In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x + 1. Its decimal expansion begins with 1.324717957244746... (sequence A060006 in the OEIS). It is the smallest Pisot number. 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 b c = a b = b + c a {\displaystyle {\frac {b}{c}}={\frac {a}{b}}={\frac {b+c}{a}}}
This ratio is commonly denoted ρ {\displaystyle \rho } (rho). Substituting b = ρ c {\displaystyle b=\rho c\,} and a = ρ b = ρ 2 c {\displaystyle a=\rho b=\rho ^{2}c\,} in the last fraction,
ρ = c ( ρ + 1 ) ρ 2 c . {\displaystyle \rho ={\frac {c(\rho +1)}{\rho ^{2}c}}.} It follows that the plastic ratio is the unique real solution of the cubic equation ρ 3 − ρ − 1 = 0. {\displaystyle \rho ^{3}-\rho -1=0.}
Solving with Cardano's formula,
w 1 , 2 = 1 2 ( 1 ± 1 3 23 3 ) ρ = w 1 3 + w 2 3 {\displaystyle {\begin{aligned}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}}}\end{aligned}}}
or, using the hyperbolic cosine,
ρ = 2 3 cosh ( 1 3 arcosh ( 3 3 2 ) ) . {\displaystyle \rho ={\frac {2}{\sqrt {3}}}\cosh \left({\frac {1}{3}}\operatorname {arcosh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}
ρ {\displaystyle \rho } is the superstable fixed point of the iteration x ← ( 2 x 3 + 1 ) / ( 3 x 2 − 1 ) , {\displaystyle x\gets (2x^{3}+1)/(3x^{2}-1),} which is the update step of Newton's method applied to x 3 − x − 1 = 0. {\displaystyle x^{3}-x-1=0.} The iteration x ← 1 + 1 x {\displaystyle x\gets {\sqrt {1+{\tfrac {1}{x}}}}} results in the continued reciprocal square root
ρ = 1 + 1 1 + 1 1 + 1 ⋱ {\displaystyle \rho ={\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\sqrt {1+{\cfrac {1}{\ddots }}}}}}}}}}}
Dividing the defining trinomial x 3 − x − 1 {\displaystyle x^{3}-x-1} by x − ρ {\displaystyle x-\rho } one obtains x 2 + ρ x + 1 / ρ , {\displaystyle x^{2}+\rho x+1/\rho ,} and the conjugate elements of ρ {\displaystyle \rho } are
x 1 , 2 = 1 2 ( − ρ ± i 3 ρ 2 − 4 ) , {\displaystyle x_{1,2}={\frac {1}{2}}\left(-\rho \pm i{\sqrt {3\rho ^{2}-4}}\right),}
with x 1 + x 2 = − ρ {\displaystyle x_{1}+x_{2}=-\rho \;} and x 1 x 2 = 1 / ρ . {\displaystyle \;x_{1}x_{2}=1/\rho .}
Properties
The plastic ratio ρ {\displaystyle \rho } and golden ratio φ {\displaystyle \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 {\displaystyle x+1=x^{m}} and x − 1 = x − n . {\displaystyle \;x-1=x^{-n}.} Morphic numbers can serve as basis for a system of measure. Properties of ρ {\displaystyle \rho } (m=3 and n=4) are related to those of φ {\displaystyle \varphi } (m=2 and n=1). For example, The plastic ratio satisfies the continued radical
ρ = 1 + 1 + | 1 + ⋯ 3 3 3 , {\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{{\phantom {|}}1+\cdots }}}}}},}
while the golden ratio satisfies the analogous
φ = 1 + 1 + | 1 + ⋯ {\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {{\phantom {|}}1+\cdots }}}}}}}
The plastic ratio can be expressed in terms of itself as the infinite geometric series
ρ 5 = ∑ n = 0 ∞ ρ − n ρ 3 = ∑ n = 0 ∞ ρ − 2 n ρ 2 = ∑ n = 0 ∞ ρ − 3 n ρ 5 ρ 2 + 1 = ∑ n = 0 ∞ ρ − 4 n ρ = ∑ n = 0 ∞ ρ − 5 n ρ 5 ρ + 2 = ∑ n = 0 ∞ ρ − 6 n ρ 3 2 = ∑ n = 0 ∞ ρ − 7 n , {\displaystyle {\begin{aligned}\rho ^{5}&=\sum _{n=0}^{\infty }\rho ^{-n}\\\rho ^{3}&=\sum _{n=0}^{\infty }\rho ^{-2n}\\\rho ^{2}&=\sum _{n=0}^{\infty }\rho ^{-3n}\\{\frac {\rho ^{5}}{\rho ^{2}+1}}&=\sum _{n=0}^{\infty }\rho ^{-4n}\\\rho &=\sum _{n=0}^{\infty }\rho ^{-5n}\\{\frac {\rho ^{5}}{\rho +2}}&=\sum _{n=0}^{\infty }\rho ^{-6n}\\{\frac {\rho ^{3}}{2}}&=\sum _{n=0}^{\infty }\rho ^{-7n},\end{aligned}}}
