Silver ratio

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: $$\backslash frac\{a\}\{b\}\; =\backslash frac\{2a+b\}\{a\}$$

The ratio $\backslash frac\{a\}\{b\}$ is here denoted {{tmath|\sigma.}}{{efn |1=Variously {{math|T(2),}}{{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}} Metallic_mean Truncated_cube{{mathworld |id=SilverRatio|title=Silver ratio}}

{{math|σ{{sub|Ag}}.}}{{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}} The last notation is adopted without the subscript, which is relevant only to the context of metallic means.}}

Based on this definition, one has

$$\backslash begin\{align\}$$

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}

It follows that the silver ratio is found as the positive solution of the quadratic equation $\backslash sigma^\{2\}\; -2\backslash sigma\; -1\; =0.$ The quadratic formula gives the two solutions $1\; \backslash pm\; \backslash sqrt\{2\},$ the decimal expansion of the positive root begins as {{tmath|2.414\,213\,562\,373\,095...}} {{OEIS|A014176}}.

Using the tangent function

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

or the hyperbolic sine

:$\backslash sigma\; =\backslash exp(\; \backslash operatorname\{arsinh\}(1)\; ).${{cite OEIS|A014176}}

{{tmath|\sigma}} is the superstable fixed point of the iteration $x\; \backslash gets\; \backslash tfrac12\; (x^2+1)\; /(x-1),\; \backslash text\{\; with\; \}\; x\_0\; \backslash in\; [2,3]$

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

File:SilverSquare_6.svg

The defining equation can be written

$$\backslash begin\{align\}$$

1 &=\frac{1}{\sigma -1} + \frac{1}{\sigma +1} \\

&=\frac{2}{\sigma +1} + \frac{1}{\sigma}.\end{align}

The silver ratio can be expressed in terms of itself as fractions

$$\backslash begin\{align\}$$

\sigma &=\frac{1}{\sigma -2} \\

\sigma^2 &=\frac{\sigma -1}{\sigma -2} +\frac{\sigma +1}{\sigma -1}.\end{align}

Similarly as the infinite geometric series

$$\backslash begin\{align\}$$

\sigma &=2\sum_{n=0}^{\infty} \sigma^{-2n} \\

\sigma^2 &=-1 +2\sum_{n=0}^{\infty} (\sigma -1)^{-n}.\end{align}

For every integer {{tmath|n}} one has

$$\backslash begin\{align\}$$

\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}

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

Continued fraction pattern of a few low powers

$$\backslash begin\{align\}$$

\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}

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

The silver ratio is a Pisot number,{{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}} the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value $\backslash sqrt\{2\}\; -1$ of the algebraic conjugate is smaller than {{math|1,}} thus powers of {{tmath|\sigma}} generate almost integers and the sequence $\backslash sigma^n\; \backslash bmod\; 1$ is dense at the borders of the unit interval.{{mathworld |id=PowerFractionalParts |title=Power Fractional Parts}}

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

The silver ratio can be used as base of a 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

:$x\; =\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a\_n\}\{\backslash sigma^n\},$ with weights {{tmath| a_n \in [0,1,2].}}

File:Sigmary_scale.svg 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

$$\backslash begin\{align\}$$

\sigma^{n+1} &=2\sigma^n +\sigma^{n-1} \\

\sigma^{n+1} +\sigma^{n-1} &=2\sigma^n +2\sigma^{n-1},\end{align}

digit blocks $21\_\backslash sigma\; \backslash text\{\; and\; \}\; 22\_\backslash sigma$ carry to the next power of {{tmath|\sigma,}} resulting in $100\_\backslash sigma\; \backslash text\{\; and\; \}\; 101\_\backslash sigma.$ The number one has finite and infinite representations $1.0\_\backslash sigma,\; 0.21\_\backslash sigma$ and $0.\backslash overline\{20\}\_\backslash sigma,\; 0.1\backslash overline\{2\}\_\backslash sigma,$ 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 number $10\; =111.12\_\backslash sigma,$ $7\backslash sigma\; +3\; =1100\_\backslash sigma\; \backslash ,$ and $\backslash tfrac\{1\}\{\backslash sigma\; -1\}\; =0.\backslash overline\{1\}\_\backslash sigma.$

