at critical point $x_{c}=\rho ^{2}t.$ Since $t+\rho ^{2}t={\sqrt {\rho }}$, this is also the distance from the point of convergence to the upper left vertex.

Plastic spiral

Two plastic spirals with different initial radii.

Chambered nautilus shell and plastic spiral.

A plastic spiral is a logarithmic spiral that gets wider by a factor of ⁠$\rho$⁠ with every quarter turn. It can be described by the polar equation $r(\theta )=a\exp(k\theta ),$ with initial radius ⁠$a$⁠ and parameter $k={\frac {2\ln(\rho )}{\pi }}.$ If drawn on a rectangle with sides in ratio ⁠$\rho$⁠, 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 ⁠$\rho ^{2}$⁠ that are perpendicular-aligned and successively scaled by a factor ⁠$1/\rho .$⁠

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." Moseley thus finds the expansion rate ${\sqrt[{4}]{3}}\approx \rho -1/116$ for a quarter turn. Since the nautilus shell is a spatial object and the three-dimensional analogue of the ubiquitous golden ratio is the plastic ratio, this seems to be the natural candidate for measuring its specific proportions.

History and names

⁠$\rho$⁠ 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 (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. In 1944 Carl Siegel showed that ρ 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, ⁠3/4⁠ and ⁠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.

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 1 + √2, one of the ratios from the family of metallic means first described by Vera W. de Spinadel. Gardner suggested referring to ρ² 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 only positive solution of the equation $x^{2}=x+1$
  • Supergolden ratio – the only real solution of the equation $x^{3}=x^{2}+1$

Further reading

• Laan, van der, Hans (1960), Le nombre plastique, Quinze leçons sur l'ordonnance architectonique, Leiden: Brill.
• Padovan, Richard; Eck, Caroline van; Scheepmaker, H.J. (1994), Dom Hans van der Laan: Modern Primitive, Amsterdam: Architectura & Natura.
• Voet, Caroline [in Dutch] (2016), "Between Looking and Making: Unravelling Dom Hans van der Laan's Plastic Number", Architectural Histories, 4 (1), London: European Architectural History Network.

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