In mathematics, the supersilver ratio is a geometrical proportion, given by the unique real solution of the equation x3 = 2x2 + 1. Its decimal expansion begins with 2.2055694304005903... (sequence A356035 in the OEIS). The name supersilver ratio is by analogy with the silver ratio, the positive solution of the equation x2 = 2x + 1, and the supergolden ratio.
Definition
Three quantities a > b > c > 0 are in the supersilver ratio if 2 a + c a = a b = b c . {\displaystyle {\frac {2a+c}{a}}={\frac {a}{b}}={\frac {b}{c}}\,.}
This ratio is commonly denoted ς {\displaystyle \varsigma } . Substituting a = ς b = ς 2 c {\displaystyle a=\varsigma \,b=\varsigma ^{2}c} in the first fraction gives
ς = 2 ς 2 c + c ς 2 c . {\displaystyle \varsigma ={\frac {2\varsigma ^{2}c+c}{\varsigma ^{2}c}}.} It follows that the supersilver ratio is the unique real solution of the cubic equation ς 3 − 2 ς 2 − 1 = 0. {\displaystyle \varsigma ^{3}-2\varsigma ^{2}-1=0.}
The minimal polynomial for the reciprocal root is the depressed cubic x 3 + 2 x − 1 , {\displaystyle x^{3}+2x-1,} thus the simplest solution with Cardano's formula,
w 1 , 2 = ( 1 ± 1 3 59 3 ) / 2 1 / ς = w 1 3 + w 2 3 {\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2\\1/\varsigma &={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}\end{aligned}}}
or, using the hyperbolic sine, 1 / ς = − 2 2 3 sinh ( 1 3 arsinh ( − 3 4 3 2 ) ) . {\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}
1 / ς {\displaystyle 1/\varsigma } is the superstable fixed point of the iteration x ← ( 2 x 3 + 1 ) / ( 3 x 2 + 2 ) . {\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}
Dividing the defining trinomial x 3 − 2 x 2 − 1 {\displaystyle x^{3}-2x^{2}-1} by x − ς {\displaystyle x-\varsigma } one obtains x 2 + x / ς 2 + 1 / ς , {\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma ,} and the conjugate elements of ς {\displaystyle \varsigma } are
x 1 , 2 = ( 2 − ς ± i 7 − ς 2 ς − 1 ) / 2 , {\displaystyle x_{1,2}=\left(2-\varsigma \pm i{\sqrt {\frac {7-\varsigma ^{2}}{\varsigma -1}}}\right)/2,}
with x 1 + x 2 = − 1 / ς 2 {\displaystyle x_{1}+x_{2}=-1/\varsigma ^{2}\;} and x 1 x 2 = 1 / ς . {\displaystyle \;x_{1}x_{2}=1/\varsigma .}
Multiply the minimal polynomial by x {\displaystyle x} , and rearrange the relation as ( x 2 + 1 ) 2 = x + 1. {\displaystyle (x^{2}+1)^{2}=x+1.} This results in the iteration x n + 1 ← − 1 + | 1 + x n {\displaystyle x_{n+1}\gets {\sqrt {-1+{\sqrt {{\phantom {|}}1+x_{n}}}}}} , with initial value x 0 > 0 {\displaystyle x_{0}>0} , and the continued radical
1 / ς = − 1 + 1 + − 1 + | 1 + ⋯ {\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {{\phantom {|}}1+\cdots }}}}}}}}}
Its counterpart is found by using polynomial f : z 3 + z 2 − z − 2 {\displaystyle f:z^{3}+z^{2}-z-2} , which has real zero ς − 1 {\displaystyle \varsigma -1} . Multiply f {\displaystyle f} by z − 1 {\displaystyle z-1} , then ( z 2 − 1 ) 2 = z − 1 , {\displaystyle (z^{2}-1)^{2}=z-1,} and the corresponding iteration with z 0 > 1 {\displaystyle z_{0}>1} gives
ς − 1 = 1 + − 1 + 1 + | − 1 + ⋯ {\displaystyle \varsigma -1={\sqrt {1+{\sqrt {-1+{\sqrt {1+{\sqrt {{\phantom {|}}-1+\cdots }}}}}}}}}
Supersilver Julia set
Both systems have linear convergence rate log 10 4 ( ς − 1 ) ς = 0.340... {\displaystyle \,\log _{10}{\tfrac {4(\varsigma -1)}{\varsigma }}=0.340...} In order to improve this constant, divide both sides of z 3 = z ( 1 − z ) + 2 by z , {\displaystyle z^{3}=z(1-z)+2\,{\text{ by }}z,} and substitute 1 / ς = 1 / ( z + 1 ) {\displaystyle 1/\varsigma =1/(z+1)\,} for 2 / z − z {\displaystyle \,2/z-z} , resulting in the iteration z n + 1 ← 1 + 1 1 + z n , {\displaystyle z_{n+1}\gets {\sqrt {1+{\frac {1}{1+z_{n}}}}},} and the continued reciprocal square root
ς − 1 = 1 + 1 1 + 1 + 1 1 + 1 + 1 1 + ⋱ {\displaystyle \varsigma -1={\sqrt {1+{\cfrac {1}{1+{\sqrt {1+{\cfrac {1}{1+{\sqrt {1+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}
For complex initial points z 0 {\displaystyle z_{0}} other than − 1 {\displaystyle -1} this method converges with linear rate log 10 2 ( ς 2 + 1 ) = 1.069... , {\displaystyle \,\log _{10}2(\varsigma ^{2}+1)=1.069...,} provided the principal root is chosen at each step. If randomly either the principal root or its negative is picked, the orbit of z 0 {\displaystyle z_{0}} is attracted to a simple closed curve, which is the Julia set J {\displaystyle J} of the backward iteration z n − 1 ← 1 z n 2 − 1 − 1. {\displaystyle z_{n-1}\gets {\frac {1}{z_{n}^{2}-1}}-1.}
