五十一角形
cos  2 π 51 = cos  ( 12 π 17 − 2 π 3 ) = cos  ( π 3 − 5 π 17 ) = cos  π 3 cos  5 π 17 + sin  π 3 sin  5 π 17 = 1 2 cos  5 π 17 + 3 2 sin  5 π 17 = 1 2 ⋅ 1 16 ( + 1 + 17 + 34 + 68 − 68 − 2448 − 2720 − 6284288 ) + 3 2 ⋅ 1 8 ( 34 + 68 − 136 + 1088 + 272 − 39168 + 43520 − 1608777728 ) = 1 32 ( + 1 + 17 + 2 ⋅ 17 + 2 2 ⋅ 17 − 2 2 ⋅ 17 − 2 4 ⋅ 153 − 2 5 ⋅ 85 − 2 10 ⋅ 6137 + 2 3 ⋅ 51 + 2 6 ⋅ 153 − 2 7 ⋅ 153 + 2 14 ⋅ 1377 + 2 8 ⋅ 153 − 2 16 ⋅ 12393 + 2 17 ⋅ 6885 − 2 34 ⋅ 40264857 ) = 1 32 ( + 1 + 17 + 2 ⋅ 17 + 2 2 ⋅ 17 − 2 2 ⋅ 17 − 2 4 ⋅ 3 2 ⋅ 17 − 2 5 ⋅ 5 ⋅ 17 − 2 10 ⋅ 19 2 ⋅ 17 + 2 3 ⋅ 3 ⋅ 17 + 2 6 ⋅ 3 2 ⋅ 17 − 2 7 ⋅ 3 2 ⋅ 17 + 2 14 ⋅ 3 4 ⋅ 17 + 2 8 ⋅ 3 2 ⋅ 17 − 2 16 ⋅ 3 6 ⋅ 17 + 2 17 ⋅ 3 4 ⋅ 5 ⋅ 17 − 2 34 ⋅ 3 8 ⋅ 19 2 ⋅ 17 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{51}}=&\cos \left({\frac {12\pi }{17}}-{\frac {2\pi }{3}}\right)\\=&\cos \left({\frac {\pi }{3}}-{\frac {5\pi }{17}}\right)\\=&\cos {\frac {\pi }{3}}\cos {\frac {5\pi }{17}}+\sin {\frac {\pi }{3}}\sin {\frac {5\pi }{17}}\\=&{\frac {1}{2}}\cos {\frac {5\pi }{17}}+{\frac {\sqrt {3}}{2}}\sin {\frac {5\pi }{17}}\\=&{\frac {1}{2}}\cdot {\frac {1}{16}}\left(+1+{\sqrt {17}}+{\sqrt {34+{\sqrt {68}}}}-{\sqrt {68-{\sqrt {2448}}-{\sqrt {2720-{\sqrt {6284288}}}}}}\right)\\&+{\frac {\sqrt {3}}{2}}\cdot {\frac {1}{8}}\left({\sqrt {34+{\sqrt {68}}-{\sqrt {136+{\sqrt {1088}}}}+{\sqrt {272-{\sqrt {39168}}+{\sqrt {43520-{\sqrt {1608777728}}}}}}}}\right)\\=&{\frac {1}{32}}\left(+1+{\sqrt {17}}+{\sqrt {2\cdot 17+{\sqrt {2^{2}\cdot 17}}}}-{\sqrt {2^{2}\cdot 17-{\sqrt {2^{4}\cdot 153}}-{\sqrt {2^{5}\cdot 85-{\sqrt {2^{10}\cdot 6137}}}}}}+{\sqrt {2^{3}\cdot 51+{\sqrt {2^{6}\cdot 153}}-{\sqrt {2^{7}\cdot 153+{\sqrt {2^{14}\cdot 1377}}}}+{\sqrt {2^{8}\cdot 153-{\sqrt {2^{16}\cdot 12393}}+{\sqrt {2^{17}\cdot 6885-{\sqrt {2^{34}\cdot 40264857}}}}}}}}\right)\\=&{\frac {1}{32}}\left(+1+{\sqrt {17}}+{\sqrt {2\cdot 17+{\sqrt {2^{2}\cdot 17}}}}-{\sqrt {2^{2}\cdot 17-{\sqrt {2^{4}\cdot 3^{2}\cdot 17}}-{\sqrt {2^{5}\cdot 5\cdot 17-{\sqrt {2^{10}\cdot {19}^{2}\cdot 17}}}}}}+{\sqrt {2^{3}\cdot 3\cdot 17+{\sqrt {2^{6}\cdot 3^{2}\cdot 17}}-{\sqrt {2^{7}\cdot 3^{2}\cdot 17+{\sqrt {2^{14}\cdot 3^{4}\cdot 17}}}}+{\sqrt {2^{8}\cdot 3^{2}\cdot 17-{\sqrt {2^{16}\cdot 3^{6}\cdot 17}}+{\sqrt {2^{17}\cdot 3^{4}\cdot 5\cdot 17-{\sqrt {2^{34}\cdot 3^{8}\cdot 19^{2}\cdot 17}}}}}}}}\right)\\\end{aligned}}}
以下のように定義すると