Supersilver ratio

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.

Three quantities a > b > c > 0 are in the supersilver ratio if $${\displaystyle {\frac {a}{b}}={\frac {a+c}{a}}={\frac {b}{c-a}}}$$ The ratio ⁠${\displaystyle {\frac {a}{b}}}$⁠ is commonly denoted ⁠${\displaystyle \varsigma .}$⁠

Substituting ${\displaystyle a=\varsigma b\,}$ and ${\displaystyle c=(\varsigma -1)a=(\varsigma ^{2}-\varsigma )b\,}$ in the third fraction, $${\displaystyle \varsigma ={\frac {b}{(\varsigma ^{2}-2\varsigma )b}}.}$$ It follows that the supersilver ratio is the unique real solution of the cubic equation ${\displaystyle \varsigma ^{3}-2\varsigma ^{2}-1=0.}$

The minimal polynomial for the reciprocal root is the depressed cubic ${\displaystyle x^{3}+2x-1,}$ thus the simplest solution with Cardano's formula, $${\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2\\1/\varsigma &={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}\end{aligned}}}$$ or, using the hyperbolic sine,

${\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}$

⁠${\displaystyle 1/\varsigma }$⁠ is the superstable fixed point of the iteration ${\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}$

Rewrite the minimal polynomial as ${\displaystyle (x^{2}+1)^{2}=1+x}$, then the iteration ${\displaystyle x\gets {\sqrt {-1+{\sqrt {1+x}}}}}$ results in the continued radical

${\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}\;}$^[2]

Dividing the defining trinomial ${\displaystyle x^{3}-2x^{2}-1}$ by ⁠${\displaystyle x-\varsigma }$⁠ one obtains ${\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma }$, and the conjugate elements of ⁠${\displaystyle \varsigma }$⁠ are $${\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2},}$$ with ${\displaystyle x_{1}+x_{2}=2-\varsigma \;}$ and ${\displaystyle \;x_{1}x_{2}=1/\varsigma .}$

The growth rate of the average value of the n-th term of a random Fibonacci sequence is ⁠${\displaystyle \varsigma -1}$⁠.^[3]

The defining equation can be written $${\displaystyle {\begin{aligned}1&={\frac {1}{\varsigma -1}}+{\frac {1}{\varsigma ^{2}+1}}\\&={\frac {1}{\varsigma }}+{\frac {\varsigma -1}{\varsigma +1}}+{\frac {\varsigma -2}{\varsigma -1}}.\end{aligned}}}$$

The supersilver ratio can be expressed in terms of itself as fractions $${\displaystyle {\begin{aligned}\varsigma &={\frac {\varsigma }{\varsigma -1}}+{\frac {\varsigma -1}{\varsigma +1}}\\\varsigma ^{2}&={\frac {1}{\varsigma -2}}.\end{aligned}}}$$

Similarly as the infinite geometric series $${\displaystyle {\begin{aligned}\varsigma &=2\sum _{n=0}^{\infty }\varsigma ^{-3n}\\\varsigma ^{2}&=-1+\sum _{n=0}^{\infty }(\varsigma -1)^{-n},\end{aligned}}}$$

in comparison to the silver ratio identities $${\displaystyle {\begin{aligned}\sigma &=2\sum _{n=0}^{\infty }\sigma ^{-2n}\\\sigma ^{2}&=-1+2\sum _{n=0}^{\infty }(\sigma -1)^{-n}.\end{aligned}}}$$

For every integer ⁠${\displaystyle n}$⁠ one has $${\displaystyle {\begin{aligned}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}\end{aligned}}}$$ From this an infinite number of further relations can be found.

Continued fraction pattern of a few low powers $${\displaystyle {\begin{aligned}\varsigma ^{-2}&=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056\;({\tfrac {5}{24}})\\\varsigma ^{-1}&=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534\;({\tfrac {5}{11}})\\\varsigma ^{0}&=[1]\\\varsigma ^{1}&=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056\;({\tfrac {53}{24}})\\\varsigma ^{2}&=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645\;({\tfrac {73}{15}})\\\varsigma ^{3}&=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729\;({\tfrac {118}{11}})\end{aligned}}}$$

The simplest rational approximations of ⁠${\displaystyle \varsigma }$⁠ are:$${\displaystyle {\tfrac {9}{4}},{\tfrac {11}{5}},{\tfrac {53}{24}},{\tfrac {75}{34}},{\tfrac {161}{73}},{\tfrac {236}{107}},{\tfrac {397}{180}},{\tfrac {633}{287}},{\tfrac {1030}{467}},{\tfrac {1663}{754}},{\tfrac {2693}{1221}},{\tfrac {7049}{3196}},...}$$

The supersilver ratio is a Pisot number.^[4] Because the absolute value ${\displaystyle 1/{\sqrt {\varsigma }}}$ of the algebraic conjugates is smaller than 1, powers of ⁠${\displaystyle \varsigma }$⁠ generate almost integers. For example: ${\displaystyle \varsigma ^{10}=2724.00146856...\approx 2724+1/681.}$ After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠${\displaystyle \pm 45\pi /82}$⁠ – nearly align with the imaginary axis.

