In mathematics, the supergolden ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x2 + 1. Its decimal expansion begins with 1.465571231876768... (sequence A092526 in the OEIS). The name supergolden ratio is by analogy with the golden ratio, the positive solution of the equation x2 = x + 1.
Definition
Three quantities a > b > c > 0 are in the supergolden ratio if
a + c a = a b = b c {\displaystyle {\frac {a+c}{a}}={\frac {a}{b}}={\frac {b}{c}}}
This common ratio is commonly denoted ψ . {\displaystyle \psi .} Substituting b = ψ c {\displaystyle b=\psi c} and a = ψ b = ψ 2 c {\displaystyle a=\psi b=\psi ^{2}c} in the first fraction,
ψ = c ( ψ 2 + 1 ) ψ 2 c . {\displaystyle \psi ={\frac {c(\psi ^{2}+1)}{\psi ^{2}c}}.} It follows that the supergolden ratio is the unique real solution of the cubic equation ψ 3 − ψ 2 − 1 = 0 {\displaystyle \psi ^{3}-\psi ^{2}-1=0} . The minimal polynomial for the reciprocal root is the depressed cubic x 3 + x − 1 {\displaystyle x^{3}+x-1} , thus the simplest solution with Cardano's formula,
w 1 , 2 = ( 1 ± 1 3 31 3 ) / 2 1 / ψ = w 1 3 + w 2 3 {\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {31}{3}}}\right)/2\\1/\psi &={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}\end{aligned}}}
or, using the hyperbolic sine,
1 / ψ = 2 3 sinh ( 1 3 arsinh ( 3 3 2 ) ) . {\displaystyle 1/\psi ={\frac {2}{\sqrt {3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left({\frac {3{\sqrt {3}}}{2}}\right)\right).}
1 / ψ {\displaystyle 1/\psi } is the superstable fixed point of the Newton's method iteration x ← ( 2 x 3 + 1 ) / ( 3 x 2 + 1 ) {\displaystyle x\gets (2x^{3}+1)/(3x^{2}+1)} . Dividing the defining trinomial x 3 − x 2 − 1 {\displaystyle x^{3}-x^{2}-1} by x − ψ {\displaystyle x-\psi } one obtains x 2 + x / ψ 2 + 1 / ψ {\displaystyle x^{2}+x/\psi ^{2}+1/\psi } , and the conjugate elements of ψ {\displaystyle \psi } are
x 1 , 2 = ( − 1 ± i 4 ψ 2 + 3 ) / 2 ψ 2 , {\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {4\psi ^{2}+3}}\right)/2\psi ^{2},}
with x 1 + x 2 = 1 − ψ {\displaystyle x_{1}+x_{2}=1-\psi } and x 1 x 2 = 1 / ψ {\displaystyle x_{1}x_{2}=1/\psi } . The iteration x ← | 1 + x 2 3 {\displaystyle x\gets {\sqrt[{3}]{{\phantom {|}}1+x^{2}}}} results in the continued radical
ψ 3 − 1 = ψ 2 = 1 + 1 + | 1 + ⋯ 3 / 2 3 / 2 3 / 2 {\displaystyle \psi ^{3}-1=\psi ^{2}={\sqrt[{3/2}]{1+{\sqrt[{3/2}]{1+{\sqrt[{3/2}]{{\phantom {|}}1+\cdots }}}}}}}
