Supersilver ratio

Template:Short description Template:Infobox non-integer number In mathematics, the supersilver ratio is a geometrical proportion, given by the unique real solution of the equation Template:Nowrap Its decimal expansion begins with Template:Math Template:OEIS.

The name supersilver ratio is by analogy with the silver ratio, the positive solution of the equation Template:Math, and the supergolden ratio.

Definition

ς = Template:Sfrac = Template:Sfrac = Template:Sfrac. For b = 1 the boxes have volumes Template:Math

Three quantities Template:Math are in the supersilver ratio if $$\frac{2a+c}{a}=\frac{a}{b}=\frac{b}{c}.$$ This ratio is commonly denoted Template:Tmath.Template:Efn

Substituting $a=\varsigma b={\varsigma }^{2}c$ in the first fraction gives $$\varsigma =\frac{2{\varsigma }^{2}c+c}{{\varsigma }^{2}c}.$$ It follows that the supersilver ratio is the unique real solution of the cubic equation ${\varsigma }^3-2{\varsigma }^2-1=0.$

The minimal polynomial for the reciprocal root is the depressed cubic $x^3+2x-1,$ thus the simplest solution with Cardano's formula, $$\begin{array}{cc}{w}_{1,2}& =(1\pm \frac{1}{3}\sqrt{\frac{59}{3}})/2\\ 1/\varsigma & =\sqrt[3]{w_1}+\sqrt[3]{w_2}\end{array}$$ or, using the hyperbolic sine, $$1/\varsigma =-2\sqrt{\frac{2}{3}}\sinh (\frac{1}{3}\mathrm{arsinh}(-\frac{3}{4}\sqrt{\frac{3}{2}})).$$

Template:Tmath is the superstable fixed point of the iteration $x\leftarrow (2x^3+1)/(3x^2+2).$

Dividing the defining trinomial $x^3-2x^2-1$ by Template:Tmath one obtains $x^2+x/{\varsigma }^2+1/\varsigma ,$ and the conjugate elements of Template:Tmath are $$x_{1,2}=(2-\varsigma \pm i\sqrt{\frac{7-{\varsigma }^2}{\varsigma -1}})/2,$$ with $x_1+x_2=-1/{\varsigma }^2$ and $x_1{x}_2=1/\varsigma .$

Multiply the minimal polynomial by Template:Tmath, and rearrange the relation as $(x^2+1{)}^2=x+1.$ This results in the iteration $x_{n+1}\leftarrow \sqrt{-1+\sqrt{\phantom{|}1+x_n}}$, with initial value Template:Tmath, and the continued radical  $$1/\varsigma =\sqrt{-1+\sqrt{1+\sqrt{-1+\sqrt{\phantom{|}1+\cdots }}}}$$

Its counterpart is found by using polynomial Template:Tmath, which has real zero Template:Tmath. Multiply Template:Tmath by Template:Tmath, then $(z^2-1{)}^2=z-1,$ and the corresponding iteration with Template:Tmath gives $$\varsigma -1=\sqrt{1+\sqrt{-1+\sqrt{1+\sqrt{\phantom{|}-1+\cdots }}}}$$

Supersilver Julia set

Julia set (in black) of the map Template:Math. Critical point Template:Math at the center of the image, critical value Template:Math on the left border, pole Template:Math at the cross, and fixed point Template:Math on the right hand side.

Both systems have linear convergence rate ${\log }_{10}\frac{4(\varsigma -1)}{\varsigma }=\mathrm{0.340...}$ In order to improve this constant, divide both sides of $z^3=z(1-z)+2\text{ by }z,$ and substitute $1/\varsigma =1/(z+1)$ for Template:Tmath, resulting in the iteration $z_{n+1}\leftarrow \sqrt{1+\frac{1}{1+z_n}},$ and the continued reciprocal square root $$\varsigma -1=\sqrt{1+\frac{1}{1+\sqrt{1+\frac{1}{1+\sqrt{1+\frac{1}{1+\ddots }}}}}}$$

For complex initial points Template:Tmath other than Template:Tmath this method converges with linear rate ${\log }_{10}2({\varsigma }^2+1)=\mathrm{1.069...},$ provided the principal root is chosen at each step. If randomly either the principal root or its negative is picked, the orbit of Template:Tmath is attracted to a simple closed curve, which is the Julia set Template:Tmath of the backward iteration $z_{n-1}\leftarrow \frac{1}{z_n^2-1}-1.$

The critical points Template:Tmath for which the derivative Template:Tmath vanishes are Template:Tmath and Template:Tmath. The latter is mapped into the right neighborhood of the pole Template:Tmath and vice versa, so Template:Tmath is the single attracting limit set.

