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{{Refimprove|date=April 2016}}
{{For|a ratio also known as the silver ratio<ref>Weisstein, Eric W. (2002). ''CRC Concise Encyclopedia of Mathematics, Second Edition'', p.1207-8. CRC Press. {{isbn|9781420035223}}.</ref>|Golden ratio conjugate}}
{{Redirects|Silver triangle||Kepler triangle}}
[[File:Silver rectangle.svg|thumb|Silver rectangle]]
{| class="infobox bordered" cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| {{gaps|10.0110|1010|0000|1001|1110|…}}
|-
| [[Decimal]]
| {{gaps|2.41421|35623|73095|0488…}}
|-
| [[Hexadecimal]]
| {{gaps|2.6A09|E667|F3BC|C908|B2F…}}
|-
| [[Continued fraction]]
| <math>\textstyle 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{\ddots}}}}</math>
|-
| [[Algebraic number|Algebraic form]]
| 1 + {{sqrt|2}}
|}
[[File:Silver ratio octagon.png|thumb|120px|Silver ratio within the octagon]]

In [[mathematics]], two quantities are in the '''silver ratio''' (also '''silver mean''' or '''silver constant''') if the [[ratio]] of the sum of the smaller and twice the larger of those quantities, to the larger quantity, is the same as the ratio of the larger one to the smaller one (see below). This defines the silver ratio as an [[irrational number|irrational]] [[mathematical constant]], whose value of one plus the [[square root of 2]] is approximately 2.4142135623. Its name is an allusion to the [[golden ratio]]; analogously to the way the golden ratio is the limiting ratio of consecutive [[Fibonacci number]]s, the silver ratio is the limiting ratio of consecutive [[Pell number]]s. The silver ratio is denoted by {{math|''δ<sub>S</sub>''}}.

[[Mathematician]]s have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, [[square triangular number]]s, [[Pell numbers]], [[octagon]]s and the like.

The relation described above can be expressed algebraically:

:<math> \frac{2a + b}{a} = \frac{a}{b} \equiv \delta_S</math>

or equivalently,

:<math> 2 + \frac{b}{a} = \frac{a}{b} \equiv \delta_S</math>

The silver ratio can also be defined by the simple [[continued fraction]] [2; 2, 2, 2, ...]:

:<math> 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}} =\delta_S </math>

The [[Convergent (continued fraction)|convergents]] of this continued fraction ({{sfrac|2|1}}, {{sfrac|5|2}}, {{sfrac|12|5}}, {{sfrac|29|12}}, {{sfrac|70|29}}, ...) are ratios of consecutive Pell numbers. These fractions provide accurate [[Diophantine approximation|rational approximations]] of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:{{math|''δ''<sub>''S''</sub>}}, and the 4 sides of the trapezoids are in a ratio of 1:1:1:{{math|''δ''<sub>''S''</sub>}}. If the edge length of a regular octagon is {{math|''t''}}, then the [[inscribed figure|inradius]] of the octagon (the distance between opposite sides) is {{math|''δ''<sub>''S''</sub>''t''}}, and the area of the octagon is {{math|2''δ''<sub>''S''</sub>''t''<sup>2</sup>}}.<ref name="kapusta"/>

== Calculation ==

For comparison, two quantities ''a'', ''b'' with ''a''&nbsp;>&nbsp;''b''&nbsp;>&nbsp;0 are said to be in the ''[[golden ratio]]'' {{math|''φ''}} if,

:<math> \frac{a+b}{a} = \frac{a}{b} = \varphi</math>

However, they are in the ''silver ratio'' {{math|''δ<sub>S</sub>''}} if,

:<math> \frac{2a+b}{a} = \frac{a}{b} = \delta_S.</math>

Equivalently,

:<math> 2+\frac{b}{a} = \frac{a}{b} = \delta_S</math>

Therefore,

:<math> 2 + \frac{1}{\delta_S} = \delta_S. </math>

Multiplying by {{math|''δ<sub>S</sub>''}} and rearranging gives

:<math>{\delta_S}^2 - 2\delta_S - 1 = 0.</math>

Using the [[quadratic formula]], two solutions can be obtained. Because {{math|''δ<sub>S</sub>''}} is the ratio of positive quantities, it is necessarily positive, so,

:<math>\delta_S = 1 + \sqrt{2} = 2.41421356237\dots</math>

== Properties ==
[[File:Silver rectangle repeats.svg|thumb|If one cuts two of the largest squares possible off of a silver rectangle one is left with a silver rectangle, to which the process may be repeated...]]
[[File:Silver spiral approximation.svg|thumb|Silver spirals within the silver rectangle]]

=== Number-theoretic properties ===
The silver ratio is a [[Pisot–Vijayaraghavan number]] (PV number), as its conjugate {{math|1 − {{sqrt|2}} {{=}} {{sfrac|−1|''δ<sub>S</sub>''}} ≈ −0.41}} has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from {{math|''δ{{su|p=&nbsp;n|b=S}}''}} to the nearest integer is {{math|{{sfrac|1|''δ{{su|p=&nbsp;n|b=S}}''}} ≈ 0.41<sup>''n''</sup>}}. Thus, the sequence of [[fractional part]]s of {{math|''δ{{su|p=&nbsp;n|b=S}}''}}, {{math|''n'' {{=}} 1, 2, 3, ...}} (taken as elements of the torus) converges. In particular, this sequence is not [[equidistributed sequence|equidistributed mod 1]].

