# 白银比例本文重定向自 白銀比例

（重定向自白銀分割）
 无理数√2 - φ - √3 - √5 - δS - e - π 二进制 10.0110101000001001111... 十进制 2.4142135623730950488... 十六进制 2.6A09E667F3BCC908B2F... 连续分数 ${\displaystyle 2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}}$ 代数形式 ${\displaystyle {1+{\sqrt {2}}}}$

${\displaystyle 1+{\sqrt {2}}=2.4142135623730950488...\,.}$

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

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

## 性质

### 乘幂

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

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

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

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

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

${\displaystyle K_{n}}$可以表示为以下的式子

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

### 三角性质

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

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

## 纸张大小及白银长方形

ISO 216纸张尺寸其长宽之间的比例为${\displaystyle 1:{\sqrt {2}}}$，若其中切掉一块边长和长方形短边相同的正方形，剩下的长方形长宽比例为${\displaystyle 1:{\sqrt {2}}-1}$，也等于${\displaystyle 1+{\sqrt {2}}:1}$，此比例和白银比例有关。若一长方形的纵横比为白银比例，此长方形有时会称为“白银长方形”，不过白银长方形也可以指纵横比为√2的长方形。