The minimal polynomial of the supersilver ratio ${\displaystyle m(x)=x^{3}-2x^{2}-1}$ has discriminant ${\displaystyle \Delta =-59}$ and factors into ${\displaystyle (x-21)^{2}(x-19){\pmod {59}};\;}$ the imaginary quadratic field ${\displaystyle K=\mathbb {Q} ({\sqrt {\Delta }})}$ has class number ⁠${\displaystyle h=3.}$⁠ Thus, the Hilbert class field of ⁠${\displaystyle K}$⁠ can be formed by adjoining ⁠${\displaystyle \varsigma .}$⁠^[5] With argument ${\displaystyle \tau =(1+{\sqrt {\Delta }})/2\,}$ a generator for the ring of integers of ⁠${\displaystyle K}$⁠, the real root  j(τ) of the Hilbert class polynomial is given by ${\displaystyle (\varsigma ^{-6}-27\varsigma ^{6}-6)^{3}.}$^[6][7]

The Weber-Ramanujan class invariant is approximated with error < 3.5 ∙ 10⁻²⁰ by

${\displaystyle {\sqrt {2}}\,{\mathfrak {f}}({\sqrt {\Delta }})={\sqrt[{4}]{2}}\,G_{59}\approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24},}$

while its true value is the single real root of the polynomial

${\displaystyle W_{59}(x)=x^{9}-4x^{8}+4x^{7}-2x^{6}+4x^{5}-8x^{4}+4x^{3}-8x^{2}+16x-8.}$

The elliptic integral singular value^[8] ${\displaystyle k_{r}=\lambda ^{*}(r){\text{ for }}r=59}$ has closed form expression

${\displaystyle \lambda ^{*}(59)=\sin(\arcsin \left(G_{59}^{-12}\right)/2)}$

(which is less than 1/294 the eccentricity of the orbit of Venus).

These numbers are related to the supersilver ratio as the Pell numbers and Pell-Lucas numbers are to the silver ratio.

The fundamental sequence is defined by the third-order recurrence relation $${\displaystyle S_{n}=2S_{n-1}+S_{n-3}{\text{ for }}n>2,}$$ with initial values $${\displaystyle S_{0}=1,S_{1}=2,S_{2}=4.}$$

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence A008998 in the OEIS). The limit ratio between consecutive terms is the supersilver ratio:${\displaystyle \lim _{n\rightarrow \infty }S_{n+1}/S_{n}=\varsigma .}$

The first 8 indices n for which ⁠${\displaystyle S_{n}}$⁠ is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 decimal digits.

The sequence can be extended to negative indices using $${\displaystyle S_{n}=S_{n+3}-2S_{n+2}.}$$

The generating function of the sequence is given by

${\displaystyle {\frac {1}{1-2x-x^{3}}}=\sum _{n=0}^{\infty }S_{n}x^{n}{\text{ for }}x<1/\varsigma \;.}$^[9]

The third-order Pell numbers are related to sums of binomial coefficients by

${\displaystyle S_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}\cdot 2^{n-3k}\;}$.^[10]

The characteristic equation of the recurrence is ${\displaystyle x^{3}-2x^{2}-1=0.}$ If the three solutions are real root ⁠${\displaystyle \alpha }$⁠ and conjugate pair ⁠${\displaystyle \beta }$⁠ and ⁠${\displaystyle \gamma }$⁠, the supersilver numbers can be computed with the Binet formula $${\displaystyle S_{n-2}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n},}$$ with real ⁠${\displaystyle a}$⁠ and conjugates ⁠${\displaystyle b}$⁠ and ⁠${\displaystyle c}$⁠ the roots of ${\displaystyle 59x^{3}+4x-1=0.}$

Since ${\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/\alpha ^{n/2}}$ and ${\displaystyle \alpha =\varsigma ,}$ the number ⁠${\displaystyle S_{n}}$⁠ is the nearest integer to ${\displaystyle a\,\varsigma ^{n+2},}$ with n ≥ 0 and ${\displaystyle a=\varsigma /(2\varsigma ^{2}+3)=}$ 0.1732702315504081807484794...