The critical points z c {\displaystyle z_{c}} for which the derivative − 2 z / ( z 2 − 1 ) 2 {\displaystyle -2z/(z^{2}-1)^{2}} vanishes are 0 {\displaystyle 0} and − ∞ {\displaystyle -\infty } . The latter is mapped into the right neighborhood of the pole z p = − 1 {\displaystyle z_{p}=-1} and vice versa, so { − ∞ , − 1 } {\displaystyle \{-\infty ,-1\}} is the single attracting limit set. On J {\displaystyle J} , the repelling fixed points z f {\displaystyle z_{f}} are the zeros of f {\displaystyle f} , namely ς − 1 {\displaystyle \varsigma -1} , and ( − ς ± i 7 − ς 2 ς − 1 ) / 2 {\displaystyle \left(-\varsigma \pm i{\sqrt {\tfrac {7-\varsigma ^{2}}{\varsigma -1}}}\right)/2\,} (the centers of the largest spirals in the left half of the image) with divergence rate 1 2 log 10 8 ς 2 + 1 = 0.067... {\displaystyle {\tfrac {1}{2}}\log _{10}{\tfrac {8}{\varsigma ^{2}+1}}=0.067...} The first preimages of z f = ς − 1 {\displaystyle z_{f}=\varsigma -1\,} are 1 − ς {\displaystyle 1-\varsigma } , and the purely imaginary zeros ± i ς 2 − 1 {\displaystyle \pm i{\sqrt {\varsigma ^{2}-1}}\,} of z 6 + z 4 − 9 z 2 + 8 {\displaystyle \,z^{6}+z^{4}-9z^{2}+8} .
Properties
The growth rate of the average value of the n-th term of a random Fibonacci sequence is ς − 1 {\displaystyle \varsigma -1} . The defining equation can be written
1 = 1 ς − 1 + 1 ς 2 + 1 = 1 ς + ς − 1 ς + 1 + ς − 2 ς − 1 . {\displaystyle {\begin{aligned}1&={\frac {1}{\varsigma -1}}+{\frac {1}{\varsigma ^{2}+1}}\\&={\frac {1}{\varsigma }}+{\frac {\varsigma -1}{\varsigma +1}}+{\frac {\varsigma -2}{\varsigma -1}}.\end{aligned}}}
The supersilver ratio can be expressed in terms of itself as fractions
ς = ς ς − 1 + ς − 1 ς + 1 ς 2 = 1 ς − 2 . {\displaystyle {\begin{aligned}\varsigma &={\frac {\varsigma }{\varsigma -1}}+{\frac {\varsigma -1}{\varsigma +1}}\\\varsigma ^{2}&={\frac {1}{\varsigma -2}}.\end{aligned}}}
Similarly as the infinite geometric series
ς ς − 1 = ∑ n = 0 ∞ ς − n ς 2 ς 2 − 1 = ∑ n = 0 ∞ ς − 2 n ς 2 = ∑ n = 0 ∞ ς − 3 n , {\displaystyle {\begin{aligned}{\frac {\varsigma }{\varsigma -1}}&=\sum _{n=0}^{\infty }\varsigma ^{-n}\\{\frac {\varsigma ^{2}}{\varsigma ^{2}-1}}&=\sum _{n=0}^{\infty }\varsigma ^{-2n}\\{\frac {\varsigma }{2}}&=\sum _{n=0}^{\infty }\varsigma ^{-3n},\end{aligned}}}
in comparison to the silver ratio identities
σ σ − 1 = ∑ n = 0 ∞ σ − n σ 2 = ∑ n = 0 ∞ σ − 2 n σ 3 σ 3 − 1 = ∑ n = 0 ∞ σ − 3 n . {\displaystyle {\begin{aligned}{\frac {\sigma }{\sigma -1}}&=\sum _{n=0}^{\infty }\sigma ^{-n}\\{\frac {\sigma }{2}}&=\sum _{n=0}^{\infty }\sigma ^{-2n}\\{\frac {\sigma ^{3}}{\sigma ^{3}-1}}&=\sum _{n=0}^{\infty }\sigma ^{-3n}.\end{aligned}}}
For every integer n {\displaystyle n} one has
ς n = 2 ς n − 1 + ς n − 3 = 4 ς n − 2 + ς n − 3 + 2 ς n − 4 = ς n − 1 + 2 ς n − 2 + ς n − 3 + ς n − 4 {\displaystyle {\begin{aligned}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}\end{aligned}}}
from this an infinite number of further relations can be found. Continued fraction pattern of a few low powers
ς − 2 = [ 0 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 0.2056 ( 5 24 ) ς − 1 = [ 0 ; 2 , 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 0.4534 ( 5 11 ) ς 0 = [ 1 ] ς 1 = [ 2 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 2.2056 ( 53 24 ) ς 2 = [ 4 ; 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , . . . ] ≈ 4.8645 ( 73 15 ) ς 3 = [ 10 ; 1 , 2 , 1 , 2 , 4 , 4 , 2 , 2 , 6 , 2 , . . . ] ≈ 10.729 ( 118 11 ) {\displaystyle {\begin{aligned}\varsigma ^{-2}&=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056\;({\tfrac {5}{24}})\\\varsigma ^{-1}&=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534\;({\tfrac {5}{11}})\\\varsigma ^{0}&=[1]\\\varsigma ^{1}&=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056\;({\tfrac {53}{24}})\\\varsigma ^{2}&=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645\;({\tfrac {73}{15}})\\\varsigma ^{3}&=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729\;({\tfra