Silver ratio

Template:Short description Template:Infobox non-integer number In mathematics, the silver ratio is a geometrical proportion with exact value Template:Math the positive solution of the equation Template:Math

The name silver ratio is by analogy with the golden ratio, the positive solution of the equation Template:Math

Although its name is recent, the silver ratio (or silver mean) 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.

File:Silver rectangle in octagon.svg

Silver rectangle in a regular octagon.

Definition

If the ratio of two quantities Template:Math is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: $$\frac{a}{b}=\frac{2a+b}{a}$$ The ratio $\frac{a}{b}$ is here denoted Template:TmathTemplate:Efn

Substituting $a=\sigma b$ in the second fraction, $$\sigma =\frac{b(2\sigma +1)}{\sigma b}.$$ It follows that the silver ratio is the positive solution of quadratic equation ${\sigma }^2-2\sigma -1=0.$ The quadratic formula gives the two solutions $1\pm \sqrt{2},$ the decimal expansion of the positive root begins with Template:Math (sequence A014176 in the OEIS).

Using the tangent function

$\sigma =\tan \left(\frac{3\pi }{8}\right)=\cot \left(\frac{\pi }{8}\right),$

or the hyperbolic sine

$\sigma =\exp (\mathrm{arsinh}(1)).$

Template:Tmath is the superstable fixed point of the iteration $x\leftarrow \frac{1}{2}(x^2+1)/(x-1),\text{ with }{x}_0\in [2,3]$

The iteration $x\leftarrow \sqrt{1+2x\phantom{/}}$ results in the continued radical $$\sigma =\sqrt{1+2\sqrt{1+2\sqrt{1+\cdots }}}.$$

Properties

File:SilverSquare 6.svg

Rectangles with aspect ratios related to Template:Math tile the square.

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

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

Similarly as the infinite geometric series $$\begin{array}{rl}\sigma & =2\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-2n}\\ {\sigma }^2& =-1+2\sum _{n=0}^{\mathrm{\infty }}(\sigma -1{)}^{-n}.\end{array}$$

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

Continued fraction pattern of a few low powers $$\begin{array}{rl}{\sigma }^{-1}& =[0;2,2,2,2,...]\approx 0.4142(17/41)\\ {\sigma }^0& =[1]\\ {\sigma }^1& =[2;2,2,2,2,...]\approx 2.4142(70/29)\\ {\sigma }^2& =[5;1,4,1,4,...]\approx 5.8284(5+29/35)\\ {\sigma }^3& =[14;14,14,14,...]\approx 14.0711(14+1/14)\\ {\sigma }^4& =[33;1,32,1,32,...]\approx 33.9706(33+33/34)\\ {\sigma }^5& =[82;82,82,82,...]\approx 82.0122(82+1/82)\end{array}$$

${\sigma }^{-n}\equiv (-1{)}^{n-1}{\sigma }^{n}mod1.$

The silver ratio is a Pisot number, the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value $\sqrt{2}-1$ of the algebraic conjugate is smaller than Template:Math thus powers of Template:Tmath generate almost integers and the sequence ${\sigma }^{n}mod1$ is dense at the borders of the unit interval.

Template:Tmath is the fundamental unit of real quadratic field $K=\mathbb{\mathbb{Q}}\left(\sqrt{2}\right).$

The silver ratio can be used as base of a numeral system, here called the sigmary scale.Template:Efn Every real number Template:Math in Template:Math can be represented as a convergent series

$x=\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{{\sigma }^n},$ with weights Template:Tmath

File:Sigmary scale.svg

The steps in the sigmary scale resemble the intervals of the mixolydian mode in log scale. Progression to the next octave is paralleled by the carry in 21 and 22.

Sigmary expansions are not unique. Due to the identities $$\begin{array}{rl}{\sigma }^{n+1}& =2{\sigma }^n+{\sigma }^{n-1}\\ {\sigma }^{n+1}+{\sigma }^{n-1}& =2{\sigma }^n+2{\sigma }^{n-1},\end{array}$$ digit blocks $21_{\sigma }\text{ and }22_{\sigma }$ carry to the next power of Template:Tmath resulting in $100_{\sigma }\text{ and }101_{\sigma }.$ The number one has finite and infinite representations $1.0_{\sigma },0.21_{\sigma }$ and $0.{\overline{20}}_{\sigma },0.1{\bar{2}}_{\sigma },$ where the first of each pair is in canonical form. The algebraic number Template:Tmath can be written Template:Tmath or non-canonically as Template:Tmath The decimal number $10=111.12_{\sigma },$ $7\sigma +3=1100_{\sigma }$ and $\frac{1}{\sigma -1}=0.{\bar{1}}_{\sigma }.$

Properties of canonical sigmary expansions, with coefficients $a,b,c,d\in \mathbb{\mathbb{Z}}:$

• Every algebraic integer $\xi =a+b\sigma \text{ in }K$ has a finite expansion.
• Every rational number $\rho =\frac{a+b\sigma }{c+d\sigma }\text{ in }K$ has a purely periodic expansion.
• All numbers that do not lie in Template:Tmath have chaotic expansions.