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

• Every algebraic integer $\backslash xi\; =a\; +b\backslash sigma\; \backslash text\{\; in\; \}\; K$ has a finite expansion.{{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}}
• Every rational number $\backslash rho\; =\backslash tfrac\{a\; +b\backslash sigma\}\{c\; +d\backslash sigma\}\; \backslash text\{\; in\; \}\; K$ has a purely periodic expansion.{{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}}
• 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.}}{{harvtxt|Schmidt|1980|p=275}}: Theorem 3.4 It follows by repeated substitution of {{tmath|x{{=}}n +\frac{1}{x} }} that all positive solutions $\backslash tfrac12\; \backslash left(n\; +\backslash sqrt\{n^2\; +\; 4\; \backslash vphantom\{/\}\; \}\; \backslash right)$ have a purely periodic continued fraction expansion $$\backslash sigma\_n\; =n\; +\backslash cfrac\{1\}\{n\; +\backslash cfrac\{1\}\{n\; +\backslash cfrac\{1\}\{\backslash ddots\}\}\}$$

Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.{{harvtxt|Spinadel|1997}}

File:SilverWord_im.png

{{main|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

$$P\_\{n\}\; =2P\_\{n-1\}\; +P\_\{n-2\}\; \backslash text\{\; for\; \}\; n\; >\; 1,$$

with initial values $$P\_\{0\}\; =0,\; P\_\{1\}\; =1.$$

The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,... {{OEIS|A000129}}. The limit ratio of consecutive terms is the silver mean.

Fractions of Pell numbers provide rational approximations of {{tmath|\sigma}} with error

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

Powers of {{tmath|\sigma}} can be written with Pell numbers as linear coefficients $$\backslash sigma^n\; =\backslash sigma\; P\_n\; +P\_\{n-1\},$$ 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

:{{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 }}

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

:$P\_n\; =a(\; \backslash sigma^n\; -\backslash bar\{\backslash sigma\}^n\; ),$ with {{tmath|a}} the positive root of $8x^2\; -1\; =0.${{br}}

Since the number {{tmath|P_{n} }} is the nearest integer to $a\backslash ,\backslash sigma^\{n\},$ with $a\; =1\; /\backslash sqrt\{8\}$ and {{math|n ≥ 0.}}

The Binet formula $\backslash sigma^n\; +\backslash bar\{\backslash sigma\}^n$ defines the companion sequence $Q\_\{n\}\; =P\_\{n+1\}\; +P\_\{n-1\}.$

The first few terms are 2, 2, 6, 14, 34, 82, 198,... {{OEIS|A002203}}.

This Pell-Lucas sequence has the Fermat property: if p is prime, $Q\_\{p\}\; \backslash equiv\; Q\_\{1\}\; \backslash bmod\; p.$ The converse does not hold, the least odd pseudoprimes $\backslash ,n\; \backslash mid\; (Q\_\{n\}\; -2)$ are 13{{sup|2}}, 385, 31{{sup|2}}, 1105, 1121, 3827, 4901.{{cite OEIS|A330276}}

{{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 Fibonacci, Pell, Lucas-Selfridge or base-2 Fermat pseudoprimes.{{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}}}}

Pell numbers are obtained as integral powers {{math|n > 2}} of a matrix with positive eigenvalue {{tmath|\sigma}}

