Silver ratio

Representations Silver rectangle 2.4142135623730950488... 1 + √2 ${\displaystyle \textstyle 2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}}$ 10.01101010000010011110... 2.6A09E667F3BCC908B2F...
Silver ratio within the octagon

In mathematics, two quantities are in the silver ratio (or silver mean)[1][2] if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an 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 numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.

Mathematicians 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 numbers, Pell numbers, octagons and the like.

The relation described above can be expressed algebraically:

${\displaystyle {\frac {2a+b}{a}}={\frac {a}{b}}\equiv \delta _{S}}$

or equivalently,

${\displaystyle 2+{\frac {b}{a}}={\frac {a}{b}}\equiv \delta _{S}}$

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

${\displaystyle 2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}=\delta _{S}}$

The convergents of this continued fraction (2/1, 5/2, 12/5, 29/12, 70/29, ...) are ratios of consecutive Pell numbers. These fractions provide accurate 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:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt2.[3]

Calculation

For comparison, two quantities a, b with a > b > 0 are said to be in the golden ratio φ if,

${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi }$

However, they are in the silver ratio δS if,

${\displaystyle {\frac {2a+b}{a}}={\frac {a}{b}}=\delta _{S}.}$

Equivalently,

${\displaystyle 2+{\frac {b}{a}}={\frac {a}{b}}=\delta _{S}}$

Therefore,

${\displaystyle 2+{\frac {1}{\delta _{S}}}=\delta _{S}.}$

Multiplying by δS and rearranging gives

${\displaystyle {\delta _{S}}^{2}-2\delta _{S}-1=0.}$

Using the quadratic formula, two solutions can be obtained. Because δS is the ratio of positive quantities, it is necessarily positive, so,

${\displaystyle \delta _{S}=1+{\sqrt {2}}=2.41421356237\dots }$

Properties

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...
Silver spirals within the silver rectangle

Number-theoretic properties

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − 2 = −1/δS ≈ −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 δ n
S
to the nearest integer is 1/δ n
S
≈ 0.41n
. Thus, the sequence of fractional parts of δ n
S
, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

Powers

The lower powers of the silver ratio are

${\displaystyle \delta _{S}^{-1}=1\delta _{S}-2=[0;2,2,2,2,2,\dots ]\approx 0.41421}$
${\displaystyle \delta _{S}^{0}=0\delta _{S}+1=[1]=1}$
${\displaystyle \delta _{S}^{1}=1\delta _{S}+0=[2;2,2,2,2,2,\dots ]\approx 2.41421}$
${\displaystyle \delta _{S}^{2}=2\delta _{S}+1=[5;1,4,1,4,1,\dots ]\approx 5.82842}$
${\displaystyle \delta _{S}^{3}=5\delta _{S}+2=[14;14,14,14,\dots ]\approx 14.07107}$
${\displaystyle \delta _{S}^{4}=12\delta _{S}+5=[33;1,32,1,32,\dots ]\approx 33.97056}$

The powers continue in the pattern

${\displaystyle \delta _{S}^{n}=K_{n}\delta _{S}+K_{n-1}}$

where

${\displaystyle K_{n}=2K_{n-1}+K_{n-2}}$

For example, using this property:

${\displaystyle \delta _{S}^{5}=29\delta _{S}+12=[82;82,82,82,\dots ]\approx 82.01219}$

Using K0 = 1 and K1 = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation

${\displaystyle K_{n}=2K_{n-1}+K_{n-2}}$

which becomes

${\displaystyle K_{n}={\frac {1}{2{\sqrt {2}}}}\left(\delta _{S}^{n+1}-{(2-\delta _{S})}^{n+1}\right)}$

Trigonometric properties

The silver ratio is intimately connected to trigonometric ratios for π/8 = 22.5°.

${\displaystyle \tan {\frac {\pi }{8}}={\sqrt {2}}-1={\frac {1}{\delta _{s}}}}$
${\displaystyle \cot {\frac {\pi }{8}}=\tan {\frac {3\pi }{8}}={\sqrt {2}}+1=\delta _{s}}$

So the area of a regular octagon with side length a is given by

${\displaystyle A=2a^{2}\cot {\frac {\pi }{8}}=2\delta _{s}a^{2}\simeq 4.828427a^{2}.}$