1. REDIRECT Template:Break

Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation Template:Tmath with integer Template:Math It follows by repeated substitution of Template:Tmath that all positive solutions $\frac{1}{2}(n+\sqrt{n^2+4\phantom{/}})$ have a purely periodic continued fraction expansion $${\sigma }_n=n+\frac{1}{n+\frac{1}{n+\frac{1}{\ddots }}}$$ Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.

Pell sequences

File:SilverWord im.png

Silver harmonics: the rectangle and its coloured subzones have areas in ratios Template:Math

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 $$P_n=2P_{n-1}+P_{n-2}\text{ for }n>1,$$ with initial values $$P_0=0,P_1=1.$$

The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,... (sequence A000129 in the OEIS). The limit ratio of consecutive terms is the silver mean.

Fractions of Pell numbers provide rational approximations of Template:Tmath with error $$|\sigma -\frac{P_{n+1}}{P_n}|<\frac{1}{\sqrt{8}{P}_n^2}$$

The sequence is extended to negative indices using $$P_{-n}=(-1{)}^{n-1}{P}_n.$$

Powers of Template:Tmath can be written with Pell numbers as linear coefficients $${\sigma }^n=\sigma P_n+P_{n-1},$$ which is proved by mathematical induction on Template:Math The relation also holds for Template:Math

The generating function of the sequence is given by

$\frac{x}{1-2x-x^2}=\sum _{n=0}^{\mathrm{\infty }}{P}_n{x}^n\text{ for }|x|<1/\sigma .$

The characteristic equation of the recurrence is $x^2-2x-1=0$ with discriminant Template:Tmath If the two solutions are silver ratio Template:Tmath and conjugate Template:Tmath so that $\sigma +\overline{\sigma }=2\text{ and }\sigma \cdot \overline{\sigma }=-1,$ the Pell numbers are computed with the Binet formula $$P_n=a({\sigma }^n-{\overline{\sigma }}^n),$$ with Template:Tmath the positive root of $8x^2-1=0.$

Since $\left|a{\overline{\sigma }}^n\right|<1/{\sigma }^{2n},$ the number Template:Tmath is the nearest integer to $a{\sigma }^n,$ with $a=1/\sqrt{8}$ and Template:Math

The Binet formula ${\sigma }^n+{\overline{\sigma }}^n$ defines the companion sequence $Q_n=P_{n+1}+P_{n-1}.$

The first few terms are 2, 2, 6, 14, 34, 82, 198,... (sequence A002203 in the OEIS).

This Pell-Lucas sequence has the Fermat property: if p is prime, $Q_p\equiv Q_{1}modp.$ The converse does not hold, the least odd pseudoprimes $n\mid (Q_n-2)$ are 13², 385, 31², 1105, 1121, 3827, 4901. Template:Efn

Pell numbers are obtained as integral powers Template:Math of a matrix with positive eigenvalue Template:Tmath $$M=\left(\begin{array}{cc}2& 1\\ 1& 0\end{array}\right),$$

$$M^n=\left(\begin{array}{cc}{P}_{n+1}& P_n\\ P_n& P_{n-1}\end{array}\right)$$

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

Geometry

Silver rectangle and regular octagon

File:Silver rectangle construction.svg

Origami construction of a silver rectangle, with creases in green.

A rectangle with edges in ratio Template:Math 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 (大和比) — the ratio is retained if the Template:Math rectangle is folded in half, parallel to the short edges. Rabatment produces a rectangle with edges in the silver ratio (according to Template:Math). Template:Efn

• Fold a square sheet of paper in half, creating a falling diagonal crease (bisect 90° angle), then unfold.
• Fold the right hand edge onto the diagonal crease (bisect 45° angle).
• Fold the top edge in half, to the back side (reduce width by Template:Sfrac), and open out the triangle. The result is a Template:Math rectangle.
• Fold the bottom edge onto the left hand edge (reduce height by Template:Sfrac). 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 Template:Math between two right triangles with angles in ratios Template:Math (left) and Template:Math (right). The unit angle is equal to Template:Math degrees.

If the octagon has edge length Template:Tmath its area is Template:Tmath and the diagonals have lengths $\sqrt{\sigma +1\phantom{/}},\sigma$ and $\sqrt{2(\sigma +1)\phantom{/}}.$ The coordinates of the vertices are given by the Template:Math permutations of $(\pm \frac{1}{2},\pm \frac{\sigma }{2}).$ The paper square has edge length Template:Tmath and area Template:Tmath The triangles have areas $1,\frac{\sigma -1}{\sigma }$ and $\frac{1}{\sigma };$ the rectangles have areas $\sigma -1\text{ and }\frac{1}{\sigma }.$

Silver whirl

File:Silver rectangle whirl.svg

A whirl of silver rectangles.

