百四十四角形
cos  ( 2 π / 144 ) {\displaystyle \cos(2\pi /144)} を平方根と立方根で表すと
cos  2 π 144 = 1 4 4 2 + 2 + 3 + i ⋅ 4 2 − 2 + 3 3 + 1 4 4 2 + 2 + 3 − i ⋅ 4 2 − 2 + 3 3 {\displaystyle \cos {\frac {2\pi }{144}}={\frac {1}{4}}{\sqrt[{3}]{4{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}+i\cdot 4{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}}}+{\frac {1}{4}}{\sqrt[{3}]{4{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}-i\cdot 4{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}}}}
三次方程式を解くと
u 1 = 2 cos  2 π 144 = cos  2 π 48 + i sin  2 π 48 3 + cos  2 π 48 − i sin  2 π 48 3 4 cos  2 π 144 = 8 cos  2 π 48 + i 8 sin  2 π 48 3 + 8 cos  2 π 48 − i 8 sin  2 π 48 3 4 cos  2 π 144 = 4 2 + 2 + 3 + i ⋅ 4 2 − 2 + 3 3 + 4 2 + 2 + 3 − i ⋅ 4 2 − 2 + 3 3 {\displaystyle {\begin{aligned}u_{1}=2\cos {\frac {2\pi }{144}}=&{\sqrt[{3}]{\cos {\frac {2\pi }{48}}+i\sin {\frac {2\pi }{48}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{48}}-i\sin {\frac {2\pi }{48}}}}\\4\cos {\frac {2\pi }{144}}=&{\sqrt[{3}]{8\cos {\frac {2\pi }{48}}+i8\sin {\frac {2\pi }{48}}}}+{\sqrt[{3}]{8\cos {\frac {2\pi }{48}}-i8\sin {\frac {2\pi }{48}}}}\\4\cos {\frac {2\pi }{144}}=&{\sqrt[{3}]{4{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}+i\cdot 4{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}}}+{\sqrt[{3}]{4{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}-i\cdot 4{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}}}\\\end{aligned}}}