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

File:Silver_rectangle_construction.svg

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) — the ratio is retained if the {{math|√2}} rectangle is folded in half, parallel to the short edges. 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.{{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}} 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 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 $\backslash sqrt\{\backslash sigma\; +1\; \backslash vphantom\{/\}\; \},\; \backslash ;\backslash sigma$ and $\backslash sqrt\{2(\backslash sigma\; +1)\; \backslash vphantom\{/\}\; \}.$ The coordinates of the vertices are given by the {{math|8}} permutations of $\backslash left(\; \backslash pm\; \backslash tfrac12,\; \backslash pm\; \backslash tfrac\{\backslash sigma\}\{2\}\; \backslash right).${{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}} The paper square has edge length {{tmath|\sigma -1}} and area {{tmath|2.}} The triangles have areas $1,\; \backslash frac\{\backslash sigma\; -1\}\{\backslash sigma\}$ and $\backslash frac\{1\}\{\backslash sigma\}\; ;$ the rectangles have areas $\backslash sigma\; -1\; \backslash text\{\; and\; \}\; \backslash frac\{1\}\{\backslash sigma\}.$

File:Silver_rectangle_whirl.svg

Divide a rectangle with sides in ratio {{math|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 {{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 whirl of converging silver rectangles.{{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}}

The logarithmic spiral through the vertices of adjacent triangles has polar slope $k\; =\backslash frac\{4\}\{\backslash pi\}\; \backslash ln(\; \backslash sigma).$ The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio {{tmath|\sigma}}, hence is a silver rhombus.

If the triangles have legs of length {{tmath|1}} then each discrete spiral has length $\backslash frac\{\backslash sigma\}\{\backslash sigma\; -1\}\; =\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash sigma^\{-n\}\; .$ The areas of the triangles in each spiral region sum to $\backslash frac\{\backslash sigma\}\{4\}\; =\backslash tfrac12\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash sigma^\{-2n\}\; ;$ the perimeters are equal to {{tmath|\sigma +2}} (light grey) and {{tmath|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 {{tmath|\tfrac{1}{\sigma -1},}} and iteration on edge length {{math|2}} gives an angular spiral of length {{tmath|\sigma +1.}}

File:Rhombicuboctahedron_by_Cutting_Rhombic_Dodecahedron.svg

The silver mean has connections to the following Archimedean solids 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 $(\; \backslash pm\; \backslash sigma,\; \backslash pm\; 1,\; \backslash pm\; 1),$ thus three mutually-perpendicular silver rectangles touch six of its square faces.{{br}}

The midradius is $\backslash sqrt\{2(\backslash sigma\; +1)\; \backslash vphantom\{/\}\; \},$ the centre radius for the square faces is {{tmath|\sigma.}}{{cite web |url=http://www.dmccooey.com/polyhedra/Rhombicuboctahedron.html |title=Rhombicuboctahedron |last=McCooey |first=David |website=Visual Polyhedra |access-date=11 December 2024}}

• Truncated cube

Coordinates: 24 permutations of $(\; \backslash pm\; \backslash sigma,\; \backslash pm\; \backslash sigma,\; \backslash pm\; 1).${{br}}

Midradius: {{tmath|\sigma +1,}} centre radius for the octagon faces: {{tmath|\sigma.}}{{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}}

• Truncated cuboctahedron

Coordinates: 48 permutations of $(\; \backslash pm\; (2\backslash sigma\; -1),\; \backslash pm\; \backslash sigma,\; \backslash pm\; 1).${{br}}

Midradius: $\backslash sqrt\{6(\backslash sigma\; +1)\; \backslash vphantom\{/\}\; \},$ centre radius for the square faces: {{tmath|\sigma +2,}} for the octagon faces: {{tmath|2\sigma -1.}}{{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}}

See also the dual Catalan solids

• Tetragonal trisoctahedron
• Trisoctahedron
• Hexakis octahedron

File:Silver triangle spiral.svg

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 {{tmath|2 :3 :3.}} The apex angle measures {{tmath|360 /8{{=}}45,}} each base angle {{tmath|67 \tfrac12}} degrees. It follows that the height to base ratio is $\backslash tfrac12\; \backslash tan(67\; \backslash tfrac12)\; =\backslash tfrac\{\backslash sigma\}\{2\}.$

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 $67\; \backslash tfrac12\; /3\; =22\; \backslash tfrac12,\; 45\; \backslash text\{\; and\; \}\; 112\; \backslash tfrac12$ degrees in the ratios {{tmath|1 :2 :5.}} From the law of sines, its edges are in ratios $1\; :\backslash sqrt\{\backslash sigma\; +1\}\; :\backslash sigma.$

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 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.Proved for the 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}}

The assumption is verified by construction, as demonstrated in the vector image.

