Silver ratio

Not to be confused with Silver constant.

In mathematics, the silver ratio is a geometrical proportion close to 70/29. Its exact value is 1 + √2, the positive solution of the equation x² = 2x + 1.

Silver ratio

Rationality	irrational algebraic
Symbol	σ
Representations
Decimal	2.4142135623730950488016887...
Algebraic form	positive root of x2 = 2x + 1
Continued fraction (linear)	[2;2,2,2,2,2,...] purely periodic infinite

The name silver ratio results from analogy with the golden ratio, the positive solution of the equation x² = x + 1.

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.

Silver rectangle in a regular octagon.

Definition

If the ratio of two quantities a > b > 0 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 ⁠$\sigma .$⁠

Based on this definition, one has $${\begin{aligned}1&=\left({\frac {2a+b}{a}}\right){\frac {b}{a}}\\&=\left({\frac {2a+b}{a}}\right)\left({\frac {2a+b}{a}}-2\right)\\&\implies \sigma \left(\sigma -2\right)=1\end{aligned}}$$

It follows that the silver ratio is found as the positive solution of the 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 as ⁠$2.414\,213\,562\,373\,095...$⁠ (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

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

⁠$\sigma$⁠ is the superstable fixed point of the iteration $x\gets {\tfrac {1}{2}}(x^{2}+1)/(x-1),{\text{ with }}x_{0}\in [2,3]$

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

Properties

Rectangles with aspect ratios related to σ tile the square.

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

The silver ratio can be expressed in terms of itself as fractions $${\begin{aligned}\sigma &={\frac {\sigma +1}{\sigma -1}},{\text{ with }}{\sqrt {2}}={\frac {\sigma +1}{\sigma }}{\text{ and }}{\frac {\sigma }{\sqrt {2}}}={\frac {\sigma }{\sigma -1}}\\\sigma ^{2}&={\frac {\sigma }{\sigma -2}},{\text{ with }}2\sigma -3={\frac {\sigma +2}{\sigma }}{\text{ and }}4\sigma +1={\frac {\sigma +2}{\sigma -2}}.\end{aligned}}$$

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

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

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

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

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 1, thus powers of ⁠$\sigma$⁠ generate almost integers and the sequence $\sigma ^{n}{\bmod {1}}$ is dense at the borders of the unit interval.

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

If the general quadratic equation ⁠$x^{2}=nx+1$⁠ with integer n > 0 is written as ⁠$x=n+{\frac {1}{x}},$⁠ it follows by repeated substitution that all positive solutions ${\tfrac {1}{2}}\left(n+{\sqrt {n^{2}+4{\vphantom {/}}}}\right)$ have a purely periodic continued fraction expansion $$\sigma _{n}=n+{\cfrac {1}{n+{\cfrac {1}{n+{\cfrac {1}{\ddots }}}}}}$$ Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.

Pell sequences

Silver harmonics: the rectangle and its coloured subzones have areas in ratios 7σ + 3 : σ³ : σ² : σ : 1.

Main article: Pell number

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 ⁠$\sigma$⁠ with error $\left\vert \sigma -{\frac {P_{n+1}}{P_{n}}}\right\vert <{\frac {1}{{\sqrt {8}}P_{n}^{2}}}$

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

Powers of ⁠$\sigma$⁠ can be written with Pell numbers as linear coefficients $$\sigma ^{n}=\sigma P_{n}+P_{n-1},$$ which is proved by mathematical induction on n. The relation also holds for n < 0.

The generating function of the sequence is given by

${\frac {x}{1-2x-x^{2}}}=\sum _{n=0}^{\infty }P_{n}x^{n}{\text{ for }}\vert x\vert <1/\sigma \;.$

The characteristic equation of the recurrence is $x^{2}-2x-1=0$ with discriminant ⁠$D=8.$⁠ If the two solutions are silver ratio ⁠$\sigma$⁠ and conjugate ⁠${\bar {\sigma }},$⁠ so that $\sigma +{\bar {\sigma }}=2\;{\text{ and }}\;\sigma \cdot {\bar {\sigma }}=-1,$ the Pell numbers are computed with the Binet formula

$P_{n}=a(\sigma ^{n}-{\bar {\sigma }}^{n}),$ with ⁠$a$⁠ the positive root of $8x^{2}-1=0.$

Since $\left\vert a\,{\bar {\sigma }}^{n}\right\vert <1/\sigma ^{2n},$ the number ⁠$P_{n}$⁠ is the nearest integer to $a\,\sigma ^{n},$ with $a=1/{\sqrt {8}}$ and n ≥ 0.

The Binet formula $\sigma ^{n}+{\bar {\sigma }}^{n}$ defines the related 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 (or companion Pell) sequence has the Fermat property: if p is prime, $Q_{p}\equiv Q_{1}{\bmod {p}}.$ The converse does not hold, the least odd pseudoprimes $\,n\mid (Q_{n}-2)$ are 13², 385, 31², 1105, 1121, 3827, 4901.

