Silver ratio

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 = −1/δ_S ≈ −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.

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

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

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

${\displaystyle \sin {\frac {\pi }{8}}={\frac {\sqrt {2-{\sqrt {2}}}}{2}}={\frac {\sqrt {1-1/\delta _{s}}}{2}}}$

${\displaystyle \cos {\frac {\pi }{8}}={\frac {\sqrt {2+{\sqrt {2}}}}{2}}={\frac {\sqrt {1+\delta _{s}}}{2}}}$

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

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

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

${\displaystyle A=2a^{2}\cot {\frac {\pi }{8}}=2(1+{\sqrt {2}})a^{2}\simeq 4.828427a^{2}.}$