in comparison to the series for the golden ratio
φ 2 = ∑ n = 0 ∞ φ − n φ = ∑ n = 0 ∞ φ − 2 n φ 2 2 = ∑ n = 0 ∞ φ − 3 n . {\displaystyle {\begin{aligned}\varphi ^{2}&=\sum _{n=0}^{\infty }\varphi ^{-n}\\\varphi &=\sum _{n=0}^{\infty }\varphi ^{-2n}\\{\frac {\varphi ^{2}}{2}}&=\sum _{n=0}^{\infty }\varphi ^{-3n}.\end{aligned}}}
Additionally, ∑ n = 0 13 ρ − n = 4 , {\displaystyle \sum _{n=0}^{13}\rho ^{-n}=4,} while ∑ n = 0 2 φ − n = 2. {\displaystyle \sum _{n=0}^{2}\varphi ^{-n}=2.}
For every integer n {\displaystyle n} one has
ρ n = ρ n − 2 + ρ n − 3 = ρ n − 1 + ρ n − 5 = ρ n − 3 + ρ n − 4 + ρ n − 5 {\displaystyle {\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}}}
from 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 the Bring radical. If y = x 5 + x {\displaystyle y=x^{5}+x} then x = B R ( y ) . {\displaystyle x=BR(y).} Since ρ − 5 + ρ − 1 = 1 , ρ = 1 / B R ( 1 ) . {\displaystyle \rho ^{-5}+\rho ^{-1}=1,\;\rho =1/BR(1).}
Continued fraction pattern of a few low powers
ρ − 1 = [ 0 ; 1 , 3 , 12 , 1 , 1 , 3 , 2 , 3 , 2 , . . . ] ≈ 0.7549 ( 25 33 ) ρ 0 = [ 1 ] ρ 1 = [ 1 ; 3 , 12 , 1 , 1 , 3 , 2 , 3 , 2 , 4 , . . . ] ≈ 1.3247 ( 45 34 ) ρ 2 = [ 1 ; 1 , 3 , 12 , 1 , 1 , 3 , 2 , 3 , 2 , . . . ] ≈ 1.7549 ( 58 33 ) ρ 3 = [ 2 ; 3 , 12 , 1 , 1 , 3 , 2 , 3 , 2 , 4 , . . . ] ≈ 2.3247 ( 79 34 ) ρ 4 = [ 3 ; 12 , 1 , 1 , 3 , 2 , 3 , 2 , 4 , 2 , . . . ] ≈ 3.0796 ( 40 13 ) ρ 5 = [ 4 ; 12 , 1 , 1 , 3 , 2 , 3 , 2 , 4 , 2 , . . . ] ≈ 4.0796 ( 53 13 ) . . . ρ 7 = [ 7 ; 6 , 3 , 1 , 1 , 4 , 1 , 1 , 2 , 1 , 1 , . . . ] ≈ 7.1592 ( 93 13 ) . . . ρ 9 = [ 12 ; 1 , 1 , 3 , 2 , 3 , 2 , 4 , 2 , 141 , . . . ] ≈ 12.5635 ( 88 7 ) {\displaystyle {\begin{aligned}\rho ^{-1}&=[0;1,3,12,1,1,3,2,3,2,...]\approx 0.7549\;({\tfrac {25}{33}})\\\rho ^{0}&=[1]\\\rho ^{1}&=[1;3,12,1,1,3,2,3,2,4,...]\approx 1.3247\;({\tfrac {45}{34}})\\\rho ^{2}&=[1;1,3,12,1,1,3,2,3,2,...]\approx 1.7549\;({\tfrac {58}{33}})\\\rho ^{3}&=[2;3,12,1,1,3,2,3,2,4,...]\approx 2.3247\;({\tfrac {79}{34}})\\\rho ^{4}&=[3;12,1,1,3,2,3,2,4,2,...]\approx 3.0796\;({\tfrac {40}{13}})\\\rho ^{5}&=[4;12,1,1,3,2,3,2,4,2,...]\approx 4.0796\;({\tfrac {53}{13}})\,...\\\rho ^{7}&=[7;6,3,1,1,4,1,1,2,1,1,...]\approx 7.1592\;({\tfrac {93}{13}})\,...\\\rho ^{9}&=[12;1,1,3,2,3,2,4,2,141,...]\approx 12.5635\;({\tfrac {88}{7}})\end{aligned}}}
The convergents of the continued fraction expansion of the plastic ratio are good rational approximations:
4 3 , 49 37 , 53 40 , 102 77 , 257 194 , 359 271 , 820 619 , 2819 2128 , 6458 4875 , 28651 21628 , 63760 48131 , … {\displaystyle {\tfrac {4}{3}},{\tfrac {49}{37}},{\tfrac {53}{40}},{\tfrac {102}{77}},{\tfrac {257}{194}},{\tfrac {359}{271}},{\tfrac {820}{619}},{\tfrac {2819}{2128}},{\tfrac {6458}{4875}},{\tfrac {28651}{21628}},{\tfrac {63760}{48131}},\ldots }
The plastic ratio is the smallest Pisot number. By definition of these numbers, the absolute value 1 / ρ {\displaystyle 1/{\sqrt {\rho }}} of the algebraic conjugates is smaller than 1, thus powers of ρ {\displaystyle \rho } generate almost integers. For example: ρ 29 = 3480.0002874... {\displaystyle \rho ^{29}=3480.0002874...} ≈ 3480 + 1 / 3479 {\displaystyle \approx 3480+1/3479} . After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to ± 45 π / 58 {\displaystyle \pm 45\pi /58} – nearly align with the imaginary axis. The infinite radicals
ρ = ( 1 3 ( 1 + ( 1 3 ( 1 + ( 1 3 ( 1 + ⋯ ) ) ) , 1 / ρ = ( − 1 2 ( 1 + ( − 1 2 ( 1 +