=== Powers ===
The lower powers of the silver ratio are
:<math> \delta_S^{-1} = 1 \delta_S - 2 = [0;2,2,2,2,2,\dots] \approx 0.41421</math>
:<math> \delta_S^0 = 0 \delta_S + 1 = [1] = 1</math>
:<math> \delta_S^1 = 1 \delta_S + 0 = [2;2,2,2,2,2,\dots] \approx 2.41421</math>
:<math> \delta_S^2 = 2 \delta_S + 1 = [5;1,4,1,4,1,\dots] \approx 5.82842</math>
:<math> \delta_S^3 = 5 \delta_S + 2 = [14;14,14,14,\dots] \approx 14.07107</math>
:<math> \delta_S^4 = 12\delta_S + 5 = [33;1,32,1,32,\dots] \approx 33.97056 </math>

The powers continue in the pattern
:<math> \delta_S^n = K_n\delta_S + K_{n-1} </math>
where
:<math> K_n = 2 K_{n-1} + K_{n-2} </math>
For example, using this property:
:<math> \delta_S^5 = 29\delta_S + 12 = [82;82,82,82,\dots] \approx 82.01219 </math>

Using {{math|''K''<sub>0</sub> {{=}} 1}} and {{math|''K''<sub>1</sub> {{=}} 2}} as initial conditions, a [[Binet's formula|Binet]]-like formula results from solving the recurrence relation
:<math> K_n = 2 K_{n-1} + K_{n-2} </math>
which becomes
:<math> K_n = \frac{1}{2\sqrt{2}} \left(\delta_S^{n+1} - {(2-\delta_S)}^{n+1}\right) </math>

=== Trigonometric properties ===
{{see also|Trigonometric constants expressed in real radicals #22.5°: regular octagon}}
The silver ratio is intimately connected to trigonometric ratios for {{math|{{sfrac|π|8}} {{=}} 22.5°}}.
:<math>\sin \frac{\pi}{8} = \frac{\sqrt{2 - \sqrt 2}}{2}=\frac{\sqrt{1-1/ \delta_s}}{2} </math>
:<math>\cos \frac{\pi}{8} = \frac{\sqrt{2 + \sqrt 2}}{2}=\frac{\sqrt{1+\delta_s}}{2} </math>
:<math>\tan \frac{\pi}{8} = \sqrt{2}-1= \frac{1}{\delta_s} </math>
:<math>\cot \frac{\pi}{8} = \tan \frac{3\pi}{8} = \sqrt{2}+1=\delta_s </math>

So the area of a regular octagon with side length {{math|''a''}} is given by
:<math>A = 2a^2 \cot \frac{\pi}{8} = 2(1+\sqrt{2})a^2 \simeq 4.828427 a^2.</math>

== Paper sizes and silver rectangles ==
<!-- linked from redirect [[Silver rectangles]] -->
{{For|the square root of {{sqrt|2}} rectangle, as in origami|ISO 216#Application}}

[[File:A size illustration2.svg|150px|right|thumb]]
A rectangle whose aspect ratio is the silver ratio (1:{{sqrt|2}}, approximately 1:1.4142135 decimal) is sometimes called a '''silver rectangle''' by analogy with [[golden rectangle]]s.  The [[paper size]]s under [[ISO 216]] are such rectangles.  The 1:{{sqrt|2}} rectangles (rectangles with the shape of ISO 216 paper) have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.

Removing a largest possible square from such a rectangle leaves a rectangle with proportions {{nowrap|1 : ({{sqrt|2}} − 1)}} which is the same as {{nowrap|(1 + {{sqrt|2}}) : 1}}, the silver ratio. Removing a largest square from the resulting rectangle leaves one again with aspect ratio 1:{{sqrt|2}}.<ref>{{cite web|url=http://www.britishorigami.info/academic/lister/a4.php|title=The A4 rectangle|last=Lister|first=David|work=The Lister List|publisher=British Origami Society|location=England|accessdate=2009-05-06}}</ref>  Removing the largest possible square from either kind of silver rectangle yields a silver rectangle of the other kind, and then repeating the process once more gives a rectangle of the original shape but smaller by a linear factor of {{nowrap|1 + {{sqrt|2}}}}.<ref name="kapusta">{{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=http://www.scipress.org/journals/forma/pdf/1904/19040293.pdf}}.</ref>

==See also==
*[[Metallic mean]]s
*[[Ammann–Beenker tiling]]

== References ==
{{Reflist}}

== Further reading ==
*Buitrago, Antonia Redondo (2008). "Polygons, Diagonals, and the Bronze Mean", ''Nexus Network Journal 9,2: Architecture and Mathematics'', p.321-2. Springer Science & Business Media. {{isbn|9783764386993}}.

== External links ==
*{{mathworld|urlname=SilverRatio|title=Silver Ratio}}
*"[http://personal.maths.surrey.ac.uk/ext/R.Knott/Fibonacci/cfINTRO.html#silver  An Introduction to Continued Fractions: The Silver Means]", '' Fibonacci Numbers and the Golden Section''.
*"[http://www.maecla.it/tartapelago/museo/oro/rettangoli/en%20silverrectangle.htm Silver rectangle and its sequence]" at Tartapelago by Giorgio Pietrocola
{{Algebraic numbers}}
{{Fractions and ratios}}
{{Irrational number}}
{{Metallic ratios}}

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[[Category:Algebraic numbers]]
<!--[[Category:Continued fractions]]-->[[Category:Periodic continued fractions]]
[[Category:Irrational numbers]]
[[Category:Quadratic irrational numbers]]
[[Category:Mathematical constants]]
[[Category:Metallic means]]
[[Category:Ratios]]