二十四角形
cos  2 π 24 = cos  π 12 = 1 4 ( 6 + 2 ) = 1 2 2 + 3 = 3 + 1 2 2 {\displaystyle \cos {\frac {2\pi }{24}}=\cos {\frac {\pi }{12}}={\frac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {3}}}}={\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}}
cos  ( 2 π / 24 ) {\displaystyle \cos(2\pi /24)} を立方根で表すと
cos  2 π 24 = 1 2 ( 1 + i 2 3 + 1 − i 2 3 ) {\displaystyle \cos {\frac {2\pi }{24}}={\frac {1}{2}}\left({\sqrt[{3}]{\frac {1+i}{\sqrt {2}}}}+{\sqrt[{3}]{\frac {1-i}{\sqrt {2}}}}\right)}