五十六角形
u 1 = 14 − 2 6 + 2 7 + 2 7 3 cos  ( 1 3 arccos  ( − 5 2 + 14 4 7 + 2 7 ) ) {\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {2{\sqrt {7+2{\sqrt {7}}}}}{3}}\cos \left({\frac {1}{3}}\arccos \left(-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}\right)\right)}
平方根、立方根で表すと
u 1 = 14 − 2 6 + 7 + 2 7 3 − 5 2 + 14 4 7 + 2 7 + i 48 + 12 7 4 7 + 2 7 3 + 7 + 2 7 3 − 5 2 + 14 4 7 + 2 7 − i 48 + 12 7 4 7 + 2 7 3 {\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}}
三次方程式の係数を求めると
2 cos  2 π 56 ⋅ 2 cos  50 π 56 + 2 cos  50 π 56 ⋅ 2 cos  18 π 56 + 2 cos  18 π 56 ⋅ 2 cos  2 π 56 = 2 cos  26 π 28 + 2 cos  22 π 28 + 2 cos  10 π 28 + 2 cos  6 π 7 + 2 cos  4 π 7 + 2 cos  2 π 7 = − 7 − 1 = 2 x 4 2 cos  2 π 56 ⋅ 2 cos  50 π 56 ⋅ 2 cos  18 π 56 = x 4 − 2 = x 4 + x 1 + x 4 = x 1 + 2 x 4 {\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}+2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}+2\cos {\frac {18\pi }{56}}\cdot 2\cos {\frac {2\pi }{56}}\\&=2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}+2\cos {\frac {10\pi }{28}}+2\cos {\frac {6\pi }{7}}+2\cos {\frac {4\pi }{7}}+2\cos {\frac {2\pi }{7}}=-{\sqrt {7}}-1={\sqrt {2}}x_{4}\\&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}=x_{4}-{\sqrt {2}}=x_{4}+x_{1}+x_{4}=x_{1}+2x_{4}\\\end{aligned}}}
cos  ( 2 π / 56 ) {\displaystyle \cos(2\pi /56)} を平方根と立方根で表すと
cos  2 π 56 = 14 − 2 12 + 7 + 2 7 6 − 5 2 + 14 4 7 + 2 7 + i 48 + 12 7 4 7 + 2 7 3 + 7 + 2 7 6 − 5 2 + 14 4 7 + 2 7 − i 48 + 12 7 4 7 + 2 7 3 {\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}}