Silver ratio explained

Rationality:	irrational algebraic
Continued Fraction Linear:	[2;2,2,2,2,2,...]
Continued Fraction Periodic:	purely periodic
Continued Fraction Finite:	infinite
Algebraic:	positive root of

In mathematics, the silver ratio is a geometrical proportion close to . Its exact value is the positive solution of the equation

The name silver ratio results from analogy with the golden ratio, the positive solution of the equation

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.

Definition

If the ratio of two quantities is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: $$\backslash frac\; =\backslash frac$$The ratio

(a)/(b) is here denoted

Based on this definition, one has

\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 .

Using the tangent function

\sigma = \tan \left( (3 \pi)/(8) \right) = \cot \left( (\pi)/(8) \right), or the hyperbolic sine\sigma = \exp( \operatorname{arsinh}( 1 ) ).

is the superstable fixed point of the iteration

x \gets \tfrac 12 (x ^{2+1)} /(x -1 ), with x _{0} \in [ 2 , 3 ]

The iteration

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

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 hasFrom this an infinite number of further relations can be found.

Continued fraction pattern of a few low powers

\sigma ^{-n} \equiv ( -1 ) ^{n-1} \sigma ^{n} \bmod 1 . \sqrt{ 2 } -1 of the algebraic conjugate is smaller than thus powers of generate almost integers and the sequence \sigma ^{n} \bmod 1 is dense at the borders of the unit interval.K = Q \left( \sqrt{ 2 } \right).

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

Sigmary expansions are not unique. Due to the identitiesdigit blocks

21 _{\sigma} and 22 _{\sigma} carry to the next power of resulting in 100 _{\sigma} and 101 _{\sigma.} The number one has finite and infinite representations 1 . 0 _{\sigma,} 0 . 21 _{\sigma} and 0 .\overline{ 20 } _{\sigma,} 0 . 1 \overline{ 2 } _{\sigma,} where the first of each pair is in canonical form. The algebraic number can be written or non-canonically as The decimal number 10 =111 . 12 _{\sigma,} 7 \sigma +3 =1100 _{\sigma} and \tfrac{ 1 }{\sigma -1 } =0 .\overline{ 1 } _{\sigma.}

Properties of canonical sigmary expansions, with coefficients

a,b,c,d \in Z : \xi = a + b\sigma in K has a finite expansion.^[2] \rho = \tfrac{a + b\sigma}{c + d\sigma} in K has a purely periodic expansion.^[3]

• All numbers that do not lie in have chaotic expansions.

Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation with integer ^[4] It follows by repeated substitution of that all positive solutions

\tfrac 12 \left(n + \sqrt{n ^{2} + 4 \vphantom{/} } \right) have a purely periodic continued fraction expansion $$\backslash sigma\_n\; =n\; +\backslash cfrac$$Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.

Pell sequences

See 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$$P\_\; =2P\_\; +P\_\; \backslash text\; n\; >\; 1,$$with initial values $$P\_\; =0,\; P\_\; =1.$$

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 $$P\_\; =(-1)^\; P\_n.$$

Powers of can be written with Pell numbers as linear coefficients $$\backslash sigma^n\; =\backslash sigma\; P\_n\; +P\_,$$ which is proved by mathematical induction on The relation also holds for

The generating function of the sequence is given by

(x)/(1-2 x - x ^{2}) = \sum ^{inf ty} _{n =0} P _{n} x ^{n} for \vert x \vert < 1 /\sigma . ^[5]

The characteristic equation of the recurrence is

x ^{2} -2 x -1 =0 with discriminant If the two solutions are silver ratio and conjugate so that \sigma + \bar{\sigma} =2 and \sigma ⋅ \bar{\sigma} =-1 , the Pell numbers are computed with the Binet formulaP _{n} = a( \sigma ^{n} - \bar{\sigma} ^{n} ), with the positive root of 8 x ^{2} -1 =0 . Since \left\vert a\bar{\sigma} ^{n} \right\vert < 1 /\sigma ^{2n} , the number is the nearest integer to a\sigma ^{n} , with a =1 /\sqrt{ 8 } and

The Binet formula

\sigma ^{n} + \bar{\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,... .

This Pell-Lucas 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 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 (大和比) — 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 (according to).

• 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), and open out the triangle. The result is a rectangle.
• Fold the bottom edge onto the left hand edge (reduce height by). 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 (left) and (right). The unit angle is equal to degrees.

If the octagon has edge length its area is 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 permutations of \left( \pm \tfrac 12 , \pm \tfrac{\sigma}{ 2 } \right). The paper square has edge length and area The triangles have areas 1 , (\sigma -1)/(\sigma) and (1)/(\sigma) ; the rectangles have areas \sigma -1 and (1)/(\sigma) .

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.^[6]

k = (4)/(\pi) ln ( \sigma). 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

(\sigma)/(\sigma -1) = \sum ^{inf ty} _{n =0} \sigma ^{-n} . The areas of the triangles in each spiral region sum to (\sigma)/(4) = \tfrac 12 \sum ^{inf ty} _{n =0} \sigma ^{-2n} ; the perimeters are equal to (light grey) and (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 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

• 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 ^[7]

• Truncated cube

Coordinates: 24 permutations of

( \pm \sigma, \pm \sigma, \pm 1 ). Midradius: centre radius for the octagon faces: ^[8]

• 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: for the octagon faces: ^[9]

See also the dual Catalan solids

• Tetragonal trisoctahedron
• Trisoctahedron
• Hexakis octahedron

Silver triangle

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

\tfrac 12 \tan( 67 \tfrac 12 ) = \tfrac{\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 \tfrac 12 / 3 =22 \tfrac 12 , 45 and 112 \tfrac 12 degrees in the ratios 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, 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.^[10] The assumption is verified by construction, as demonstrated in the vector image.