Pell numbers are obtained as integral powers n > 2 of a matrix with positive eigenvalue ⁠$\sigma$⁠ $$M={\begin{pmatrix}2&1\\1&0\end{pmatrix}},$$

$$M^{n}={\begin{pmatrix}P_{n+1}&P_{n}\\P_{n}&P_{n-1}\end{pmatrix}}$$

The trace of ⁠$M^{n}$⁠ gives the above ⁠$Q_{n}.$⁠

Geometry

Silver rectangle and regular octagon

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

A rectangle with edges in a ratio of √2 : 1 can be created from a square piece of paper with a basic origami folding sequence. Considered a proportion of great harmony in Japanese aesthetics — Yamato-hi (大和比) — the ratio is retained if the √2 rectangle is folded in half, parallel to the short edges. Rabatment produces a rectangle with edges in the silver ratio (according to ⁠1/σ⁠ = √2 − 1).

• 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 ⁠1/σ + 1⁠), and open out the triangle. The result is a √2 rectangle.
• Fold the bottom edge onto the left hand edge (reduce height by ⁠1/σ − 1⁠). The horizontal piece 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 5 : 2 : 1, between two right triangles with angles in ratios 4 : 2 : 2 (left) and 4 : 3 : 1 (right). The unit angle is equal to 22.5 degrees.

If the octagon has edge length ⁠$1,$⁠ its area is ⁠$2\sigma$⁠ and the diagonals have lengths ${\sqrt {\sigma +1{\vphantom {/}}}},\;\sigma$ and ${\sqrt {2(\sigma +1){\vphantom {/}}}}.$ The coordinates of the vertices are given by the 8 permutations of $\left(\pm {\tfrac {1}{2}},\pm {\tfrac {\sigma }{2}}\right).$ The paper square has edge length ⁠$\sigma -1$⁠ and area ⁠$2.$⁠ 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

A whirl of silver rectangles.

Divide a rectangle with sides in ratio 1 : 2 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 ⁠${\tfrac {1}{\sigma }}$⁠ and rotated about the centre by ⁠${\tfrac {\pi }{4}}.$⁠ 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 $k={\frac {4\ln(\sigma )}{\pi }}.$ The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio ⁠$\sigma$⁠, hence is a silver rhombus.

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

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 ⁠${\tfrac {1}{\sigma -1}},$⁠ and iteration on edge length 2 gives an angular spiral of length ⁠$\sigma +1.$⁠

Polyhedra

Dimensions of the rhombi­cuboctahedron are linked to σ.

The silver mean has connections to the following Archimedean solids with octahedral symmetry; all values are based on edge length = 2.

• 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.
The midradius is ${\sqrt {2(\sigma +1){\vphantom {/}}}},$ the centre radius for the square faces is ⁠$\sigma .$⁠

• Truncated cube

Coordinates: 24 permutations of $(\pm \sigma ,\pm \sigma ,\pm 1).$
Midradius: ⁠$\sigma +1,$⁠ centre radius for the octagon faces: ⁠$\sigma .$⁠

• Truncated cuboctahedron

Coordinates: 48 permutations of $(\pm (2\sigma -1),\pm \sigma ,\pm 1).$
Midradius: ${\sqrt {6(\sigma +1){\vphantom {/}}}},$ centre radius for the square faces: ⁠$\sigma +2,$⁠ for the octagon faces: ⁠$2\sigma -1.$⁠

See also the dual Catalan solids

• Tetragonal trisoctahedron
• Trisoctahedron
• Hexakis octahedron

Silver rectangle and silver triangle

Powers of σ within a silver rectangle.

Assume a silver rectangle has been constructed as indicated above, with height 1, length ⁠$\sigma$⁠ 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 ⁠$\sigma ^{2}.$⁠

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

Relative to vertex A, the coordinates of feet of altitudes U and V are $\left({\frac {\sigma }{\sigma ^{2}+1}},{\frac {1}{\sigma ^{2}+1}}\right),$ $\left({\frac {\sigma }{1+\sigma ^{-2}}},{\frac {1}{1+\sigma ^{-2}}}\right).$

If the diagram is further subdivided by perpendicular lines through U and V, the lengths of the diagonal and its subsections can be expressed as trigonometric functions of argument ⁠$\alpha =67.5$⁠ degrees. This is the base angle of an isosceles triangle formed by connecting two adjacent vertices of a regular octagon to its centre point; here called the silver triangle.

Diagonal segments of the silver rectangle measure the silver triangle. The ratio AB:AS is σ.

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

with ⁠$\sigma =\tan(\alpha ).$⁠

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

The silver rhombus with edge ⁠$1$⁠ has diagonal lengths equal to ⁠${\overline {UV}}$⁠ and ⁠$2{\overline {AU}}.$⁠ The regular octagon with edge ⁠$2$⁠ has long diagonals of length ⁠$2{\overline {AB}}$⁠ 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 σ and the diagonal segments of the silver rectangle, as illustrated above, pars pro toto on a single triangle.

The leg to base ratio ⁠${\overline {AB}}/2\approx 1.306563$⁠ 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

Silver spirals with different initial angles on a σ− rectangle.

A silver spiral is a logarithmic spiral that gets wider by a factor of ⁠$\sigma$⁠ for every quarter turn. It is described by the polar equation $r(\theta )=a\exp(k\theta ),$ with initial radius ⁠$a$⁠ and parameter $k={\frac {2\ln(\sigma )}{\pi }}.$ 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 $1/\sigma .$

Ammann–Beenker tiling

Patch inflation of Amman A5-tiles with factor σ².

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 ⁠$\sigma ^{2},$⁠ the dominant eigenvalue of substitution matrix $$M={\begin{pmatrix}3&2\\4&3\end{pmatrix}}.$$

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