Silver ratio
Template:Short description Template:Distinguish Template:Infobox non-integer number In mathematics, the silver ratio is a geometrical proportion close to Template:Math. Its exact value is Template:Math the positive solution of the equation Template:Math
The name silver ratio results from 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.
DefinitionEdit
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: <math display="block"> \frac{a}{b} =\frac{2a+b}{a}</math> The ratio <math> \frac{a}{b} </math> is here denoted Template:TmathTemplate:Efn
Based on this definition, one has <math display="block"> \begin{align} 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{align} </math>
It follows that the silver ratio is found as the positive solution of the quadratic equation <math>\sigma^{2} -2\sigma -1 =0.</math> The quadratic formula gives the two solutions <math>1 \pm \sqrt{2},</math> the decimal expansion of the positive root begins as Template:Tmath (sequence A014176 in the OEIS).
Using the tangent function
- <math> \sigma =\tan \left( \frac{3\pi}{8} \right) =\cot \left( \frac{\pi}{8} \right),</math>
or the hyperbolic sine
- <math> \sigma =\exp( \operatorname{arsinh}(1) ).</math><ref>Template:Cite OEIS</ref>
Template:Tmath is the superstable fixed point of the iteration <math> x \gets \tfrac12 (x^2+1) /(x-1), \text{ with } x_0 \in [2,3]</math>
The iteration <math> x \gets \sqrt{1 +2x \vphantom{/} } </math> results in the continued radical <math display="block">\sigma =\sqrt{1 +2\sqrt{1 +2\sqrt{1 +\cdots}}} \;.</math>
PropertiesEdit
The defining equation can be written <math display="block"> \begin{align}
1 &=\frac{1}{\sigma -1} + \frac{1}{\sigma +1} \\
&=\frac{2}{\sigma +1} + \frac{1}{\sigma}.\end{align} </math>
The silver ratio can be expressed in terms of itself as fractions <math display="block"> \begin{align}
\sigma &=\frac{1}{\sigma -2} \\
\sigma^2 &=\frac{\sigma -1}{\sigma -2} +\frac{\sigma +1}{\sigma -1}.\end{align} </math>
Similarly as the infinite geometric series <math display="block"> \begin{align}
\sigma &=2\sum_{n=0}^{\infty} \sigma^{-2n} \\
\sigma^2 &=-1 +2\sum_{n=0}^{\infty} (\sigma -1)^{-n}.\end{align} </math>
For every integer Template:Tmath one has <math display="block"> \begin{align}
\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{align}</math>
From this an infinite number of further relations can be found.
Continued fraction pattern of a few low powers <math display="block"> \begin{align}
\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{align}</math>
- <math> \sigma^{-n} \equiv (-1)^{n-1} \sigma^n \bmod 1.</math>
The silver ratio is a Pisot number,<ref>Template:Cite journal</ref> the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value <math>\sqrt{2} -1</math> of the algebraic conjugate is smaller than Template:Math thus powers of Template:Tmath generate almost integers and the sequence <math> \sigma^n \bmod 1 </math> is dense at the borders of the unit interval.<ref>Template:Mathworld</ref>
Template:Tmath is the fundamental unit of real quadratic field <math>K =\mathbb{Q}\left( \sqrt{2} \right).</math>
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
- <math> x =\sum_{n=1}^{\infty} \frac{a_n}{\sigma^n},</math> with weights Template:Tmath
Sigmary expansions are not unique. Due to the identities <math display="block"> \begin{align}
\sigma^{n+1} &=2\sigma^n +\sigma^{n-1} \\
\sigma^{n+1} +\sigma^{n-1} &=2\sigma^n +2\sigma^{n-1},\end{align}</math>
digit blocks <math> 21_\sigma \text{ and } 22_\sigma </math> carry to the next power of Template:Tmath resulting in <math> 100_\sigma \text{ and } 101_\sigma.</math> The number one has finite and infinite representations <math> 1.0_\sigma, 0.21_\sigma </math> and <math> 0.\overline{20}_\sigma, 0.1\overline{2}_\sigma,</math> 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 <math> 10 =111.12_\sigma,</math> <math> 7\sigma +3 =1100_\sigma \,</math> and <math> \tfrac{1}{\sigma -1} =0.\overline{1}_\sigma.</math>
Properties of canonical sigmary expansions, with coefficients <math>a,b,c,d \in \mathbb{Z}:</math>
- Every algebraic integer <math> \xi =a +b\sigma \text{ in } K </math> has a finite expansion.<ref>Template:Cite journal</ref>
- Every rational number <math> \rho =\tfrac{a +b\sigma}{c +d\sigma} \text{ in } K </math> has a purely periodic expansion.<ref>Template:Cite journal</ref>
- All numbers that do not lie in Template:Tmath have chaotic expansions.
