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{{Short description|Number, approximately 2.41421}}
{{Distinguish|Silver constant}}
{{infobox non-integer number
| image=Silver rectangle repeats.svg
| rationality=irrational algebraic
| symbol={{math|σ}}
| decimal={{gaps|2.41421|35623|73095|04880|16887|...}}
| continued_fraction_linear=[2;2,2,2,2,2,...]
| continued_fraction_periodic=purely periodic
| continued_fraction_finite=infinite
| algebraic=positive root of {{math|1=''x''{{sup|2}} = 2''x'' + 1}}
}}
In mathematics, the '''silver ratio''' is a geometrical [[aspect ratio|proportion]] close to {{math|70/29}}. Its exact value is {{math|1 + √2,}} the positive [[polynomial root|solution]] of the equation {{math|1=''x''{{sup|2}} = 2''x'' + 1.}}
The name ''silver ratio'' results from analogy with the [[golden ratio]], the positive solution of the equation {{math|1=''x''{{sup|2}} = ''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 triple#Special cases and related equations|Pythagorean triple]]s, [[square triangular number]]s, [[Pell number]]s, the [[octagon]], and six [[polyhedron|polyhedra]] with [[octahedral symmetry]].
[[File:silver_rectangle_in_octagon.svg |thumb |upright=.83 |Silver rectangle in a regular octagon.]]
==Definition==
If the ratio of two quantities {{math|a > b > 0}} 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 {{tmath|\sigma.}}{{efn |1=Variously {{math|T(2),}}<ref>{{Cite web |url=https://r-knott.surrey.ac.uk/Fibonacci/cfINTRO.html#silver |title=An introduction to Continued Fractions |last=Knott |first=Ron |date=2015 |website=Dr Ron Knott's web pages on Mathematics |publisher=University of Surrey |access-date=December 11, 2024}}</ref> [[Metallic_mean|{{math|S{{sub|2}},}}]] [[Truncated_cube|{{math|δ{{sub|S}},}}]]<ref>{{mathworld |id=SilverRatio|title=Silver ratio}}</ref>
{{math|σ{{sub|Ag}}.}}<ref>{{cite conference |url=https://zenodo.org/records/9055 |title=New Smarandache sequences: the family of metallic means |last=Spinadel |first=Vera W. de |author-link=Vera W. de Spinadel |date=1997 |publisher=American Research Press |___location=Rehoboth, NM |conference=Proceedings of the first international conference on Smarandache type notions in number theory (Craiova, Romania) |pages=79–114 |doi=10.5281/ZENODO.9055|doi-access=free}}</ref> The last notation is adopted without the subscript, which is relevant only to the context of [[metallic mean]]s.}}
<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 [[Zero of a function|root]] begins as {{tmath|2.414\,213\,562\,373\,095...}} {{OEIS|A014176}}.
Using the [[Exact_trigonometric_values|tangent function]]
:<math> \sigma =\tan \left( \frac{3\pi}{8} \right) =\cot \left( \frac{\pi}{8} \right),</math>
or the [[Hyperbolic functions|hyperbolic sine]]
:<math> \sigma =\exp( \operatorname{arsinh}(1) ).</math><ref>{{cite OEIS|A014176}}</ref>
{{tmath|\sigma}} is the superstable [[Fixed-point iteration|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 [[Nested radical#Infinitely nested radicals|continued radical]] <math display="block">\sigma =\sqrt{1 +2\sqrt{1 +2\sqrt{1 +\cdots}}} \;.</math>
==Properties==
[[File:SilverSquare_6.svg |thumb |upright=1.25 |Rectangles with aspect ratios related to {{math|σ}} tile the square.]]
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 {{tmath|n}} 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} \\
From this an infinite number of further relations can be found.