Coefficients ${\displaystyle a=b=c=1}$ result in the Binet formula for the related sequence ${\displaystyle A_{n}=S_{n}+2S_{n-3}.}$

The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence A332647 in the OEIS).

This third-order Pell-Lucas sequence has the Fermat property: if p is prime, ${\displaystyle A_{p}\equiv A_{1}{\bmod {p}}.}$ The converse does not hold, but the small number of odd pseudoprimes ${\displaystyle \,n\mid (A_{n}-2)}$ makes the sequence special. The 14 odd composite numbers below 10⁸ to pass the test are n = 3², 5², 5³, 315, 99297, 222443, 418625, 9122185, 3257², 11889745, 20909625, 24299681, 64036831, 76917325.^[11]

The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue ⁠${\displaystyle \varsigma }$⁠ $${\displaystyle Q={\begin{pmatrix}2&0&1\\1&0&0\\0&1&0\end{pmatrix}},}$$

$${\displaystyle Q^{n}={\begin{pmatrix}S_{n}&S_{n-2}&S_{n-1}\\S_{n-1}&S_{n-3}&S_{n-2}\\S_{n-2}&S_{n-4}&S_{n-3}\end{pmatrix}}}$$

The trace of ⁠${\displaystyle Q^{n}}$⁠ gives the above ⁠${\displaystyle A_{n}.}$⁠

Alternatively, ⁠${\displaystyle Q}$⁠ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet ⁠${\displaystyle \{a,b,c\}}$⁠ with corresponding substitution rule $${\displaystyle {\begin{cases}a\;\mapsto \;aab\\b\;\mapsto \;c\\c\;\mapsto \;a\end{cases}}}$$ and initiator ⁠${\displaystyle w_{0}=b}$⁠. The series of words ⁠${\displaystyle w_{n}}$⁠ produced by iterating the substitution have the property that the number of c's, b's and a's are equal to successive third-order Pell numbers. The lengths of these words are given by ${\displaystyle l(w_{n})=S_{n-2}+S_{n-3}+S_{n-4}.}$^[12]

Associated to this string rewriting process is a compact set composed of self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence.^[13]

Given a rectangle of height 1, length ⁠${\displaystyle \varsigma }$⁠ and diagonal length ${\displaystyle \varsigma {\sqrt {\varsigma -1}}.}$ The triangles on the diagonal have altitudes ${\displaystyle 1/{\sqrt {\varsigma -1}}\,;}$ each perpendicular foot divides the diagonal in ratio ⁠${\displaystyle \varsigma ^{2}}$⁠.

On the right-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio ${\displaystyle 1+1/\varsigma ^{2}:1}$ (according to ${\displaystyle \varsigma =2+1/\varsigma ^{2}}$). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.^[14]

The parent supersilver rectangle and the two scaled copies along the diagonal have linear sizes in the ratios ${\displaystyle \varsigma :\varsigma -1:1.}$ The areas of the rectangles opposite the diagonal are both equal to ${\displaystyle (\varsigma -1)/\varsigma ,}$ with aspect ratios ${\displaystyle \varsigma (\varsigma -1)}$ (below) and ${\displaystyle \varsigma /(\varsigma -1)}$ (above).

If the diagram is further subdivided by perpendicular lines through the feet of the altitudes, the lengths of the diagonal and its seven distinct subsections are in ratios ${\displaystyle \varsigma ^{2}+1:\varsigma ^{2}:\varsigma ^{2}-1:\varsigma +1:}$ ${\displaystyle \,\varsigma (\varsigma -1):\varsigma :2/(\varsigma -1):1.}$

A supersilver spiral is a logarithmic spiral that gets wider by a factor of ⁠${\displaystyle \varsigma }$⁠ for every quarter turn. It is described by the polar equation ${\displaystyle r(\theta )=a\exp(k\theta ),}$ with initial radius ⁠${\displaystyle a}$⁠ and parameter ${\displaystyle k={\frac {2\ln(\varsigma )}{\pi }}.}$ If drawn on a supersilver rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio ${\displaystyle \varsigma (\varsigma -1)}$ which are perpendicularly aligned and successively scaled by a factor ${\displaystyle 1/\varsigma .}$

• Solutions of equations similar to ${\displaystyle x^{3}=2x^{2}+1}$:
  • Silver ratio – the only positive solution of the equation ${\displaystyle x^{2}=2x+1}$
  • Golden ratio – the only positive solution of the equation ${\displaystyle x^{2}=x+1}$
  • Supergolden ratio – the only real solution of the equation ${\displaystyle x^{3}=x^{2}+1}$