Better convergence is attained by using relation y 3 = y ( 1 − 2 y ) + 3 {\displaystyle y^{3}=y(1-2y)+3\,} with real zero ψ 2 − 1 {\displaystyle \psi ^{2}-1} . Divide both sides by y {\displaystyle y} , and substitute ψ − 3 = 1 / ( y + 2 ) for 3 / y − 2 y . {\displaystyle \psi ^{-3}=1/(y+2)\,{\text{ for }}\,3/y-2y.} This gives the iteration y ← 1 + 1 2 + y , {\displaystyle y\gets {\sqrt {1+{\tfrac {1}{2+y}}}},} and the continued reciprocal square root
ψ 2 − 1 = 1 + 1 2 + 1 + 1 2 + 1 + 1 2 + ⋱ {\displaystyle \psi ^{2}-1={\sqrt {1+{\cfrac {1}{2+{\sqrt {1+{\cfrac {1}{2+{\sqrt {1+{\cfrac {1}{2+\ddots }}}}}}}}}}}}}
Properties
The supergolden ratio can be expressed in terms of itself in many different ways, for example as fractions
2 ψ − 1 = ψ 2 + 2 ψ 2 = ψ 3 + 1 ψ 3 − 1 = 2 ψ 7 − 1 ψ 7 , {\displaystyle 2\psi -1={\frac {\psi ^{2}+2}{\psi ^{2}}}={\frac {\psi ^{3}+1}{\psi ^{3}-1}}={\frac {2\psi ^{7}-1}{\psi ^{7}}},} along with
1 + ψ − 1 = ψ 2 + 2 ψ + 1 = 2 ψ 3 − 1 ψ 3 1 + ψ − 3 = ψ 2 + 2 ψ 2 + 1 1 + ψ − 5 = ψ + 1 ψ 2 = ψ 2 − 1 = ψ 7 + 1 ψ 7 − 1 ψ 4 − 1 = ψ 2 + 2 ψ 2 − 1 ψ 6 − 1 = ψ 2 + 2 ψ − 1 . {\displaystyle {\begin{aligned}1+\psi ^{-1}&={\frac {\psi ^{2}+2}{\psi +1}}={\frac {2\psi ^{3}-1}{\psi ^{3}}}\\1+\psi ^{-3}&={\frac {\psi ^{2}+2}{\psi ^{2}+1}}\\1+\psi ^{-5}&={\frac {\psi +1}{\psi ^{2}}}\\=\psi ^{2}-1&={\frac {\psi ^{7}+1}{\psi ^{7}-1}}\\\psi ^{4}-1&={\frac {\psi ^{2}+2}{\psi ^{2}-1}}\\\psi ^{6}-1&={\frac {\psi ^{2}+2}{\psi -1}}.\end{aligned}}}
Similarly as infinite geometric series
ψ 3 = ∑ n = 0 ∞ ψ − n ψ 2 ψ 2 − 1 = ∑ n = 0 ∞ ψ − 2 n ψ = ∑ n = 0 ∞ ψ − 3 n ψ ψ 2 − 1 = ∑ n = 0 ∞ ψ − 4 n 1 3 − ψ 2 = ∑ n = 0 ∞ ψ − 5 n 1 ψ 2 − 1 + 1 ψ 2 + 2 = ∑ n = 0 ∞ ψ − 6 n ψ 2 2 = ∑ n = 0 ∞ ψ − 7 n . {\displaystyle {\begin{aligned}\psi ^{3}&=\sum _{n=0}^{\infty }\psi ^{-n}\\{\frac {\psi ^{2}}{\psi ^{2}-1}}&=\sum _{n=0}^{\infty }\psi ^{-2n}\\\psi &=\sum _{n=0}^{\infty }\psi ^{-3n}\\{\frac {\psi }{\psi ^{2}-1}}&=\sum _{n=0}^{\infty }\psi ^{-4n}\\{\frac {1}{3-\psi ^{2}}}&=\sum _{n=0}^{\infty }\psi ^{-5n}\\{\frac {1}{\psi ^{2}-1}}+{\frac {1}{\psi ^{2}+2}}&=\sum _{n=0}^{\infty }\psi ^{-6n}\\{\frac {\psi ^{2}}{2}}&=\sum _{n=0}^{\infty }\psi ^{-7n}.\end{aligned}}}
Additionally, ∑ n = 0 7 ψ − n = 3. {\displaystyle \sum _{n=0}^{7}\psi ^{-n}=3.}
For every integer n {\displaystyle n} one has
ψ n = ψ n − 1 + ψ n − 3 = ψ n − 2 + ψ n − 3 + ψ n − 4 = ψ n − 2 + 2 ψ n − 4 + ψ n − 6 {\displaystyle {\begin{aligned}\psi ^{n}&=\psi ^{n-1}+\psi ^{n-3}\\&=\psi ^{n-2}+\psi ^{n-3}+\psi ^{n-4}\\&=\psi ^{n-2}+2\psi ^{n-4}+\psi ^{n-6}\end{aligned}}}