Template:Br Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation Template:Tmath with integer Template:Math<ref>Template:Harvtxt: Theorem 3.4</ref> It follows by repeated substitution of Template:Tmath that all positive solutions <math> \tfrac12 \left(n +\sqrt{n^2 + 4 \vphantom{/} } \right) </math> have a purely periodic continued fraction expansion <math display="block"> \sigma_n =n +\cfrac{1}{n +\cfrac{1}{n +\cfrac{1}{\ddots}}}</math> Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.<ref>Template:Harvtxt</ref>
Pell sequencesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
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 <math display="block"> P_{n} =2P_{n-1} +P_{n-2} \text{ for } n > 1,</math> with initial values <math display="block"> P_{0} =0, P_{1} =1.</math>
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 <math display="block"> \left\vert \sigma - \frac{P_{n+1}}{P_n} \right\vert < \frac{1}{\sqrt{8} P_n^2}</math>
The sequence is extended to negative indices using <math display="block"> P_{-n} =(-1)^{n-1} P_n.</math>
Powers of Template:Tmath can be written with Pell numbers as linear coefficients <math display="block"> \sigma^n =\sigma P_n +P_{n-1},</math> 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
- <math> \frac{x}{1 - 2x - x^2} = \sum_{n=0}^{\infty} P_{n}x^{n} \text{ for } \vert x \vert <1 /\sigma \;.</math><ref>Template:Cite journal</ref>
The characteristic equation of the recurrence is <math>x^2 -2x -1 =0</math> with discriminant Template:Tmath If the two solutions are silver ratio Template:Tmath and conjugate Template:Tmath so that <math>\sigma +\bar{\sigma} =2 \;\text{ and } \;\sigma \cdot \bar{\sigma} =-1,</math> the Pell numbers are computed with the Binet formula
- <math> P_n =a( \sigma^n -\bar{\sigma}^n ),</math> with Template:Tmath the positive root of <math>8x^2 -1 =0.</math>Template:Br
Since <math> \left\vert a\,\bar{\sigma}^n \right\vert < 1 /\sigma^{2n},</math> the number Template:Tmath is the nearest integer to <math> a\,\sigma^{n},</math> with <math> a =1 /\sqrt{8} </math> and Template:Math
The Binet formula <math> \sigma^n +\bar{\sigma}^n </math> defines the companion sequence <math> Q_{n} =P_{n+1} +P_{n-1}.</math>
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, <math> Q_{p} \equiv Q_{1} \bmod p.</math> The converse does not hold, the least odd pseudoprimes <math>\,n \mid (Q_{n} -2) </math> are 132, 385, 312, 1105, 1121, 3827, 4901.<ref>Template:Cite OEIS</ref> Template:Efn
Pell numbers are obtained as integral powers Template:Math of a matrix with positive eigenvalue Template:Tmath <math display="block"> M = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix} ,</math>
<math display="block"> M^{n} = \begin{pmatrix} P_{n+1} & P_{n} \\ P_{n} & P_{n-1} \end{pmatrix} </math>
The trace of Template:Tmath gives the above Template:Tmath
GeometryEdit
Silver rectangle and regular octagonEdit