The centre of rotation has barycentric coordinates

$$\backslash left(\; \backslash tfrac\{\backslash sigma\; +1\}\{\backslash sigma\; +5\}\; :\backslash tfrac\{2\}\{\backslash sigma\; +5\}\; :\backslash tfrac\{2\}\{\backslash sigma\; +5\}\; \backslash right)\; \backslash sim\; \backslash left(\; \backslash tfrac\{\backslash sigma\; +1\}\{2\}\; :1\; :1\; \backslash right),$$

the three whorls of stacked gnomons have areas in ratios

$$\backslash left(\; \backslash tfrac\{\backslash sigma\; +1\}\{2\}\; \backslash right)^\{2\}\; :\backslash tfrac\{\backslash sigma\; +1\}\{2\}\; :1.$$

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

:$k\; =\backslash frac\{4\}\{5\backslash pi\}\; \backslash ln\; \backslash left(\; \backslash tfrac\{\backslash sigma\}\{\backslash sigma-1\}\; \backslash right),$ or an expansion rate of {{tmath| \tfrac{\sigma +1}{2} }} for every {{tmath|225}} degrees of rotation.

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

File:Silver_rectangle_segmented.svg

Assume a silver rectangle has been constructed as indicated above, with height {{math|1}}, length {{tmath|\sigma}} and diagonal length $\backslash sqrt\{\backslash sigma^2\; +1\}$. The triangles on the diagonal have altitudes $1\; /\backslash sqrt\{1\; +\backslash sigma^\{-2\}\}\backslash ,;$ 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 of a rabatment square, the parent silver rectangle and the two scaled copies along the diagonal have areas in the ratios $\backslash sigma^2\; :2\; :1\backslash ,,$ the rectangles opposite the diagonal both have areas equal to $\backslash tfrac\{2\}\{\backslash sigma\; +1\}.$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}}

Relative to vertex {{math|A}}, the coordinates of feet of altitudes {{math|U}} and {{math|V}} are

$$\backslash left(\; \backslash tfrac\{\backslash sigma\}\{\backslash sigma^2\; +1\},\; \backslash tfrac\{1\}\{\backslash sigma^2\; +1\}\; \backslash right)\; \backslash text\{\; and\; \}\; \backslash left(\; \backslash tfrac\{\backslash sigma\}\{1\; +\backslash sigma^\{-2\}\},\; \backslash tfrac\{1\}\{1\; +\backslash sigma^\{-2\}\}\; \backslash right).$$

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 $\backslash alpha\; =67\; \backslash tfrac12$ degrees, the base angle of the silver triangle:

File:Silver_triangle.svg

$$\backslash begin\{align\}$$

\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}

:with {{tmath|1=\sigma =\tan(\alpha).}}

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

The 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 architecture and intricate decorations of the mediæval Mosque of Córdoba, Andalusia.{{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= |publisher=Mathematical Institute |publication-place=Belgrade |issn=1821-1437 |access-date=December 11, 2024}}

File:Silver_spiral.svg

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 $r(\; \backslash theta)\; =a\; \backslash exp(k\; \backslash theta),$ with initial radius {{tmath|a}} and parameter $k\; =\backslash frac\{2\}\{\backslash pi\}\; \backslash ln(\; \backslash sigma).$ 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 $1/\; \backslash sigma.$

{{br}}

File:AmmanSubstRules.png

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.{{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}}

If the squares are cut into two triangles, the inflation factor for Ammann A5-tiles is {{tmath|\sigma^2,}} the dominant eigenvalue of substitution matrix

• 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 $x^3\; =2x^2\; +1$

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{{Reflist}}

• [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

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{{Irrational numbers}}

{{Metallic ratios}}

Category:Quadratic irrational numbers

Category:Mathematical constants

Category:History of geometry

Category:Metallic means