On Template:Tmath, the repelling fixed points Template:Tmath are the zeros of Template:Tmath, namely Template:Tmath, and $(-\varsigma \pm i\sqrt{\frac{7-{\varsigma }^2}{\varsigma -1}})/2$ (the centers of the largest spirals in the left half of the image) with divergence rate $\frac{1}{2}{\log }_{10}\frac{8}{{\varsigma }^2+1}=\mathrm{0.067...}$ The first preimages of Template:Tmath are Template:Tmath, and the purely imaginary zeros Template:Tmath of Template:Tmath.

Properties

Rectangles with aspect ratios Template:Math, Template:Sfrac, and Template:Math tile the square.

The growth rate of the average value of the n-th term of a random Fibonacci sequence is Template:Tmath.

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

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

Similarly as the infinite geometric series $$\begin{array}{cc}\frac{\varsigma }{\varsigma -1}& =\sum _{n=0}^{\mathrm{\infty }}{\varsigma }^{-n}\\ \frac{{\varsigma }^2}{{\varsigma }^2-1}& =\sum _{n=0}^{\mathrm{\infty }}{\varsigma }^{-2n}\\ \frac{\varsigma }{2}& =\sum _{n=0}^{\mathrm{\infty }}{\varsigma }^{-3n},\end{array}$$

in comparison to the silver ratio identities $$\begin{array}{cc}\frac{\sigma }{\sigma -1}& =\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-n}\\ \frac{\sigma }{2}& =\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-2n}\\ \frac{{\sigma }^3}{{\sigma }^3-1}& =\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-3n}.\end{array}$$

For every integer Template:Tmath one has $$\begin{array}{cc}{\varsigma }^n& =2{\varsigma }^{n-1}+{\varsigma }^{n-3}\\ & =4{\varsigma }^{n-2}+{\varsigma }^{n-3}+2{\varsigma }^{n-4}\\ & ={\varsigma }^{n-1}+2{\varsigma }^{n-2}+{\varsigma }^{n-3}+{\varsigma }^{n-4}\end{array}$$ from this an infinite number of further relations can be found.

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

As derived from its continued fraction expansion, the simplest rational approximations of Template:Tmath are:$$\frac{9}{4},\frac{11}{5},\frac{53}{24},\frac{75}{34},\frac{161}{73},\frac{236}{107},\frac{397}{180},\frac{633}{287},\frac{1030}{467},\frac{1663}{754},\frac{2693}{1221},\frac{7049}{3196},\dots$$

Newton's method for Template:Math Template:Nowrap and its complex conjugates at the nuclei of their basins of attraction. Julia set of the Newton map in orange, with unit circle and real curve for reference.

The supersilver ratio is a Pisot number. By definition of these numbers, the absolute value $1/\sqrt{\varsigma }$ of the algebraic conjugates is smaller than 1, so powers of Template:Tmath generate almost integers. For example: ${\varsigma }^{10}=\mathrm{2724.00146856...}$ Template:Tmath. After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to Template:Tmath – nearly align with the imaginary axis.

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

The Weber-Ramanujan class invariant is approximated with error Template:Math by $$\sqrt{2}𝔣(\sqrt{\mathrm{\Delta }})=\sqrt[4]{2}{G}_{59}\approx (e^{\pi \sqrt{-\mathrm{\Delta }}}+24{)}^{1/24},$$ while its true value is the single real root of the polynomial $$W_{59}(x)=x^9-4x^8+4x^7-2x^6+4x^5-8x^4+4x^3-8x^2+16x-8.$$

The elliptic integral singular value  $k_r={\lambda }^{\ast }(r)\text{ for }r=59$ has closed form expression $${\lambda }^{\ast }(59)=\sin (\arcsin \left(G_{59}^{-12}\right)/2)$$ (which is less than 1/294 the eccentricity of the orbit of Venus).

Third-order Pell sequences

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 $$S_n=2S_{n-1}+S_{n-3}\text{ for }n>2,$$ with initial values $$S_0=1,S_1=2,S_2=4.$$

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... Template:OEIS. The limit ratio between consecutive terms is the supersilver ratio:$\underset{n\to \mathrm{\infty }}{lim}{S}_{n+1}/S_n=\varsigma .$

The first 8 indices n for which Template:Tmath 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 $$S_n=S_{n+3}-2S_{n+2}.$$

Powers of the supersilver ratio can be written with third-order Pell numbers as quadratic coefficients $${\varsigma }^n={\varsigma }^2{S}_{n-2}+\varsigma S_{n-4}+S_{n-3},$$ which is proved by mathematical induction on Template:Tmath. This relation also holds for Template:Tmath. The order of the coefficients corresponds to the bottom row of matrix Template:Tmath below.

