Supersilver ratio
In mathematics, the supersilver ratio is a geometrical proportion, given by the unique real solution of the equation Its decimal expansion begins with .
The name supersilver ratio is by analogy with the silver ratio, the positive solution of the equation, and the supergolden ratio.
Definition
Three quantities are in the supersilver ratio ifThis ratio is commonly denoted.
Substituting in the first fraction gives
It follows that the supersilver ratio is the unique real solution of the cubic equation
The minimal polynomial for the reciprocal root is the depressed cubic thus the simplest solution with Cardano's formula,
or, using the hyperbolic sine,
is the superstable fixed point of the iteration
Rewrite the minimal polynomial as ; then the iteration results in the continued radical
Dividing the defining trinomial by one obtains and the conjugate elements of are
with and
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
The supersilver ratio can be expressed in terms of itself as fractions
Similarly as the infinite geometric series
in comparison to the silver ratio identities
For every integer one has
from this an infinite number of further relations can be found.
Continued fraction pattern of a few low powers
As derived from its continued fraction expansion, the simplest rational approximations of are:
File:Supersilver Newton map.svg |thumb|upright=1.25 |Newton's method for 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 of the algebraic conjugates is smaller than 1, so powers of generate almost integers.
For example: 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 has discriminant and factors into the imaginary quadratic field has class number Thus, the Hilbert class field of can be formed by adjoining
With argument a generator for the ring of integers of, the real root J-invariant| of the Hilbert class polynomial is given by
The Weber-Ramanujan class invariant is approximated with error by while its true value is the single real root of the polynomial
The elliptic integral singular value has closed form expression
.
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
with initial values
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:
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
The generating function of the sequence is given by
The third-order Pell numbers are related to sums of binomial coefficients by
The characteristic equation of the recurrence is If the three solutions are real root and conjugate pair and, the supersilver numbers can be computed with the Binet formula
with real and conjugates and the roots of
Since and the number is the nearest integer to with and
Coefficients result in the Binet formula for the related sequence
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, The converse does not hold, but the small number of odd pseudoprimes 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
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
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
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 The triangles on the diagonal have altitudes 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 . 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 The areas of the rectangles opposite the diagonal are both equal to with aspect ratios and .
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