[[Simple continued fraction|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>{{cite journal |last=Panju |first=Maysum |date=2011 |title=A systematic construction of almost integers |url=https://mathreview.uwaterloo.ca/archive/voli/2/panju.pdf |journal=The Waterloo Mathematics Review |volume=1 |issue=2 |pages=35–43}}</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 [[Conjugate element (field theory)|algebraic conjugate]] is smaller than {{math|1,}} thus powers of {{tmath|\sigma}} generate [[almost integer]]s and the sequence <math> \sigma^n \bmod 1 </math> is dense at the borders of the [[unit interval]].<ref>{{mathworld |id=PowerFractionalParts |title=Power Fractional Parts}}</ref>
{{tmath|\sigma}} is the [[Quadratic integer#Units|fundamental unit]] of real [[quadratic field]] <math>K =\mathbb{Q}\left( \sqrt{2} \right).</math>
The silver ratio can be used as [[Radix|base]] of a [[Non-integer base of numeration|numeral system]], here called the ''sigmary scale''.{{efn|In what follows, it is assumed that {{math|0 ≤ ''x'' ≤ 1.}} Negative numbers are multiplied by {{math|−1}} first, and numbers {{math|> 1}} divided by the least power of {{math|σ ≥ ''x''.}} The sigmary digits are then obtained by successive multiplications with {{math|σ}}, clearing the integer part at each step. Lastly, the 'sigmary point' is restored.}} Every [[real number]] {{math|''x''}} in {{math|[0,1]}} can be represented as a [[convergent series]]
:<math> x =\sum_{n=1}^{\infty} \frac{a_n}{\sigma^n},</math> with [[Weight function#Discrete weights|weights]] {{tmath| a_n \in [0,1,2].}}
[[File:Sigmary_scale.svg |thumb |upright=1.6 |The steps in the sigmary scale resemble the intervals of the [[Mixolydian mode#Modern Mixolydian|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
<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 (arithmetic)|carry]] to the next power of {{tmath|\sigma,}} 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]] {{tmath| 2(3\sigma -7) }} can be written {{tmath| 0.101_\sigma,}} or non-canonically as {{tmath| 0.022_\sigma.}} The [[decimal|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>{{Cite journal |last1=Frougny |first1=Christiane |last2=Solomyak |first2=Boris |url=https://www.researchgate.net/publication/232019477_Finite_beta-expansions |title=Finite beta-expansions |year=1992 |journal=Ergodic Theory and Dynamical Systems |volume=12 |issue=4 |pages=713–723 [721: Proposition 1] |access-date=January 19, 2025 |doi=10.1017/S0143385700007057}}</ref>
* Every [[Field of fractions|rational number]] <math> \rho =\tfrac{a +b\sigma}{c +d\sigma} \text{ in } K </math> has a purely periodic expansion.<ref>{{Cite journal |last=Schmidt |first=Klaus |title=On periodic expansions of Pisot numbers and Salem numbers |year=1980 |journal=Bulletin of the London Mathematical Society |volume=12 |issue=4 |pages=269–278 [274: Theorem 3.1] |doi=10.1112/blms/12.4.269 |hdl=10338.dmlcz/141479 |hdl-access=free}}</ref>
* All numbers that do not lie in {{tmath|K}} have chaotic expansions.
{{br}}
Remarkably, the same holds ''[[mutatis mutandis]]'' for all quadratic Pisot numbers that satisfy the general equation {{tmath|x^2{{=}}nx +1,}} with integer {{math|''n'' > 0.}}<ref>{{harvtxt|Schmidt|1980|p=275}}: Theorem 3.4</ref> It follows by repeated substitution of {{tmath|x{{=}}n +\frac{1}{x} }} 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_W._de_Spinadel|Vera de Spinadel]] described the properties of these irrationals and introduced the moniker [[metallic mean]]s.<ref>{{harvtxt|Spinadel|1997}}</ref>
==Pell sequences==
[[File:SilverWord_im.png |thumb |upright=1.6 |Silver harmonics: the rectangle and its coloured subzones have areas in ratios {{math|7σ + 3 : σ{{sup|3}} : σ{{sup|2}} : σ : 1.}}]]
{{main|Pell number}}
<!-- IT MAY BE tempting to add more interesting Pell number properties from the main article, but for this section a selection of only those that illustrate the connection with the Silver ratio will do. Thank you.-->
These numbers are related to the silver ratio as the [[Fibonacci number]]s and [[Lucas number]]s 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,... {{OEIS|A000129}}. The limit ratio of consecutive terms is the silver mean.
