五十四角形
x 3 − 3 x − 1 = 0 {\displaystyle x^{3}-3x-1=0}
三角関数、逆三角関数を用いた解を求め、立方根を使った解を求めると
α = 2 cos  ( 1 3 arccos  1 2 ) = 1 2 + i 3 2 3 + 1 2 − i 3 2 3 = − ω 2 3 + − ω 3 {\displaystyle \alpha =2\cos \left({\frac {1}{3}}\arccos {\frac {1}{2}}\right)={\sqrt[{3}]{{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}}}+{\sqrt[{3}]{{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}}}={\sqrt[{3}]{-\omega ^{2}}}+{\sqrt[{3}]{-\omega }}}
三次方程式の係数を求めると
2 cos  2 π 54 + 2 cos  34 π 54 + 2 cos  38 π 54 = 0 2 cos  2 π 54 ⋅ 2 cos  34 π 54 + 2 cos  34 π 54 ⋅ 2 cos  38 π 54 + 2 cos  38 π 54 ⋅ 2 cos  2 π 54 = − 3 2 cos  2 π 54 ⋅ 2 cos  34 π 54 ⋅ 2 cos  38 π 54 = α {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{54}}+2\cos {\frac {34\pi }{54}}+2\cos {\frac {38\pi }{54}}=0\\&2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}+2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}+2\cos {\frac {38\pi }{54}}\cdot 2\cos {\frac {2\pi }{54}}=-3\\&2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}=\alpha \\\end{aligned}}}