The centre of rotation has barycentric coordinates$$\backslash left(\backslash tfrac\; :\backslash tfrac\; :\backslash tfrac\; \backslash right)\; \backslash sim\; \backslash left(\backslash tfrac\; :1\; :1\; \backslash right),$$the three whorls of stacked gnomons have areas in ratios$$\backslash left(\backslash tfrac\; \backslash right)^\; :\backslash tfrac\; :1.$$

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

k = (4)/(5 \pi) ln \left( \tfrac{\sigma}{\sigma -1 } \right), or an expansion rate of for every degrees of rotation.

Silver triangle centers: affine coordinates on the axis of symmetry

	\left( \tfrac{ 2 }{\sigma +1 } :\tfrac{ 1 }{\sigma} \right) \sim ( \sigma -1 : 1 )
	\left( \tfrac 23 :\tfrac 13 \right) \sim ( 2 : 1 )
	\left( \tfrac{ 1 }{\sigma -1 } :\tfrac{ 1 }{\sigma +1 } \right) \sim ( \sigma : 1 )
incenter,	\left( [ 1 + \cos(\alpha)] ^{-1} :[ 1 + \sec(\alpha)] ^{-1} \right) \sim ( \sec(\alpha) : 1 )
	\left( \tfrac{\sigma +1 }{\sigma +2 } :\tfrac{ 1 }{\sigma +2 } \right) \sim ( \sigma +1 : 1 )
	\left( \tfrac{ 2 }{\sigma} :\tfrac{ 1 }{\sigma ^{2}} \right) \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

Assume a silver rectangle has been constructed as indicated above, with height, length 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

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 \tfrac{ 2 }{\sigma +1 }. ^[11]

Relative to vertex, the coordinates of feet of altitudes and are$$\backslash left(\backslash tfrac,\; \backslash tfrac\; \backslash right)\; \backslash text\; \backslash left(\backslash tfrac,\; \backslash tfrac\; \backslash right).$$

If the diagram is further subdivided by perpendicular lines through and, the lengths of the diagonal and its subsections can be expressed as trigonometric functions of argument

\alpha =67 \tfrac 12 degrees, the base angle of the silver triangle:

with

K = Q \left( \sqrt{ 2 + \sqrt{ 2 }} \right).

The silver rhombus with edge has diagonal lengths equal to and The regular octagon with edge has long diagonals of length 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 and the diagonal segments of the silver rectangle, as illustrated above, pars pro toto on a single triangle.

The leg to base ratio 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.^[12]

Silver spiral

r( \theta) = a \exp(k \theta), with initial radius and parameter k = (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.

Ammann–Beenker tiling

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.^[13] If the squares are cut into two triangles, the inflation factor for Ammann A5-tiles is the dominant eigenvalue of substitution matrix

See also

• Solutions of equations similar to

x ^{2} =2 x +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} =2 x ^{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

Notes and References

1. Panju . Maysum . 2011 . A systematic construction of almost integers . The Waterloo Mathematics Review . 1 . 2 . 35–43.
2. Frougny . Christiane . Solomyak . Boris . Finite beta-expansions . 1992 . Ergodic Theory and Dynamical Systems . 12 . 4 . 713–723 [721: Proposition 1] . January 19, 2025 . 10.1017/S0143385700007057.
3. Schmidt . Klaus . On periodic expansions of Pisot numbers and Salem numbers . 1980 . Bulletin of the London Mathematical Society . 12 . 4 . 269–278 [274: Theorem 3.1] . 10.1112/blms/12.4.269 . 10338.dmlcz/141479 . free.

4. Theorem 3.4

5. Horadam . A. F. . 1971 . Pell identities . . 9 . 3 . 245–252, 263 [248] . 10.1080/00150517.1971.12431004 .
6. Book: Walser, Hans . Spiralen, Schraubenlinien und spiralartige Figuren . de . 2022 . . Berlin, Heidelberg . 77–78 . 10.1007/978-3-662-65132-2 . 978-3-662-65131-5.
7. Web site: Rhombicuboctahedron . McCooey . David . Visual Polyhedra . 11 December 2024.
8. Web site: Truncated Cube . McCooey . David . Visual Polyhedra . 11 December 2024.
9. Web site: Truncated Cuboctahedron . McCooey . David . Visual Polyhedra . 11 December 2024.
10. Proved for the golden triangle in:Book: Loeb . Arthur L. . Varney . William . Hargittai . István . Pickover . Clifford A. . 1992 . Spiral Symmetry . World Scientific . Singapore . 47–61 . Does the golden spiral exist, and if not, where is its center? . https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA47 . January 14, 2025 . 981-02-0615-1 . 10.1142/9789814343084_0002.
11. Analogue to the construction in: Crilly . Tony . 1994 . A supergolden rectangle . . 78 . 483 . 320–325 . 10.2307/3620208 . 3620208.
12. Redondo Buitrago . Antonia . Reyes Iglesias . Encarnación . 2008 . The Geometry of the Cordovan Polygons . Visual Mathematics . 10 . 4 . . Mathematical Institute . Belgrade . 1821-1437 . December 11, 2024.
13. Images of the Ammann-Beenker Tiling . Harriss . Edmund . Edmund Harriss . 2007 . The Bridges Organization . 377–378 . San Sebastián . Bridges Donostia: Mathematics, music, art, architecture, culture.