Fractions of Pell numbers provide [[Diophantine approximation#Upper bounds for Diophantine approximations|rational approximations]] of {{tmath|\sigma}} 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 {{tmath|\sigma}} 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 {{math|n.}} The relation also holds for {{math|n < 0.}}
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>{{Cite journal |last=Horadam |first=A. F. |date=1971 |title=Pell identities |journal=[[The Fibonacci Quarterly]] |volume=9 |issue=3 |pages=245–252, 263 [248] |doi=10.1080/00150517.1971.12431004 }}</ref>
The [[Characteristic equation (calculus)|characteristic equation]] of the recurrence is <math>x^2 -2x -1 =0</math> with [[discriminant]] {{tmath|D{{=}}8.}} If the two solutions are silver ratio {{tmath|\sigma}} and conjugate {{tmath|\bar{\sigma},}} so that <math>\sigma +\bar{\sigma} =2 \;\text{ and } \;\sigma \cdot \bar{\sigma} =-1,</math> the Pell numbers are computed with the [[Fibonacci sequence#Binet's formula|Binet formula]]
:<math> P_n =a( \sigma^n -\bar{\sigma}^n ),</math> with {{tmath|a}} the positive root of <math>8x^2 -1 =0.</math>{{br}}
Since <math> \left\vert a\,\bar{\sigma}^n \right\vert < 1 /\sigma^{2n},</math> the number {{tmath|P_{n} }} is the nearest integer to <math> a\,\sigma^{n},</math> with <math> a =1 /\sqrt{8} </math> and {{math|''n'' ≥ 0.}}
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,... {{OEIS|A002203}}.
This [[Pell number#Pell–Lucas numbers|Pell-Lucas]] sequence has the [[Fermat's little theorem|Fermat property]]: if p is prime, <math> Q_{p} \equiv Q_{1} \bmod p.</math> The converse does not hold, the least odd [[pseudoprime]]s <math>\,n \mid (Q_{n} -2) </math> are 13{{sup|2}}, 385, 31{{sup|2}}, 1105, 1121, 3827, 4901.<ref>{{cite OEIS|A330276}}</ref>
{{efn |There are 3360 odd composite numbers below {{math|10{{sup|9}}}} that pass the Pell-Lucas test. This compares favourably to the number of odd [[Lucas pseudoprime#Fibonacci pseudoprimes|Fibonacci]], [[Lucas pseudoprime#Pell pseudoprimes|Pell]], [[Lucas pseudoprime#Lucas probable primes and pseudoprimes|Lucas-Selfridge]] or base-2 [[Fermat pseudoprime|Fermat]] pseudoprimes.<ref>{{cite web |url=https://ntheory.org/pseudoprimes.html |title=Pseudoprime statistics and tables |last=Jacobsen |first=Dana |date=2020 |website=ntheory.org |access-date=18 December 2024}}</ref>}}
Pell numbers are obtained as integral powers {{math|''n'' > 2}} of a [[Matrix (mathematics)|matrix]] with positive [[Eigenvalues and eigenvectors#Eigenvalues and eigenvectors of matrices|eigenvalue]] {{tmath|\sigma}} <math display="block"> M = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix} ,</math>
The [[Trace (linear algebra)#Relationship to eigenvalues|trace]] of {{tmath|M^{n} }} gives the above {{tmath|Q_{n}.}}
==Geometry==
===Silver rectangle and regular octagon===
[[File:Silver_rectangle_construction.svg |thumb |upright=1.25 |Origami construction of a silver rectangle, with creases in green.]]
A rectangle with edges in ratio {{math|√2 ∶ 1}} 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) — [[Square root of 2#Applications|the ratio is retained]] if the {{math|√2}} rectangle is folded in half, parallel to the short edges. [[Rabatment of the rectangle|Rabatment]] produces a rectangle with edges in the silver ratio (according to {{math| {{sfrac|1|σ}} {{=}} √2 − 1}}).
{{efn|In 1979 the [[British Origami Society]] proposed the alias ''silver rectangle'' for the {{math|√2}} rectangle, which is commonly used now.<ref>{{cite web |url=https://www.britishorigami.org/cp-lister-list/a4-silver-rectangles/ |title=A4 (Silver) Rectangles |last=Lister |first=David |date=2021 |website=The Lister List |publisher=British Origami Society |access-date=December 15, 2024}}</ref> In this article the name is reserved for the {{math|σ}} rectangle.}}
* 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 {{sfrac|1|σ + 1}}), and open out the triangle. The result is a {{math|√2}} rectangle.
* Fold the bottom edge onto the left hand edge (reduce height by {{sfrac|1|σ − 1}}). 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 [[#gnomon|silver gnomon]] with angles in the ratios {{math|5 ∶ 2 ∶ 1,}} between two right triangles with angles in ratios {{math|4 ∶ 2 ∶ 2}} (left) and {{math|4 ∶ 3 ∶ 1}} (right). The unit angle is equal to {{math|{{sfrac|22|1|2}} }} degrees.