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 <math>\sqrt{\sigma +1 \vphantom{/} }, \;\sigma</math> and <math>\sqrt{2(\sigma +1) \vphantom{/} }.</math> The coordinates of the vertices are given by the Template:Math permutations of <math>\left( \pm \tfrac12, \pm \tfrac{\sigma}{2} \right).</math><ref>Template:Citation</ref> The paper square has edge length Template:Tmath and area Template:Tmath The triangles have areas <math>1, \frac{\sigma -1}{\sigma} </math> and <math>\frac{1}{\sigma} ;</math> the rectangles have areas <math>\sigma -1 \text{ and } \frac{1}{\sigma}.</math>
Silver whirlEdit
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.<ref>Template:Cite book</ref>
The logarithmic spiral through the vertices of adjacent triangles has polar slope <math> k =\frac{4}{\pi} \ln( \sigma).</math> 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 <math>\frac{\sigma}{\sigma -1} =\sum_{n=0}^{\infty} \sigma^{-n} .</math> The areas of the triangles in each spiral region sum to <math>\frac{\sigma}{4} =\tfrac12 \sum_{n=0}^{\infty} \sigma^{-2n} ;</math> 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. 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
PolyhedraEdit
The silver mean has connections to the following Archimedean solids with octahedral symmetry; all values are based on edge length Template:Math
The coordinates of the vertices are given by 24 distinct permutations of <math>( \pm \sigma, \pm 1, \pm 1),</math> thus three mutually-perpendicular silver rectangles touch six of its square faces.Template:Br The midradius is <math> \sqrt{2(\sigma +1) \vphantom{/} },</math> the centre radius for the square faces is Template:Tmath<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Coordinates: 24 permutations of <math>( \pm \sigma, \pm \sigma, \pm 1).</math>Template:Br Midradius: Template:Tmath centre radius for the octagon faces: Template:Tmath<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Coordinates: 48 permutations of <math>( \pm (2\sigma -1), \pm \sigma, \pm 1).</math>Template:Br Midradius: <math> \sqrt{6(\sigma +1) \vphantom{/} },</math> centre radius for the square faces: Template:Tmath for the octagon faces: Template:Tmath<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
See also the dual Catalan solids
Silver triangleEdit
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 <math> \tfrac12 \tan(67 \tfrac12) =\tfrac{\sigma}{2}.</math>
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 <math> 67 \tfrac12 /3 =22 \tfrac12, 45 \text{ and } 112 \tfrac12 </math> degrees in the ratios Template:Tmath From the law of sines, its edges are in ratios <math> 1 :\sqrt{\sigma +1} :\sigma.</math>
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.<ref>Proved for the golden triangle in: Template:Cite book</ref> The assumption is verified by construction, as demonstrated in the vector image.
The centre of rotation has barycentric coordinates <math display="block"> \left( \tfrac{\sigma +1}{\sigma +5} :\tfrac{2}{\sigma +5} :\tfrac{2}{\sigma +5} \right) \sim \left( \tfrac{\sigma +1}{2} :1 :1 \right),</math> the three whorls of stacked gnomons have areas in ratios <math display="block"> \left( \tfrac{\sigma +1}{2} \right)^{2} :\tfrac{\sigma +1}{2} :1.</math>
The logarithmic spiral through the vertices of all nested triangles has polar slope
- <math> k =\frac{4}{5\pi} \ln \left( \tfrac{\sigma}{\sigma-1} \right),</math> or an expansion rate of Template:Tmath for every Template:Tmath degrees of rotation.
