三十一角形
cos  2 π 31 = x 1 6 + 6 − x 1 + x 2 − x 3 6 129 − 10 x 1 + 81 x 2 − 48 x 3 − 30 x 4 134 6 − x 1 + x 2 − x 3 + i 27 ( 1091 − 96 x 1 − 348 x 2 − 367 x 3 + 114 x 4 ) 134 6 − x 1 + x 2 − x 3 3 + 6 − x 1 + x 2 − x 3 6 129 − 10 x 1 + 81 x 2 − 48 x 3 − 30 x 4 134 6 − x 1 + x 2 − x 3 − i 27 ( 1091 − 96 x 1 − 348 x 2 − 367 x 3 + 114 x 4 ) 134 6 − x 1 + x 2 − x 3 3 {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{31}}={\frac {x_{1}}{6}}+{\frac {\sqrt {6-x_{1}+x_{2}-x_{3}}}{6}}{\sqrt[{3}]{{\frac {129-10x_{1}+81x_{2}-48x_{3}-30x_{4}}{134{\sqrt {6-x_{1}+x_{2}-x_{3}}}}}+i{\frac {\sqrt {27(1091-96x_{1}-348x_{2}-367x_{3}+114x_{4})}}{134{\sqrt {6-x_{1}+x_{2}-x_{3}}}}}}}\\+{\frac {\sqrt {6-x_{1}+x_{2}-x_{3}}}{6}}{\sqrt[{3}]{{\frac {129-10x_{1}+81x_{2}-48x_{3}-30x_{4}}{134{\sqrt {6-x_{1}+x_{2}-x_{3}}}}}-i{\frac {\sqrt {27(1091-96x_{1}-348x_{2}-367x_{3}+114x_{4})}}{134{\sqrt {6-x_{1}+x_{2}-x_{3}}}}}}}\end{aligned}}}