If the octagon has edge length {{tmath|1,}} its area is {{tmath|2\sigma}} 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 {{math|8}} [[permutation]]s of <math>\left( \pm \tfrac12, \pm \tfrac{\sigma}{2} \right).</math><ref>{{citation |last=Kapusta |first=Janos |title=The square, the circle, and the golden proportion: a new class of geometrical constructions |journal=Forma |volume=19 |year=2004 |pages=293–313 |url=https://archive.bridgesmathart.org/2000/bridges2000-247.pdf}}</ref> The paper square has edge length {{tmath|\sigma -1}} and area {{tmath|2.}} 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 whirl===
[[File:Silver_rectangle_whirl.svg |thumb |upright=1.6 |A whirl of silver rectangles.]]
Divide a rectangle with sides in ratio {{math|1 ∶ 2}} into four congruent [[right triangle]]s with legs of equal length and arrange these in the shape of a silver rectangle, enclosing a [[Similarity (geometry)|similar]] rectangle that is scaled by factor {{tmath|\tfrac{1}{\sigma} }} and rotated about the centre by {{tmath|\tfrac{\pi}{4}.}} Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing a [[mice problem|whirl]] of converging silver rectangles.<ref>{{cite book |last=Walser |first=Hans |title=Spiralen, Schraubenlinien und spiralartige Figuren |language=de |date=2022 |publisher=[[Springer Spektrum]] |___location=Berlin, Heidelberg |pages=77–78 |doi=10.1007/978-3-662-65132-2 |isbn=978-3-662-65131-5}}</ref>
The logarithmic spiral through the vertices of adjacent triangles has [[Pitch angle of a spiral|polar slope]] <math> k =\frac{4}{\pi} \ln( \sigma).</math> <span id="rhomb">The [[parallelogram]] between the pair of grey triangles on the sides has perpendicular diagonals in ratio {{tmath|\sigma}}, hence is a ''silver [[rhombus]]''.</span>
If the triangles have legs of length {{tmath|1}} 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 {{tmath|\sigma +2}} (light grey) and {{tmath|2\sigma -1}} (silver regions).
Arranging the tiles with the four [[hypotenuse]]s facing inward results in the [[:File:Terme Taurine Opus Sectile.jpg|diamond-in-a-square]] shape. Roman architect [[Vitruvius]] recommended the implied [[wikisource:Page%3AVitruvius the Ten Books on Architecture.djvu/213|''ad quadratura'']] ratio as one of three for proportioning a town house [[Cavaedium|''atrium'']]. The scaling factor is {{tmath|\tfrac{1}{\sigma -1},}} and iteration on edge length {{math|2}} gives an angular spiral of length {{tmath|\sigma +1.}}
===Polyhedra===
[[File:Rhombicuboctahedron_by_Cutting_Rhombic_Dodecahedron.svg |thumb |upright=1.1 |Dimensions of the rhombi­cuboctahedron are linked to {{math|σ.}}]]
The silver mean has connections to the following [[Archimedean solid]]s with [[octahedral symmetry]]; all values are based on edge length {{math|{{=}} 2.}}
* [[Rhombicuboctahedron]]
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.{{br}}
The [[Midsphere|midradius]] is <math> \sqrt{2(\sigma +1) \vphantom{/} },</math> the centre radius for the square faces is {{tmath|\sigma.}}<ref>{{cite web |url=http://www.dmccooey.com/polyhedra/Rhombicuboctahedron.html |title=Rhombicuboctahedron |last=McCooey |first=David |website=Visual Polyhedra |access-date=11 December 2024}}</ref>
* [[Truncated cube]]
Coordinates: 24 permutations of <math>( \pm \sigma, \pm \sigma, \pm 1).</math>{{br}}
Midradius: {{tmath|\sigma +1,}} centre radius for the octagon faces: {{tmath|\sigma.}}<ref>{{cite web |url=http://www.dmccooey.com/polyhedra/TruncatedCube.html |title=Truncated Cube |last=McCooey |first=David |website=Visual Polyhedra |access-date=11 December 2024}}</ref>
* [[Truncated cuboctahedron]]
Coordinates: 48 permutations of <math>( \pm (2\sigma -1), \pm \sigma, \pm 1).</math>{{br}}
Midradius: <math> \sqrt{6(\sigma +1) \vphantom{/} },</math> centre radius for the square faces: {{tmath|\sigma +2,}} for the octagon faces: {{tmath|2\sigma -1.}}<ref>{{cite web |url=http://www.dmccooey.com/polyhedra/TruncatedCuboctahedron.html |title=Truncated Cuboctahedron |last=McCooey |first=David |website=Visual Polyhedra |access-date=11 December 2024}}</ref>
See also the dual [[Catalan solid]]s
* [[Deltoidal icositetrahedron#Cartesian coordinates|Tetragonal trisoctahedron]]
* [[Triakis octahedron#Cartesian coordinates|Trisoctahedron]]
* [[Disdyakis dodecahedron#Cartesian coordinates|Hexakis octahedron]]
===Silver triangle===
[[File:Silver triangle spiral.svg |thumb |upright=1.25 |Silver triangle and whirling gnomons.]]
