Silver ratio


In mathematics, the silver ratio is a geometrical proportion with exact value the positive solution of the equation
The name silver ratio is by analogy with the golden ratio, the positive solution of the equation
Although its name is recent, the silver ratio has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry.

Definition

If the ratio of two quantities is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio:
The ratio is here denoted
Substituting in the second fraction,
It follows that the silver ratio is the positive solution of quadratic equation The quadratic formula gives the two solutions the decimal expansion of the positive root begins with .
Using the tangent function
or the hyperbolic sine
and its algebraic conjugate can be written as sums of eighth roots of unity:
which is guaranteed by the Kronecker–Weber theorem.
is the superstable fixed point of the Newton iteration
The iteration results in the continued radical

Properties

The defining equation can be written
The silver ratio can be expressed in terms of itself as fractions
Similarly as the infinite geometric series
For every integer one has
from this an infinite number of further relations can be found.
Continued fraction pattern of a few low powers
The silver ratio is a Pisot number, the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value of the algebraic conjugate is smaller than thus powers of generate almost integers and the sequence is dense at the borders of the unit interval.

Quadratic field \mathbb{Q}(√2)

is the fundamental unit of real quadratic field with discriminant The integers are the numbers with conjugate norm and trace
The first few positive numbers occurring as norm are 1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25. Arithmetic in the ring resembles that of the rational integers, i.e. the elements of Prime factorization is unique up to order and unit factors and there is a Euclidean function on the absolute value of the norm. The primes of are of three types:
  • with norm the single rational prime that divides
  • the factors of rational primes with norm
  • the rational primes with
and any one of these numbers multiplied by a unit.
The silver ratio can be used as base of a numeral system, here called the sigmary scale. Every real number in can be represented as a convergent series
with weights
Sigmary expansions are not unique. Due to the identities
digit blocks carry to the next power of resulting in The number one has finite and infinite representations and where the first of each pair is in canonical form. The algebraic number can be written or non-canonically as The decimal number and
Properties of canonical sigmary expansions, with coefficients
Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation with integer It follows by repeated substitution of that all positive solutions have a purely periodic continued fraction expansion
Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.
The silver ratio is related to the central Delannoy numbers = 1, 3, 13, 63, 321, 1683, 8989,... that count the number of "king walks" between one pair of opposite corners of a square lattice. The sequence has generating function
from which are obtained the integral representation
and asymptotic formula
For an application of the sigmary scale, consider the problem of writing a possible third-order coefficient in terms of the silver ratio. The decimal value of is approximately which can be found with the method of dominant balance using the recurrence relation for the central Delannoy numbers, with "The coefficients all lie in and have denominators equal to some power of the prime "
Choosing denominator the approximate numerator has sigmary expansion and is truncated to a quadratic integer by dropping all digits of order Write the remaining powers in linear form with Pell numbers as coefficients, take the weighted sum and simplify, giving term A certified value for is however as yet unknown.

Pell sequences

These numbers are related to the silver ratio as the Fibonacci numbers and Lucas numbers are to the golden ratio.
The fundamental sequence is defined by the recurrence relation
with initial values
The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,....
The limit ratio of consecutive terms is the silver mean.
Fractions of Pell numbers provide rational approximations of with error
The sequence is extended to negative indices using
Powers of can be written with Pell numbers as linear coefficients which is proved by mathematical induction on The relation also holds for
The generating function of the sequence is given by
File:Silver Newton map.svg |thumb|upright=1.33 |Newton's method for the silver ratio and its conjugate with perturbing complex roots 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 characteristic equation of the recurrence is with discriminant If the two solutions are silver ratio and conjugate so that the Pell numbers are computed with the Binet formula
with the positive root of
Since the number is the nearest integer to with and
The Binet formula defines the companion sequence
The first few terms are 2, 2, 6, 14, 34, 82, 198,....
This Pell-Lucas sequence has the Fermat property: if p is prime, The converse does not hold, the least odd pseudoprimes are 13, 385, 31, 1105, 1121, 3827, 4901.
Pell numbers are obtained as integral powers of a matrix with positive eigenvalue
The trace of gives the above

Geometry

Silver rectangle and regular octagon

A rectangle with edges in ratio can be created from a square piece of paper with an origami folding sequence. Considered a proportion of great harmony in Japanese aesthetics — Yamato-hi — the ratio is retained if the rectangle is folded in half, parallel to the short edges. Rabatment produces a rectangle with edges in the silver ratio.
  • Fold a square sheet of paper in half, creating a falling diagonal crease, then unfold.
  • Fold the right hand edge onto the diagonal crease.
  • Fold the top edge in half, to the back side, and open out the triangle. The result is a rectangle.
  • Fold the bottom edge onto the left hand edge. The horizontal part on top is a silver rectangle.
If the folding paper is opened out, the creases coincide with diagonal sections of a regular octagon. The first two creases divide the square into a silver gnomon with angles in the ratios between two right triangles with angles in ratios and . The unit angle is equal to degrees.
If the octagon has edge length its area is and the diagonals have lengths and The coordinates of the vertices are given by the permutations of The paper square has edge length and area The triangles have areas and the rectangles have areas

Silver whirl

Divide a rectangle with sides in ratio into four congruent right triangles with legs of equal length and arrange these in the shape of a silver rectangle, enclosing a similar rectangle that is scaled by factor and rotated about the centre by Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing a whirl of converging silver rectangles.
The logarithmic spiral through the vertices of adjacent triangles has polar slope The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio, hence is a silver rhombus.
If the triangles have legs of length then each discrete spiral has length The areas of the triangles in each spiral region sum to the perimeters are equal to and .
Arranging the tiles with the four hypotenuses facing inward results in the diamond-in-a-square shape. Roman architect Vitruvius recommended the implied ad quadratura ratio as one of three for proportioning a town house atrium. The scaling factor is and iteration on edge length gives an angular spiral of length

Polyhedra

The silver mean has connections to the following Archimedean solids with octahedral symmetry; all values are based on edge length
The coordinates of the vertices are given by 24 distinct permutations of thus three mutually-perpendicular silver rectangles touch six of its square faces.
The midradius is the centre radius for the square faces is
Coordinates: 24 permutations of
Midradius: centre radius for the octagon faces:
Coordinates: 48 permutations of
Midradius: centre radius for the square faces: for the octagon faces:
See also the dual Catalan solids
The acute isosceles triangle formed by connecting two adjacent vertices of a regular octagon to its centre point, is here called the silver triangle. It is uniquely identified by its angles in ratios The apex angle measures each base angle degrees. It follows that the height to base ratio is
By trisecting one of its base angles, the silver triangle is partitioned into a similar triangle and an obtuse silver gnomon. The trisector is collinear with a medium diagonal of the octagon. Sharing the apex of the parent triangle, the gnomon has angles of degrees in the ratios From the law of sines, its edges are in ratios
The similar silver triangle is likewise obtained by scaling the parent triangle in base to leg ratio, accompanied with an degree rotation. Repeating the process at decreasing scales results in an infinite sequence of silver triangles, which converges at the centre of rotation. It is assumed without proof that the centre of rotation is the intersection point of sequential median lines that join corresponding legs and base vertices.
The assumption is verified by construction, as demonstrated in the vector image.
The centre of rotation has barycentric coordinates
the three whorls of stacked gnomons have areas in ratios
The logarithmic spiral through the vertices of all nested triangles has polar slope
or an expansion rate of for every degrees of rotation.
circumcenter
centroid
nine-point center
incenter,
symmedian point
orthocenter

The long, medium and short diagonals of the regular octagon concur respectively at the apex, the circumcenter and the orthocenter of a silver triangle.