三十三角形
ここで、以下の関係式を使って
2 cos  22 π 33 = 2 cos  2 π 3 = − 1 2 cos  6 π 33 + 2 cos  12 π 33 + 2 cos  18 π 33 + 2 cos  24 π 33 + 2 cos  30 π 33 = 2 cos  2 π 11 + 2 cos  4 π 11 + 2 cos  6 π 11 + 2 cos  8 π 11 + 2 cos  10 π 11 = − 1 {\displaystyle {\begin{aligned}&2\cos {\frac {22\pi }{33}}=2\cos {\frac {2\pi }{3}}=-1\,\\&2\cos {\frac {6\pi }{33}}+2\cos {\frac {12\pi }{33}}+2\cos {\frac {18\pi }{33}}+2\cos {\frac {24\pi }{33}}+2\cos {\frac {30\pi }{33}}\\&=2\cos {\frac {2\pi }{11}}+2\cos {\frac {4\pi }{11}}+2\cos {\frac {6\pi }{11}}+2\cos {\frac {8\pi }{11}}+2\cos {\frac {10\pi }{11}}=-1\\\end{aligned}}}