| Rationality: | irrational algebraic |
| Continued Fraction Linear: | [1;3,12,1,1,3,2,3,2,4,2,141,80,...] |
| Continued Fraction Periodic: | not periodic |
| Continued Fraction Finite: | infinite |
| Algebraic: | real root of |
In mathematics, the plastic ratio is a geometrical proportion close to . Its true value is the real solution of the equation
The adjective plastic does not refer to the artificial material, but to the formative and sculptural qualities of this ratio, as in plastic arts.
Three quantities are in the plastic ratio if
| a | |
| b |
=
| b+c | |
| a |
=
| b | |
| c |
| a | |
| b |
Let and, then
\rho2=1+c\land\rho=1/c
\implies\rho2-1=\rho-1
\rho3-\rho-1=0.
1.324717957244746...
Solving the equation with Cardano's formula,
w1,2=\left(1\pm
| 1 | \sqrt{ | |
| 3 |
| 23 | |
| 3 |
\rho=\sqrt[3]{w1}+\sqrt[3]{w2}
\rho=
| 2 | |
| \sqrt{3 |
is the superstable fixed point of the iteration
x\gets(2x3+1)/(3x2-1)
The iteration
x\gets\sqrt{1+\tfrac{1}{x}}
\rho=\sqrt{1+\cfrac{1}{\sqrt{1+\cfrac{1}{\sqrt{1+\cfrac{1}{\ddots}}}}}}
Dividing the defining trinomial
x3-x-1
x2+\rhox+1/\rho
x1,2=\left(-\rho\pmi\sqrt{3\rho2-4}\right)/2,
x1+x2=-\rho
x1x2=1/\rho.
The plastic ratio and golden ratio are the only morphic numbers: real numbers for which there exist natural numbers m and n such that
x+1=xm
x-1=x-n
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
\rho=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+ … }}}
\varphi=\sqrt{1+\sqrt{1+\sqrt{1+ … }}}
The plastic ratio can be expressed in terms of itself as the infinite geometric series
\rho=
| infty | |
| \sum | |
| n=0 |
\rho-5n
\rho2=
| infty | |
| \sum | |
| n=0 |
\rho-3n,
\varphi=
| infty | |
| \sum | |
| n=0 |
\varphi-2n
1+\varphi-1+\varphi-2=2
| 13 | |
| \sum | |
| n=0 |
\rho-n=4.
For every integer
n
\begin{align} \rhon&=\rhon-2+\rhon-3\\ &=\rhon-1+\rhon-5\\ &=\rhon-3+\rhon-4+\rhon-5.\end{align}
The algebraic solution of a reduced quintic equation can be written in terms of square roots, cube roots and the Bring radical. If
y=x5+x
x=BR(y)
\rho-5+\rho-1=1, \rho=1/BR(1).
Continued fraction pattern of a few low powers
\rho-1=[0;1,3,12,1,1,3,2,3,2,...] ≈ 0.7549
\rho0=[1]
\rho1=[1;3,12,1,1,3,2,3,2,4,...] ≈ 1.3247
\rho2=[1;1,3,12,1,1,3,2,3,2,...] ≈ 1.7549
\rho3=[2;3,12,1,1,3,2,3,2,4,...] ≈ 2.3247
\rho4=[3;12,1,1,3,2,3,2,4,2,...] ≈ 3.0796
\rho5=[4;12,1,1,3,2,3,2,4,2,...] ≈ 4.0796
\rho7=[7;6,3,1,1,4,1,1,2,1,1,...] ≈ 7.1592
\rho9=[12;1,1,3,2,3,2,4,2,141,...] ≈ 12.5635
1/\sqrt{\rho}
\rho29=3480.0002874... ≈ 3480+1/3479.
The minimal polynomial of the plastic ratio
m(x)=x3-x-1
\Delta=-23
K=Q(\sqrt{\Delta})
\tau=(1+\sqrt{\Delta})/2
\rho=
| e\piη(\tau) | |
| \sqrt{2 |
η(2\tau)}
Expressed in terms of the Weber-Ramanujan class invariant Gn
\rho=
| ak{f | |
| ( |
\sqrt{\Delta})}{\sqrt{2}}=
| G23 | |
| \sqrt[4]{2 |
}
Properties of the related Klein j-invariant result in near identity
e\pi
kr=λ*(r)
λ*(23)=\sin(\arcsin\left((\sqrt[4]{2}\rho)-12\right)/2)
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, spanning a single order of size.[5] 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
Vn=Vn-2+Vn-3
V1=0,V0=V2=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,... .The limit ratio between consecutive terms is the plastic ratio.
| 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
Vn=Vn+3-Vn+1
The generating function of the Van der Laan sequence is given by
| 1 | |
| 1-x2-x3 |
=
| infty | |
| \sum | |
| n=0 |
Vnxn
x<1/\rho .
The sequence is related to sums of binomial coefficients by
Vn=
| \lfloorn/2\rfloor | |
| \sum | |
| k=\lfloor(n+2)/3\rfloor |
{k\choosen-2k}
The characteristic equation of the recurrence is
Vn-1=a\alphan+b\betan+c\gamman
23x3+x-1=0
\left\vertb\betan+c\gamman\right\vert<1/\sqrt{\alphan
\alpha=\rho
a\rhon+1
a=\rho/(3\rho2-1)=
Coefficients
a=b=c=1
Pn=2Vn+Vn-3
The first few terms are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119,... .
This Perrin sequence has the Fermat property: if p is prime,
Pp\equivP1\bmodp
n\midPn
Q=\begin{pmatrix}0&1&1\ 1&0&0\ 0&1&0\end{pmatrix},
The trace of gives the Perrin numbers.
Alternatively, can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet with corresponding substitution rule
\begin{cases} a \mapsto b\\ b \mapsto ac\\ c \mapsto a \end{cases}
l(wn)=Vn+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.[7]
There are precisely three ways of partitioning a square into three similar rectangles:[8]
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
| 1 | |
| 2 |
\sqrt{
| 2\rho-1 | |
| \rho-1 |
was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919.[3] 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; Flemish: de grondverhouding), using the plastic number (Dutch; Flemish: 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.
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.[9] 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.[10]
The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé[11] and subsequently used by Martin Gardner,[12] 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").
x3=x+1
x2=x+1
x3=x2+1
nl:Caroline Voet
. 2019. The digital study room of Dom Hans van der Laan. Van der Laan Foundation. 28 November 2023. en.