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,...] [1] not periodic infinite

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.

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

${\displaystyle {\frac {a}{b}}={\frac {b+c}{a}}={\frac {b}{c}}}$.

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

Let ${\displaystyle a=\rho \,}$ and ${\displaystyle \,b=1}$, then

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

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

It follows that the plastic ratio is found as the unique real solution of the cubic equation ${\displaystyle \rho ^{3}-\rho -1=0.}$ The decimal expansion of the root begins as ${\displaystyle 1.324\,717\,957\,244\,746...}$ (sequence A060006 in the OEIS).

Solving the equation with Cardano's formula,

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

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

or, using the hyperbolic cosine,^[2]

${\displaystyle \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}$

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

The iteration ${\displaystyle x\gets {\sqrt {1+{\tfrac {1}{x}}}}}$ results in the continued reciprocal square root

${\displaystyle \rho ={\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\ddots }}}}}}}}}}}$

Dividing the defining trinomial ${\displaystyle x^{3}-x-1}$ by ${\displaystyle x-\rho }$ one obtains ${\displaystyle x^{2}+\rho x+1/\rho }$, and the conjugate elements of ${\displaystyle \rho }$ are

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

Properties

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

The plastic ratio ${\displaystyle \rho }$ and golden ratio ${\displaystyle \varphi }$ are the only morphic numbers: real numbers x > 1 for which there exist natural numbers m and n such that

${\displaystyle x+1=x^{m}}$ and ${\displaystyle x-1=x^{-n}}$.^[3]

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

Properties of ${\displaystyle \rho }$ (m=3 and n=4) are related to those of ${\displaystyle \varphi }$ (m=2 and n=1). For example, The plastic ratio satisfies the continued radical

${\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}$,

while the golden ratio satisfies the analogous

${\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}$

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

${\displaystyle \rho ^{3}=\sum _{n=0}^{\infty }\rho ^{-2n}}$ and ${\displaystyle \,\rho ^{5}=\sum _{n=0}^{\infty }\rho ^{-n}}$,

in comparison to the golden ratio identity

${\displaystyle \varphi =\sum _{n=0}^{\infty }\varphi ^{-2n}}$ or ${\displaystyle \,\varphi ^{2}=\sum _{n=0}^{\infty }\varphi ^{-n}}$.

Additionally, ${\displaystyle 1+\varphi ^{-1}+\varphi ^{-2}=2}$, while ${\displaystyle \sum _{n=0}^{13}\rho ^{-n}=4.}$

For all powers

${\displaystyle \rho ^{n}=\rho ^{n-2}+\rho ^{n-3}=\rho ^{n-1}+\rho ^{n-5}=\rho ^{n-3}+\rho ^{n-4}+\rho ^{n-5}.}$

The algebraic solution of a reduced quintic equation can be written in terms of square roots, cube roots and the Bring radical. If ${\displaystyle y=x^{5}+x}$ then ${\displaystyle x=BR(y)}$. Since ${\displaystyle \rho ^{-5}+\rho ^{-1}=1,\quad \rho =1/BR(1)}$.

Continued fraction pattern of a few low powers

${\displaystyle \rho ^{-1}=[0;1,3,12,1,1,3,2,3,2,...]\approx 0.7549}$ (25/33)

${\displaystyle \ \rho ^{0}=[1]}$

${\displaystyle \ \rho ^{1}=[1;3,12,1,1,3,2,3,2,4,...]\approx 1.3247}$ (45/34)

${\displaystyle \ \rho ^{2}=[1;1,3,12,1,1,3,2,3,2,...]\approx 1.7549}$ (58/33)

${\displaystyle \ \rho ^{3}=[2;3,12,1,1,3,2,3,2,4,...]\approx 2.3247}$ (79/34)

${\displaystyle \ \rho ^{4}=[3;12,1,1,3,2,3,2,4,2,...]\approx 3.0796}$ (40/13)

${\displaystyle \ \rho ^{5}=[4;12,1,1,3,2,3,2,4,2,...]\approx 4.0796}$ (53/13) ...

${\displaystyle \ \rho ^{7}=[7;6,3,1,1,4,1,1,2,1,1,...]\approx 7.1592}$ (93/13) ...