| circumcenter | <math> \left( \tfrac{2}{\sigma +1} :\tfrac{1}{\sigma} \right) \sim ( \sigma -1 :1) </math> |
| centroid | <math> \left( \tfrac23 :\tfrac13 \right) \sim (2 :1)</math> |
| nine-point center | <math> \left( \tfrac{1}{\sigma -1} :\tfrac{1}{\sigma +1} \right) \sim ( \sigma :1) </math> |
| incenter, Template:Math | <math> \left( [ 1 +\cos(\alpha)]^{-1} :[ 1 +\sec(\alpha)]^{-1} \right) \sim ( \sec(\alpha) :1) </math> |
| symmedian point | <math> \left( \tfrac{\sigma +1}{\sigma +2} :\tfrac{1}{\sigma +2} \right) \sim ( \sigma +1 :1) </math> |
| orthocenter | <math> \left( \tfrac{2}{\sigma} :\tfrac{1}{\sigma^2} \right) \sim ( 2\sigma :1) </math> |
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 triangleEdit
Assume a silver rectangle has been constructed as indicated above, with height Template:Math, length Template:Tmath and diagonal length <math> \sqrt{\sigma^2 +1}</math>. The triangles on the diagonal have altitudes <math>1 /\sqrt{1 +\sigma^{-2}}\,;</math> 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 <math> \sigma^2 :2 :1\,,</math> the rectangles opposite the diagonal both have areas equal to <math> \tfrac{2}{\sigma +1}.</math><ref>Analogue to the construction in: Template:Cite journal</ref>
Relative to vertex Template:Math, the coordinates of feet of altitudes Template:Math and Template:Math are <math display="block">\left( \tfrac{\sigma}{\sigma^2 +1}, \tfrac{1}{\sigma^2 +1} \right) \text{ and } \left( \tfrac{\sigma}{1 +\sigma^{-2}}, \tfrac{1}{1 +\sigma^{-2}} \right).</math>
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 <math>\alpha =67 \tfrac12 </math> degrees, the base angle of the silver triangle:
<math display="block"> \begin{align} \overline{A B} =\sqrt{\sigma^2 +1} &=\sec(\alpha) \\ \overline{A V} =\sigma^2 /\overline{A B} &=\sigma\sin(\alpha) \\ \overline{U V} =2 /\overline{A S} &=2\sin(\alpha) \\ \overline{S B} =4 /\overline{A B} &=4\cos(\alpha) \\ \overline{S V} =3 /\overline{A B} &=3\cos(\alpha) \\ \overline{A S} =\sqrt{1 +\sigma^{-2}} &=\csc(\alpha) \\ \overline{h} =1 /\overline{A S} &=\sin(\alpha) \\ \overline{U S} =\overline{A V} -\overline{S B} &=(2\sigma -3)\cos(\alpha) \\ \overline{A U} =1 /\overline{A B} &=\cos(\alpha),\end{align}</math>
- with Template:Tmath
Both the lengths of the diagonal sections and the trigonometric values are elements of biquadratic number field <math>K =\mathbb{Q}\left( \sqrt{2 +\sqrt{2}} \right).</math>
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.<ref>Template:Cite journal</ref>
Silver spiralEdit
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 <math>r( \theta) =a \exp(k \theta),</math> with initial radius Template:Tmath and parameter <math>k =\frac{2}{\pi} \ln( \sigma).</math> 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 <math>1/ \sigma.</math>
Ammann–Beenker tilingEdit
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.<ref>Template:Cite conference</ref> If the squares are cut into two triangles, the inflation factor for Ammann A5-tiles is Template:Tmath the dominant eigenvalue of substitution matrix <math display="block"> M =\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}.</math>
See alsoEdit
- Solutions of equations similar to <math> x^2 =2x +1 </math>:
- Golden ratio – the real positive solution of the equation <math> x^2 =x +1 </math>
- Metallic means – real positive solutions of the general equation <math> x^2 =nx +1 </math>
- Supersilver ratio – the only real solution of the equation <math> x^3 =2x^2 +1</math>
NotesEdit
ReferencesEdit
External linksEdit
- YouTube lecture on the silver ratio, Pell sequence and metallic means
- Silver rectangle and Pell sequence at Tartapelago by Giorgio Pietrocola
Template:Algebraic numbers Template:Irrational numbers Template:Metallic ratios