The [[Acute_and_obtuse_triangles|acute]] [[isosceles triangle]] formed by connecting two adjacent vertices of a [[Octagon#Regularity|regular octagon]] to its centre point, is here called the ''silver triangle''. It is uniquely identified by its angles in ratios {{tmath|2 :3 :3.}} The [[Apex (geometry)|apex]] angle measures {{tmath|360 /8{{=}}45,}} each [[Base (geometry)|base]] angle {{tmath|67 \tfrac12}} degrees. It follows that the [[Altitude (triangle)|height]] to base ratio is <math> \tfrac12 \tan(67 \tfrac12) =\tfrac{\sigma}{2}.</math>
By [[angle trisection|trisecting]] one of its base angles, the silver triangle is partitioned into a similar triangle and <span id="gnomon">an [[Acute_and_obtuse_triangles|obtuse]] ''silver [[Gnomon (figure)#Isosceles triangles|gnomon]]''.</span> The trisector is collinear with a [[Octagon#Diagonality|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 {{tmath|1 :2 :5.}} 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 {{tmath|2\cos(67 \tfrac12)}}, accompanied with an {{tmath|112 \tfrac12}} degree rotation. Repeating the process at decreasing scales results in an infinite sequence of silver triangles, which converges at the [[Rotation (mathematics)#Related definitions and terminology|centre of rotation]]. It is assumed without proof that the centre of rotation is the intersection point of sequential [[Median (geometry)|median lines]] that join corresponding legs and base vertices.<ref>Proved for the [[Golden triangle (mathematics)|golden triangle]] in:
{{cite book |last1=Loeb |first1=Arthur L. |last2=Varney |first2=William |editor1-last=Hargittai |editor1-first=István |editor2-last=Pickover |editor2-first=Clifford A. |year=1992 |title=Spiral Symmetry |publisher=World Scientific |___location=Singapore |pages=47–61 |chapter=Does the golden spiral exist, and if not, where is its center? |chapter-url=https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA47 |access-date=January 14, 2025 |isbn=981-02-0615-1 |doi=10.1142/9789814343084_0002}}</ref>
The assumption is verified by construction, as demonstrated in the vector image.
The centre of rotation has [[Barycentric coordinate system|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 [[Pitch angle of a spiral|polar slope]]
:<math> k =\frac{4}{5\pi} \ln \left( \tfrac{\sigma}{\sigma-1} \right),</math> or an expansion rate of {{tmath| \tfrac{\sigma +1}{2} }} for every {{tmath|225}} degrees of rotation.
{| class="wikitable"
|+ Silver [[triangle center]]s: [[Affine space#Coordinates|affine coordinates]] on the [[Reflection symmetry|axis of symmetry]]
|-
| [[Circumcircle|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]], {{math|α {{=}} {{sfrac|3π|8}} }} || <math> \left( [ 1 +\cos(\alpha)]^{-1} :[ 1 +\sec(\alpha)]^{-1} \right) \sim ( \sec(\alpha) :1) </math>
|-
| [[Lemoine point|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 triangle===
[[File:Silver_rectangle_segmented.svg |thumb |upright=1.6 |Powers of {{math|σ}} within a silver rectangle.]]
