白銀比例

（重定向自白銀分割

 數表—无理数√2 - φ - √3 - √5 - δS - e - π

${\displaystyle \delta _{S}\approx }$2.41421356...

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

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

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

性質

乘幂

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

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

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

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

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

${\displaystyle K_{n}}$ 可以表示為以下的式子

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

三角性質

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

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

紙張大小及白銀長方形

ISO 216紙張尺寸其長寬之間的比例為${\displaystyle 1:{\sqrt {2}}}$ ，若其中切掉一塊邊長和長方形短邊相同的正方形，剩下的長方形長寬比例為${\displaystyle 1:{\sqrt {2}}-1}$ ，也等於${\displaystyle 1+{\sqrt {2}}:1}$ ，此比例和白銀比例有關。若一長方形的縱橫比為白銀比例，此長方形有時會稱為「白銀長方形」，不過白銀長方形也可以指縱橫比為√2的長方形。

參考資料

1. Kapusta, Janos, The square, the circle, and the golden proportion: a new class of geometrical constructions (PDF), Forma, 2004, 19: 293–313 [2012-08-09], （原始内容存档 (PDF)于2020-09-18）