五十四角形
以下のようにα、β、γを定義すると
α = 2 cos  2 π 54 ⋅ 2 cos  34 π 54 ⋅ 2 cos  38 π 54 β = 2 cos  10 π 54 ⋅ 2 cos  46 π 54 ⋅ 2 cos  26 π 54 γ = 2 cos  50 π 54 ⋅ 2 cos  14 π 54 ⋅ 2 cos  22 π 54 {\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{54}}\cdot 2\cos {\frac {34\pi }{54}}\cdot 2\cos {\frac {38\pi }{54}}\\&\beta =2\cos {\frac {10\pi }{54}}\cdot 2\cos {\frac {46\pi }{54}}\cdot 2\cos {\frac {26\pi }{54}}\\&\gamma =2\cos {\frac {50\pi }{54}}\cdot 2\cos {\frac {14\pi }{54}}\cdot 2\cos {\frac {22\pi }{54}}\\\end{aligned}}}
三次方程式の係数を求めると
α + β + γ = 0 α β + β γ + γ α = − 3 α β γ = 1 {\displaystyle {\begin{aligned}&\alpha +\beta +\gamma =0\\&\alpha \beta +\beta \gamma +\gamma \alpha =-3\\&\alpha \beta \gamma =1\\\end{aligned}}}
三次方程式は