Divide a rectangle with sides in ratio Template:Math 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 Template:Tmath and rotated about the centre by Template:Tmath 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 $k=\frac{4}{\pi }\ln (\sigma ).$ The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio Template:Tmath, hence is a silver rhombus.

If the triangles have legs of length Template:Tmath then each discrete spiral has length $\frac{\sigma }{\sigma -1}=\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-n}.$ The areas of the triangles in each spiral region sum to $\frac{\sigma }{4}=\frac{1}{2}\sum _{n=0}^{\mathrm{\infty }}{\sigma }^{-2n};$ the perimeters are equal to Template:Tmath (light grey) and Template:Tmath (silver regions).

Arranging the tiles with the four hypotenuses facing inward results in the diamond-in-a-square shape. Ancient Roman tile work. Roman architect Vitruvius recommended the implied ad quadratura ratio as one of three for proportioning a town house atrium. The scaling factor is Template:Tmath and iteration on edge length Template:Math gives an angular spiral of length Template:Tmath

Polyhedra

File:Rhombicuboctahedron by Cutting Rhombic Dodecahedron.svg

Dimensions of the rhombi­cuboctahedron are linked to Template:Math

The silver mean has connections to the following Archimedean solids with octahedral symmetry; all values are based on edge length Template:Math

• Rhombicuboctahedron

The coordinates of the vertices are given by 24 distinct permutations of $(\pm \sigma ,\pm 1,\pm 1),$ thus three mutually-perpendicular silver rectangles touch six of its square faces.

1. REDIRECT Template:Break

The midradius is $\sqrt{2(\sigma +1)\phantom{/}},$ the centre radius for the square faces is Template:Tmath

• Truncated cube

Coordinates: 24 permutations of $(\pm \sigma ,\pm \sigma ,\pm 1).$

1. REDIRECT Template:Break

Midradius: Template:Tmath centre radius for the octagon faces: Template:Tmath

• Truncated cuboctahedron

Coordinates: 48 permutations of $(\pm (2\sigma -1),\pm \sigma ,\pm 1).$

1. REDIRECT Template:Break

Midradius: $\sqrt{6(\sigma +1)\phantom{/}},$ centre radius for the square faces: Template:Tmath for the octagon faces: Template:Tmath

See also the dual Catalan solids

• Tetragonal trisoctahedron
• Trisoctahedron
• Hexakis octahedron

Silver triangle

File:Silver triangle spiral.svg

Silver triangle and whirling gnomons.

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 Template:Tmath The apex angle measures Template:Tmath each base angle Template:Tmath degrees. It follows that the height to base ratio is $\frac{1}{2}\tan (67\frac{1}{2})=\frac{\sigma }{2}.$

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 $67\frac{1}{2}/3=22\frac{1}{2},45\text{ and }112\frac{1}{2}$ degrees in the ratios Template:Tmath From the law of sines, its edges are in ratios $1:\sqrt{\sigma +1}:\sigma .$

The similar silver triangle is likewise obtained by scaling the parent triangle in base to leg ratio Template:Tmath, accompanied with an Template:Tmath 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 $$(\frac{\sigma +1}{\sigma +5}:\frac{2}{\sigma +5}:\frac{2}{\sigma +5})\sim (\frac{\sigma +1}{2}:1:1),$$ the three whorls of stacked gnomons have areas in ratios $${\left(\frac{\sigma +1}{2}\right)}^2:\frac{\sigma +1}{2}:1.$$

The logarithmic spiral through the vertices of all nested triangles has polar slope

$k=\frac{4}{5\pi }\ln \left(\frac{\sigma }{\sigma -1}\right),$ or an expansion rate of Template:Tmath for every Template:Tmath degrees of rotation.

Silver triangle centers: affine coordinates on the axis of symmetry

circumcenter	$(\frac{2}{\sigma +1}:\frac{1}{\sigma })\sim (\sigma -1:1)$
centroid	$(\frac{2}{3}:\frac{1}{3})\sim (2:1)$
nine-point center	$(\frac{1}{\sigma -1}:\frac{1}{\sigma +1})\sim (\sigma :1)$
incenter, Template:Math	$([1+\cos (\alpha ){]}^{-1}:[1+\sec (\alpha ){]}^{-1})\sim (\sec (\alpha ):1)$
symmedian point	$(\frac{\sigma +1}{\sigma +2}:\frac{1}{\sigma +2})\sim (\sigma +1:1)$
orthocenter	$(\frac{2}{\sigma }:\frac{1}{{\sigma }^2})\sim (2\sigma :1)$

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

Silver rectangle and silver triangle

File:Silver rectangle segmented.svg

Powers of Template:Math within a silver rectangle.

