Supersilver ratio

Template:Short description Template:Infobox non-integer number In mathematics, the supersilver ratio is a geometrical proportion close to Template:Math. Its true value is the real solution of the equation Template:Math

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

Definition

Two quantities Template:Math are in the supersilver ratio-squared if $${\left(\frac{2a+b}{a}\right)}^2=\frac{a}{b}.$$ The ratio $\frac{2a+b}{a}$ is here denoted Template:Tmath

Based on this definition, one has $$\begin{array}{cc}1& ={\left(\frac{2a+b}{a}\right)}^2\frac{b}{a}\\ & ={\left(\frac{2a+b}{a}\right)}^2(\frac{2a+b}{a}-2)\\ & ⟹{\varsigma }^2(\varsigma -2)=1\end{array}$$

It follows that the supersilver ratio is found as the unique real solution of the cubic equation ${\varsigma }^3-2{\varsigma }^2-1=0.$ The decimal expansion of the root begins as $2.205569430400590...$ Template:OEIS.

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 $x\leftarrow (2x^3+1)/(3x^2+2).$

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

$1/\varsigma =\sqrt{-1+\sqrt{1+\sqrt{-1+\sqrt{1+\cdots }}}}$

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 $$x_{1,2}=(-1\pm i\sqrt{8{\varsigma }^2+3})/2{\varsigma }^2,$$ with $x_1+x_2=2-\varsigma$ and $x_1{x}_2=1/\varsigma .$

Properties

Rectangles with aspect ratios related to powers of 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 $$\begin{array}{cc}\varsigma & =2\sum _{n=0}^{\mathrm{\infty }}{\varsigma }^{-3n}\\ {\varsigma }^2& =-1+\sum _{n=0}^{\mathrm{\infty }}(\varsigma -1{)}^{-n},\end{array}$$

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

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 $$\begin{array}{cc}{\varsigma }^{-2}& =[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056(5/24)\\ {\varsigma }^{-1}& =[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534(5/11)\\ {\varsigma }^0& =[1]\\ {\varsigma }^1& =[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056(53/24)\\ {\varsigma }^2& =[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645(73/15)\\ {\varsigma }^3& =[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729(118/11)\end{array}$$

The supersilver ratio is a Pisot number. Because the absolute value $1/\sqrt{\varsigma }$ of the algebraic conjugates is smaller than 1, powers of Template:Tmath generate almost integers. For example: ${\varsigma }^{10}=2724.00146856...\approx 2724+1/681.$ 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 $k_r={\lambda }^{*}(r)\text{ for }r=59$ has closed form expression

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

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

Third-order Pell sequences

Hop o' my Thumb: a supersilver Rauzy fractal of type Template:Nowrap The fractal boundary has box-counting dimension 1.22

A supersilver Rauzy fractal of type Template:Nowrap with areas in the ratios Template:Math

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.

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 $$S_n=S_{n+3}-2S_{n+2}.$$

The generating function of the sequence is given by

$\frac{1}{1-2x-x^3}=\sum _{n=0}^{\mathrm{\infty }}{S}_n{x}^n\text{ for }x<1/\varsigma .$

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 can be computed with the Binet formula

$S_{n-2}=a{\alpha }^n+b{\beta }^n+c{\gamma }^n,$ with real Template:Tmath and conjugates Template:Tmath and Template:Tmath the roots of $59x^3+4x-1=0.$

Since $|b{\beta }^n+c{\gamma }^n|<1/{\alpha }^{n/2}$ and $\alpha =\varsigma ,$ the number Template:Tmath is the nearest integer to $a{\varsigma }^{n+2},$ with Template:Math and $a=\varsigma /(2{\varsigma }^2+3)=$ 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 $k=\frac{2\ln (\varsigma )}{\pi }.$ 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 $\varsigma (\varsigma -1)$ which are perpendicularly aligned and successively scaled by a factor $1/\varsigma .$

See also

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