This article is about the irrational number. For numbers stamped on
plastics for recycling identification, see
Resin identification code.
Plastic number
Triangles with sides in ratio $\rho$ form a closed spiral
Rationality
irrational algebraic
Symbol
$\rho$
Representations
Decimal
1.32471795724474602596...
Algebraic form
real root of $x^{3}=x+1$
Continued fraction (linear)
[1;3,12,1,1,3,2,3,2,4,2,141,80,...] ^{
[1]} not periodic infinite
Binary
1.01010011001000001011...
Hexadecimal
1.5320B74ECA44ADAC1788...
In
mathematics, the plastic number is a geometrical
proportion close to 53/40. Its true value is the real
solution of the equation $x^{3}=x+1$.
The adjective plastic does not refer to the artificial material, but to the formative and sculptural qualities of this ratio, as in plastic arts.
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 from $\,\rho ={\frac {1+c}{\rho }}$ and $\,\rho ={\frac {1}{c}}$
one has $\rho ^{2}-1=c\,$ and $\,c=\rho ^{-1}$, thus $\,\rho ^{2}-1=\rho ^{-1}$.
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).
Using formulas for the cubic equation, one can show that
$\rho$ is the superstable
fixed point of the iteration $x\gets (2x^{3}+1)/(3x^{2}-1)$.
Properties
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 $x-1=x^{-n}$. ^{
[4]}
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
infinitely nested radical
The plastic ratio is the smallest
Pisot number.^{
[5]} Because the
absolute value$1/{\sqrt {\rho }}$ of both
algebraic conjugates is smaller than 1, powers of $\rho$ generate
almost integers. For example: $\rho ^{29}=3480.00028744...\approx 3480+1/3479$. After 29 rotation steps the
phases of the inward spiraling conjugate pair – initially close to ±45π/58 – nearly align with the imaginary axis.
The
minimal polynomial of the plastic ratio $f(x)=x^{3}-x-1$ has
discriminant$\Delta =-23$. Its
splitting field over rationals is $\mathbb {Q} ({\sqrt {\Delta }},\rho )$. This field is also the
Hilbert class field of $\mathbb {Q} ({\sqrt {\Delta }})$. Thus, $\rho$ can be expressed in terms of the
Dedekind eta function$\eta (\tau )$ with argument $\tau =(1+{\sqrt {\Delta }})/2\,$
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.
Van der Laan sequence
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 percieve them as distinct. Also, what is the maximum ratio of two sizes, so that we can still relate them and percieve nearness. According to his observations, the answers are 1/4 and 7/1, spanning a single order of size.^{
[7]} 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. Expressed in countable values, this architectonic system of measure consists of 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 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
$V_{n}/V_{m}$
err$(\rho ^{k})$
interval name
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).^{
[8]} The last number has 154 decimal digits.
The
characteristic equation of the recurrence is $x^{3}-x-1=0$. If the three solutions are real root α and conjugate pair β and γ, the Van der Laan numbers can be computed with the
Binet formula^{
[10]}
$V_{n-3}={\frac {\alpha ^{n}}{2\alpha +3}}+{\frac {\beta ^{n}}{2\beta +3}}+{\frac {\gamma ^{n}}{2\gamma +3}}$ for n > 2,
Since the magnitude of the two last terms is $<1/{\sqrt {\alpha ^{n}}}$ for all n > 4 and $\alpha =\rho$, the number $V_{n}$ is the nearest integer to $a\,\rho ^{n}$, with $a\approx 0.4115$.
There are precisely three ways of partitioning a square into three similar rectangles:^{
[11]}^{
[12]}
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 length 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 ρ^{2}. The ratios of the linear sizes of the three rectangles are: ρ (large:medium); ρ^{2} (medium:small); and ρ^{3} (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 ρ^{4}.
The fact that a rectangle of aspect ratio ρ^{2} can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ^{2} related to the
Routh–Hurwitz theorem: all of its conjugates have positive real part.^{
[13]}^{
[14]}
$\rho$ was first studied by
Axel Thue in 1912 and by
G. H. Hardy in 1919.^{
[5]} French high school student
Gérard Cordonnier discovered the number for himself in 1924 and referred to it as the radiant number (
French: le nombre radiant). Hans van der Laan gave it the name plastic number (
Dutch: het plastische getal) in 1928.
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.^{
[16]} 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.^{
[17]}
The plastic number is also sometimes called the silver number, a name given to it by
Midhat J. Gazalé^{
[18]} and subsequently used by
Martin Gardner,^{
[19]} but that name is more commonly used for the
silver ratio$1+{\sqrt {2}},$ one of the ratios from the family of
metallic means first described by
Vera W. de Spinadel in 1998.^{
[20]}
Martin Gardner has suggested referring to $\rho ^{2}$ 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").^{
[21]}
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 $x^{3}=x^{2}+1$
^Voet, Caroline[in Dutch] (2019).
"1:7 and a series of 8". The digital study room of Dom Hans van der Laan. Van der Laan Foundation. Retrieved 28 November 2023.
^Freiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", Mathematical Research Letters, 1 (5): 547–558,
doi:10.4310/MRL.1994.v1.n5.a3,
MR1295549
^Gazalé, Midhat J. (April 19, 1999), "Chapter VII: The Silver Number", Gnomon: From Pharaohs to Fractals, Princeton, N.J.: Princeton University Press, pp. 135–150,
ISBN9780691005140,
OCLC40298400
^Martin Gardner, A Gardner's Workout (2001), Chapter 16, pp. 121–128.
Shannon, A. G.; Anderson, P. G.; Horadam, A. F. (2006), "Properties of Cordonnier, Perrin and Van der Laan numbers", International Journal of Mathematical Education in Science and Technology, 37 (7): 825–831,
doi:
10.1080/00207390600712554,
S2CID119808971.