Supersilver Ratio

In mathematics, the supersilver ratio is a geometrical proportion, given by the unique real solution of the equation Its decimal expansion begins as .

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

Definition

Three quantities are in the supersilver ratio if
$\frac =\frac =\frac$
The ratio is commonly denoted

Substituting $a=\varsigma b \,$ and $c=(\varsigma -1)a =(\varsigma^2 -\varsigma)b \,$ in the third fraction,
$\varsigma =\frac.$ 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 $x^ +2x -1,$ thus the simplest solution with Cardano's formula,
\begin
w_ &=\left( 1 \pm \frac \sqrt \right) /2 \\
1 /\varsigma &=\sqrt +\sqrt \end
or, using the hyperbolic sine,
:$1 /\varsigma =-2 \sqrt\frac \sinh \left( \frac \operatorname \left( -\frac \sqrt\frac \right) \right).$

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

Rewrite the minimal polynomial as $(x^2+1)^2 =1+x$, then the iteration $x \gets \sqrt$ results in the continued radical
:$1/\varsigma =\sqrt \;$

Dividing the defining trinomial $x^ -2x^ -1$ by one obtains $x^ +x /\varsigma^2 +1 /\varsigma$, and the conjugate elements of are
$x_ = \left( -1 \pm i \sqrt \right) /2 \varsigma^2,$
with $x_1 +x_2 = 2 -\varsigma \;$ and $\; x_1x_2 =1 /\varsigma.$

Properties

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

The defining equation can be written
$\begin 1 &=\frac +\frac \\ &=\frac +\frac +\frac.\end$

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

Similarly as the infinite geometric series
$\begin \varsigma &=2\sum_^ \varsigma^ \\ \varsigma^2 &=-1 +\sum_^ (\varsigma -1)^,\end$

in comparison to the silver ratio identities
$\begin \sigma &=2\sum_^ \sigma^ \\ \sigma^2 &=-1 +2\sum_^ (\sigma -1)^.\end$

For every integer one has
$\begin \varsigma^ &=2\varsigma^ +\varsigma^ \\ &=4\varsigma^ +\varsigma^ +2\varsigma^ \\ &=\varsigma^ +2\varsigma^ +\varsigma^ +\varsigma^ \end$
From this an infinite number of further relations can be found.

Continued fraction pattern of a few low powers
\begin
\varsigma^ &= ;4,1,6,2,1,1,1,1,1,1,...\approx 0.2056 \;(\tfrac) \\
\varsigma^ &= ;2,4,1,6,2,1,1,1,1,1,...\approx 0.4534 \;(\tfrac) \\
\varsigma^0 &= \\
\varsigma^1 &= ;4,1,6,2,1,1,1,1,1,1,...\approx 2.2056 \;(\tfrac) \\
\varsigma^2 &= ;1,6,2,1,1,1,1,1,1,2,...\approx 4.8645 \;(\tfrac) \\
\varsigma^3 &= 0;1,2,1,2,4,4,2,2,6,2,...\approx 10.729 \;(\tfrac) \end

The simplest rational approximations of are:$\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,\tfrac,...$

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

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

The Weber-Ramanujan class invariant is approximated with error by
:\sqrt\,\mathfrak( \sqrt ) = \sqrt ,G_ \approx (e^ + 24)^,
while its true value is the single real root of the polynomial
:$W_(x) = x^9 -4x^8 +4x^7 -2x^6 +4x^5 -8x^4 +4x^3 -8x^2 +16x -8.$

The elliptic integral singular value $k_ =\lambda^(r) \text r =59$ has closed form expression
:$\lambda^(59) =\sin ( \arcsin \left( G_^ \right) /2)$
(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
$S_ =2S_ +S_ \text n > 2,$
with initial values $S_ =1, S_ =2, S_ =4.$

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... .
The limit ratio between consecutive terms is the supersilver ratio:$\lim_ S_/S_n =\varsigma.$

The first 8 indices n for which 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
$S_ =S_ -2S_.$

The generating function of the sequence is given by
:$\frac = \sum_^ S_x^ \text x <1 /\varsigma \;.$

The third-order Pell numbers are related to sums of binomial coefficients by
:$S_ =\sum_^ \cdot 2^ \;$.

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

Since $\left\vert b \beta^ +c \gamma^ \right\vert < 1 /\alpha^$ and $\alpha = \varsigma,$ the number is the nearest integer to $a\,\varsigma^,$ with and $a =\varsigma /( 2\varsigma^ +3) =$

Coefficients $a =b =c =1$ result in the Binet formula for the related sequence $A_ =S_ +2S_.$

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

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

The third-order Pell numbers are obtained as integral powers of a matrix with real eigenvalue $Q = \begin 2 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end ,$

$Q^ = \begin S_ & S_ & S_ \\ S_ & S_ & S_ \\ S_ & S_ & S_ \end$

The trace of gives the above

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

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

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

On the right-hand side, cut off a square of side length 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.Analogue to the construction in:

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

A supersilver spiral is a logarithmic spiral that gets wider by a factor of for every quarter turn. It is described by the polar equation $r( \theta) =a \exp(k \theta),$ with initial radius and parameter $k =\frac.$ 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 $x^ =2x^ +1$:
** Silver ratio – the only positive solution of the equation $x^=2x+1$
** Golden ratio – the only positive solution of the equation $x^=x+1$
** Supergolden ratio – the only real solution of the equation $x^=x^+1$

References

{{Algebraic numbers

Cubic irrational numbers
Mathematical constants
History of geometry
Integer sequences