The generating function of the sequence is given by  $$\frac{1}{1-2x-x^3}=\sum _{n=0}^{\mathrm{\infty }}{S}_n{x}^n\text{ for }x<\frac{1}{\varsigma }$$

The third-order Pell numbers are related to sums of binomial coefficients by  $$S_n=\sum _{k=0}^{\lfloor n/3\rfloor }(\genfrac{}{}{0ex}{}{n-2k}{k})\cdot 2^{n-3k}$$

The characteristic equation of the recurrence is $x^3-2x^2-1=0.$ If the three solutions are real root Template:Tmath and conjugate pair Template:Tmath and Template:Tmath, the supersilver numbers are given by the Binet formula $$S_{n-1}=a{\varsigma }^n+b{\beta }^n+c{\gamma }^n,$$ with real Template:Tmath and conjugates Template:Tmath and Template:Tmath the roots of $59x^3-6x-1=0.$

Since $|b{\beta }^n+c{\gamma }^n|<{\varsigma }^{-n/2}$, the number Template:Tmath is the nearest integer to Template:Tmath, with Template:Tmath and coefficient $a={\varsigma }^2/({\varsigma }^3+2)=$ Template:Gaps

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

The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... Template:OEIS.

This third-order Pell-Lucas sequence has the Fermat property: if p is prime, $A_p\equiv A_{1}modp.$ The converse does not hold, but the small number of odd pseudoprimes $n\mid (A_n-2)$ makes the sequence special. The 14 odd composite numbers below Template:Math to pass the test are n = 3Template:Sup, 5Template:Sup, 5Template:Sup, 315, 99297, 222443, 418625, 9122185, 3257Template:Sup, 11889745, 20909625, 24299681, 64036831, 76917325.

The Pilgrim: a supersilver Rauzy fractal of type Template:Nowrap The three subtiles have areas in ratio Template:Math

The third-order Pell numbers are obtained as integral powers Template:Math of a matrix with real eigenvalue Template:Tmath $$Q=\left(\begin{array}{ccc}2& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),$$

$$Q^n=\left(\begin{array}{ccc}{S}_n& S_{n-2}& S_{n-1}\\ S_{n-1}& S_{n-3}& S_{n-2}\\ S_{n-2}& S_{n-4}& S_{n-3}\end{array}\right)$$

The trace of Template:Tmath gives the above Template:Tmath.

Alternatively, Template:Tmath can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet Template:Tmath with corresponding substitution rule $$\{\begin{array}{c}a\mapsto aab\\ b\mapsto c\\ c\mapsto a\end{array}$$ and initiator Template:Tmath. The series of words Template:Tmath produced by iterating the substitution have the property that the number of Template:Math and Template:Math are equal to successive third-order Pell numbers. The lengths of these words are given by $l(w_n)=S_{n-2}+S_{n-3}+S_{n-4}.$

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.

Supersilver rectangle

Powers of Template:Math within a supersilver rectangle.

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

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

The parent supersilver rectangle and the two scaled copies along the diagonal have linear sizes in the ratios $\varsigma :\varsigma -1:1.$ The areas of the rectangles opposite the diagonal are both equal to $(\varsigma -1)/\varsigma ,$ with aspect ratios $\varsigma (\varsigma -1)$ (below) and $\varsigma /(\varsigma -1)$ (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 ${\varsigma }^2+1:{\varsigma }^2:{\varsigma }^2-1:\varsigma +1:$ $\varsigma (\varsigma -1):\varsigma :2/(\varsigma -1):1.$

Supersilver spiral

Supersilver spirals with different initial angles on a Template:Math− rectangle.

A supersilver spiral is a logarithmic spiral that gets wider by a factor of Template:Tmath for every quarter turn. It is described by the polar equation $r(\theta )=a\exp (k\theta ),$ with initial radius Template:Tmath and parameter $k=\frac{2}{\pi }\ln (\varsigma ).$ 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 $\varsigma (\varsigma -1)$ which are perpendicularly aligned and successively scaled by a factor Template:Tmath.

See also

Solutions of equations similar to $x^3=2x^2+1$:

• Silver ratio – the positive solution of the equation $x^2=2x+1$
• Golden ratio – the positive solution of the equation $x^2=x+1$
• Supergolden ratio – the real solution of the equation $x^3=x^2+1$