Assume a silver rectangle has been constructed as indicated above, with height Template:Math, length Template:Tmath and diagonal length $\sqrt{{\sigma }^2+1}$. The triangles on the diagonal have altitudes $1/\sqrt{1+{\sigma }^{-2}};$ each perpendicular foot divides the diagonal in ratio Template:Tmath

If an horizontal line is drawn through the intersection point of the diagonal and the internal edge of a rabatment square, the parent silver rectangle and the two scaled copies along the diagonal have areas in the ratios ${\sigma }^2:2:1,$ the rectangles opposite the diagonal both have areas equal to $\frac{2}{\sigma +1}.$

Relative to vertex Template:Math, the coordinates of feet of altitudes Template:Math and Template:Math are $$(\frac{\sigma }{{\sigma }^2+1},\frac{1}{{\sigma }^2+1})\text{ and }(\frac{\sigma }{1+{\sigma }^{-2}},\frac{1}{1+{\sigma }^{-2}}).$$

If the diagram is further subdivided by perpendicular lines through Template:Math and Template:Math, the lengths of the diagonal and its subsections can be expressed as trigonometric functions of argument $\alpha =67\frac{1}{2}$ degrees, the base angle of the silver triangle:

File:Silver triangle.svg

Diagonal segments of the silver rectangle measure the silver triangle. The ratio Template:Math is Template:Math

$$\begin{array}{rl}\overline{AB}=\sqrt{{\sigma }^2+1}& =\sec (\alpha )\\ \overline{AV}={\sigma }^2/\overline{AB}& =\sigma \sin (\alpha )\\ \overline{UV}=2/\overline{AS}& =2\sin (\alpha )\\ \overline{SB}=4/\overline{AB}& =4\cos (\alpha )\\ \overline{SV}=3/\overline{AB}& =3\cos (\alpha )\\ \overline{AS}=\sqrt{1+{\sigma }^{-2}}& =\csc (\alpha )\\ \bar{h}=1/\overline{AS}& =\sin (\alpha )\\ \overline{US}=\overline{AV}-\overline{SB}& =(2\sigma -3)\cos (\alpha )\\ \overline{AU}=1/\overline{AB}& =\cos (\alpha ),\end{array}$$

with Template:Tmath

Both the lengths of the diagonal sections and the trigonometric values are elements of biquadratic number field $K=\mathbb{\mathbb{Q}}\left(\sqrt{2+\sqrt{2}}\right).$

The silver rhombus with edge Template:Tmath has diagonal lengths equal to Template:Tmath and Template:Tmath The regular octagon with edge Template:Tmath has long diagonals of length Template:Tmath that divide it into eight silver triangles. Since the regular octagon is defined by its side length and the angles of the silver triangle, it follows that all measures can be expressed in powers of Template:Math and the diagonal segments of the silver rectangle, as illustrated above, pars pro toto on a single triangle.

The leg to base ratio Template:Tmath has been dubbed the Cordovan proportion by Spanish architect Rafael de la Hoz Arderius. According to his observations, it is a notable measure in the architecture and intricate decorations of the mediæval Mosque of Córdoba, Andalusia.

Silver spiral

File:Silver spiral.svg

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

A silver 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 $k=\frac{2}{\pi }\ln (\sigma ).$ If drawn on a silver rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of paired squares which are perpendicularly aligned and successively scaled by a factor $1/\sigma .$

1. REDIRECT Template:Break

Ammann–Beenker tiling

File:AmmanSubstRules.png

Patch inflation of Ammann A5-tiles with factor Template:Math

The silver ratio appears prominently in the Ammann–Beenker tiling, a non-periodic tiling of the plane with octagonal symmetry, build from a square and silver rhombus with equal side lengths. Discovered by Robert Ammann in 1977, its algebraic properties were described by Frans Beenker five years later. If the squares are cut into two triangles, the inflation factor for Ammann A5-tiles is Template:Tmath the dominant eigenvalue of substitution matrix $$M=\left(\begin{array}{cc}3& 2\\ 4& 3\end{array}\right).$$

See also

• Solutions of equations similar to $x^2=2x+1$:
  • Golden ratio – the real positive solution of the equation $x^2=x+1$
  • Metallic means – real positive solutions of the general equation $x^2=nx+1$
  • Supersilver ratio – the only real solution of the equation $x^3=2x^2+1$

External links

• YouTube lecture on the silver ratio, Pell sequence and metallic means
• Silver rectangle and Pell sequence at Tartapelago by Giorgio Pietrocola