Assume a silver rectangle has been constructed as indicated above, with height {{math|1}}, length {{tmath|\sigma}} and [[diagonal]] length <math> \sqrt{\sigma^2 +1}</math>. The triangles on the diagonal have [[Altitude (triangle)|altitudes]] <math>1 /\sqrt{1 +\sigma^{-2}}\,;</math> each perpendicular foot divides the diagonal in ratio {{tmath|\sigma^2.}}
If an horizontal line is drawn through the intersection point of the diagonal and the internal [[Edge (geometry)|edge]] of a [[Rabatment of the rectangle|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: {{cite journal |last=Crilly |first=Tony |date=1994 |title=A supergolden rectangle |journal=[[The Mathematical Gazette]] |volume=78 |issue=483 |pages=320–325 |doi=10.2307/3620208 |jstor=3620208}}</ref>
Relative to [[Vertex (geometry)|vertex]] {{math|A}}, the coordinates of feet of altitudes {{math|U}} and {{math|V}} 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 {{math|U}} and {{math|V}}, 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:
[[File:Silver_triangle.svg |thumb |upright=1.5 |Diagonal segments of the silver rectangle measure the silver triangle. The ratio {{math|size=94%|AB:AS}} is {{math|σ.}}]]
<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 {{tmath|1=\sigma =\tan(\alpha).}}
Both the lengths of the diagonal sections and the trigonometric values are elements of [[Quartic equation#Biquadratic equations|biquadratic]] [[Algebraic number field|number field]] <math>K =\mathbb{Q}\left( \sqrt{2 +\sqrt{2}} \right).</math>
The [[#rhomb|silver rhombus]] with edge {{tmath|1}} has diagonal lengths equal to {{tmath|\overline{U V} }} and {{tmath|2\overline{A U}.}} The regular [[octagon]] with edge {{tmath|2}} has long diagonals of length {{tmath|2\overline{A B} }} 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 {{math|σ}} and the diagonal segments of the silver rectangle, as illustrated above, ''pars pro toto'' on a single triangle.
The leg to base ratio {{tmath|\overline{A B} /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 [[Moorish architecture|architecture]] and [[Zellij|intricate decorations]] of the [[Middle Ages|mediæval]] [[Mosque–Cathedral of Córdoba|Mosque of Córdoba]], [[Andalusia]].<ref>{{cite journal |last1=Redondo Buitrago |first1=Antonia |last2=Reyes Iglesias |first2=Encarnación |date=2008 |title=The Geometry of the Cordovan Polygons |url=https://www.mi.sanu.ac.rs/vismath/redondo2009/cordovan.pdf |journal=Visual Mathematics |volume=10 |issue=4 |pages=<!-- E journal --> |publisher=Mathematical Institute |publication-place=Belgrade |issn=1821-1437 |access-date=December 11, 2024}}</ref>
===Silver spiral===
[[File:Silver_spiral.svg |thumb |upright=1.5 |Silver spirals with different initial angles on a {{math|σ}}− rectangle.]]
A silver spiral is a [[logarithmic spiral]] that gets wider by a factor of {{tmath|\sigma}} for every quarter turn. It is described by the [[polar equation]] <math>r( \theta) =a \exp(k \theta),</math> with initial radius {{tmath|a}} 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>
{{br}}
===Ammann–Beenker tiling===
[[File:AmmanSubstRules.png |thumb |upright=1.3 |Patch inflation of Ammann A5-tiles with factor {{math|σ{{sup|2}}.}}]]
The silver ratio appears prominently in the [[Ammann–Beenker tiling]], a [[Aperiodic_tiling#Constructions|non-periodic tiling]] of the plane with octagonal symmetry, build from a square and [[#rhomb|silver rhombus]] with equal side lengths. Discovered by [[Robert Ammann]] in 1977, its algebraic properties were described by Frans Beenker five years later.<ref>{{cite conference |url=https://archive.bridgesmathart.org/2007/bridges2007-377.pdf |title=Images of the Ammann-Beenker Tiling |last=Harriss |first=Edmund |author-link=Edmund Harriss |date=2007 |publisher=The Bridges Organization |pages=377–378 |___location=San Sebastián |conference=Bridges Donostia: Mathematics, music, art, architecture, culture}}</ref>
If the squares are cut into two triangles, the inflation factor for [[Ammann–Beenker tiling#Description of the tiles|Ammann A5-tiles]] is {{tmath|\sigma^2,}} the dominant [[Eigenvalues and eigenvectors#Eigenvalues and eigenvectors of matrices|eigenvalue]] of substitution [[Matrix (mathematics)|matrix]] <math display="block"> M =\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}.</math>
==
* 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 mean]]s – 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>
==Notes==
{{Notelist}}
==References==
{{Reflist}}
==External links==
*[https://www.youtube.com/watch?v=7lRgeTmxnlg YouTube lecture on the silver ratio, Pell sequence and metallic means]
*[http://www.maecla.it/tartapelago/museo/oro/rettangoli/en%20silverrectangle.htm Silver rectangle and Pell sequence] at Tartapelago by Giorgio Pietrocola
{{Algebraic numbers}}
{{Irrational numbers}}
{{Metallic ratios}}
[[Category:
[[Category:Mathematical constants]]
[[Category:History of geometry]]
[[Category:Metallic means]]
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