${\displaystyle \ \rho ^{9}=[12;1,1,3,2,3,2,4,2,141,...]\approx 12.5635}$ (88/7)

The plastic ratio is the smallest Pisot number.^[4] Because the absolute value ${\displaystyle 1/{\sqrt {\rho }}}$ of the algebraic conjugates is smaller than 1, powers of ${\displaystyle \rho }$ generate almost integers. For example: ${\displaystyle \rho ^{29}=3480.0002874...\approx 3480+1/3479}$. After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to ${\displaystyle \pm 45\pi /58}$ – nearly align with the imaginary axis.

The minimal polynomial of the plastic ratio ${\displaystyle m(x)=x^{3}-x-1}$ has discriminant ${\displaystyle \Delta =-23}$. The Hilbert class field of imaginary quadratic field ${\displaystyle K=\mathbb {Q} ({\sqrt {\Delta }})}$ can be formed by adjoining ${\displaystyle \rho }$. With argument ${\displaystyle \tau =(1+{\sqrt {\Delta }})/2\,}$ a generator for the ring of integers of ${\displaystyle K}$, one has the special value of Dedekind eta quotient

${\displaystyle \rho ={\frac {e^{\pi i/24}\,\eta (\tau )}{{\sqrt {2}}\,\eta (2\tau )}}}$.^[5]

Expressed in terms of the Weber-Ramanujan class invariant G_n

${\displaystyle \rho ={\frac {{\mathfrak {f}}({\sqrt {\Delta }})}{\sqrt {2}}}={\frac {G_{23}}{\sqrt[{4}]{2}}}}$.^[6]

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

The elliptic integral singular value^[7] ${\displaystyle k_{r}=\lambda ^{*}(r)}$ for ${\displaystyle r=23}$ has closed form expression

${\displaystyle \lambda ^{*}(23)=\sin(\arcsin \left(({\sqrt[{4}]{2}}\,\rho )^{-12}\right)/2)}$

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

Van der Laan sequence

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.

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.^[8] 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

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

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 (sequence A112882 in the OEIS).^[9] The last number has 154 decimal digits.

The sequence can be extended to negative indices using

${\displaystyle V_{n}=V_{n+3}-V_{n+1}}$.

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

${\displaystyle {\frac {1}{1-x^{2}-x^{3}}}=\sum _{n=0}^{\infty }V_{n}x^{n}}$ for ${\displaystyle x<1/\rho }$ ^[10]

The sequence is related to sums of binomial coefficients by

${\displaystyle V_{n}=\sum _{k=\lfloor (n+2)/3\rfloor }^{\lfloor n/2\rfloor }{k \choose n-2k}}$.^[11]

The characteristic 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 the Binet formula ^[11]

${\displaystyle V_{n-1}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n}}$, with real ${\displaystyle a}$ and conjugates ${\displaystyle b}$ and ${\displaystyle c}$ the roots of ${\displaystyle 23x^{3}+x-1=0}$.

Since ${\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}}$ and ${\displaystyle \alpha =\rho }$, the number ${\displaystyle V_{n}}$ is the nearest integer to ${\displaystyle a\,\rho ^{n+1}}$, with n > 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,... (sequence A001608 in the OEIS).

This Perrin sequence has the Fermat property: if p is prime, ${\displaystyle P_{p}\equiv P_{1}{\bmod {p}}}$. The converse does not hold, but the small number of pseudoprimes ${\displaystyle \,n\mid P_{n}}$ makes the sequence special.^[12] The only 7 composite numbers below 10⁸ to pass the test are n = 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.^[13]

The Van der Laan numbers are obtained as integral powers n > 2 of a matrix with real eigenvalue ${\displaystyle \rho }$ ^[10]

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

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

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

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:^[14][15]

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 length 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.^[16][17]

The circumradius of the snub icosidodecadodecahedron for unit edge length is

${\displaystyle {\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}}}$.^[18]

History and names

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

${\displaystyle \rho }$ was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919.^[4] 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 1950's onward.^[19] 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.^[20] 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.^[21]

The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé^[22] and subsequently used by Martin Gardner,^[23] 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 ${\displaystyle x^{3}=x+1}$:
  • Golden ratio – the only positive solution of the equation ${\displaystyle x^{2}=x+1}$
  • Supergolden ratio – the only real solution of the equation ${\displaystyle x^{3}=x^{2}+1}$

References

Further reading

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