from this an infinite number of further relations can be found. Argument θ = arcsec ( 2 ψ 4 ) {\displaystyle \theta =\operatorname {arcsec}(2\psi ^{4})} satisfies the identity tan ( θ ) − 4 sin ( θ ) = 3 3 {\displaystyle \tan(\theta )-4\sin(\theta )=3{\sqrt {3}}} . Continued fraction pattern of a few low powers
ψ − 1 = [ 0 ; 1 , 2 , 6 , 1 , 3 , 5 , 4 , 22 , . . . ] ≈ 0.6823 ( 13 19 ) ψ 0 = [ 1 ] ψ 1 = [ 1 ; 2 , 6 , 1 , 3 , 5 , 4 , 22 , 1 , . . . ] ≈ 1.4656 ( 22 15 ) ψ 2 = [ 2 ; 6 , 1 , 3 , 5 , 4 , 22 , 1 , 1 , . . . ] ≈ 2.1479 ( 15 7 ) ψ 3 = [ 3 ; 6 , 1 , 3 , 5 , 4 , 22 , 1 , 1 , . . . ] ≈ 3.1479 ( 22 7 ) ψ 4 = [ 4 ; 1 , 1 , 1 , 1 , 2 , 2 , 1 , 2 , 2 , . . . ] ≈ 4.6135 ( 60 13 ) ψ 5 = [ 6 ; 1 , 3 , 5 , 4 , 22 , 1 , 1 , 4 , . . . ] ≈ 6.7614 ( 115 17 ) {\displaystyle {\begin{aligned}\psi ^{-1}&=[0;1,2,6,1,3,5,4,22,...]\approx 0.6823\;({\tfrac {13}{19}})\\\psi ^{0}&=[1]\\\psi ^{1}&=[1;2,6,1,3,5,4,22,1,...]\approx 1.4656\;({\tfrac {22}{15}})\\\psi ^{2}&=[2;6,1,3,5,4,22,1,1,...]\approx 2.1479\;({\tfrac {15}{7}})\\\psi ^{3}&=[3;6,1,3,5,4,22,1,1,...]\approx 3.1479\;({\tfrac {22}{7}})\\\psi ^{4}&=[4;1,1,1,1,2,2,1,2,2,...]\approx 4.6135\;({\tfrac {60}{13}})\\\psi ^{5}&=[6;1,3,5,4,22,1,1,4,...]\approx 6.7614\;({\tfrac {115}{17}})\end{aligned}}}
Notably, the continued fraction of ψ 2 {\displaystyle \psi ^{2}} begins as permutation of the first six natural numbers; the next term is equal to their sum + 1. As derived from its continued fraction expansion, the simplest rational approximations of ψ {\displaystyle \psi } are:
3 2 , 19 13 , 22 15 , 85 58 , 277 189 , 447 305 , 1873 1278 , 41653 28421 , 43526 29699 , 85179 58120 , … {\displaystyle {\tfrac {3}{2}},{\tfrac {19}{13}},{\tfrac {22}{15}},{\tfrac {85}{58}},{\tfrac {277}{189}},{\tfrac {447}{305}},{\tfrac {1873}{1278}},{\tfrac {41653}{28421}},{\tfrac {43526}{29699}},{\tfrac {85179}{58120}},\ldots }
The supergolden ratio is the fourth smallest Pisot number. By definition of these numbers, the absolute value 1 / ψ {\displaystyle 1/{\sqrt {\psi }}} of the algebraic conjugates is smaller than 1, thus powers of ψ {\displaystyle \psi } generate almost integers. For example: ψ 11 = 67.000222765... {\displaystyle \psi ^{11}=67.000222765...} ≈ 67 + 1 / 4489 {\displaystyle \approx 67+1/4489} . After eleven rotation steps the phases of the inward spiraling conjugate pair – initially close to ± 13 π / 22 {\displaystyle \pm 13\pi /22} – nearly align with the imaginary axis. In the complex plane, if ψ