正五十一角形
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/08/16 17:01 UTC 版)
正五十一角形においては、中心角と外角は7.058…°で、内角は172.941…°となる。一辺の長さが a の正五十一角形の面積 S は S = 51 4 a 2 cot π 51 ≃ 206.71914 a 2 {\displaystyle S={\frac {51}{4}}a^{2}\cot {\frac {\pi }{51}}\simeq 206.71914a^{2}} cos ( 2 π / 51 ) {\displaystyle \cos(2\pi /51)} を有理数と平方根で表すことが可能である。 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 x k = 2 cos 2 k π 51 {\displaystyle x_{k}=2\cos {\frac {2k\pi }{51}}} 根号の内の2のべき乗以外の値は x k {\displaystyle x_{k}} の和または差の2乗の組み合わせで求められる。 ( ( ( ( x 1 + x 16 ) + ( x 13 + x 4 ) ) + ( ( x 25 + x 8 ) + ( x 19 + x 2 ) ) ) + ( ( ( x 5 + x 22 ) + ( x 14 + x 20 ) ) + ( ( x 23 + x 11 ) + ( x 7 + x 10 ) ) ) ) = 1 ( ( ( ( x 1 + x 16 ) + ( x 13 + x 4 ) ) + ( ( x 25 + x 8 ) + ( x 19 + x 2 ) ) ) − ( ( ( x 5 + x 22 ) + ( x 14 + x 20 ) ) + ( ( x 23 + x 11 ) + ( x 7 + x 10 ) ) ) ) 2 = 17 ⋮ ( ( ( ( x 1 − x 16 ) 2 − ( x 13 − x 4 ) 2 ) 2 − ( ( x 25 − x 8 ) 2 − ( x 19 − x 2 ) 2 ) 2 ) 2 + ( ( ( x 5 − x 22 ) 2 − ( x 14 − x 20 ) 2 ) 2 − ( ( x 23 − x 11 ) 2 − ( x 7 − x 10 ) 2 ) 2 ) 2 ) = 6885 ( ( ( ( x 1 − x 16 ) 2 − ( x 13 − x 4 ) 2 ) 2 − ( ( x 25 − x 8 ) 2 − ( x 19 − x 2 ) 2 ) 2 ) 2 − ( ( ( x 5 − x 22 ) 2 − ( x 14 − x 20 ) 2 ) 2 − ( ( x 23 − x 11 ) 2 − ( x 7 − x 10 ) 2 ) 2 ) 2 ) 2 = 40264857 {\displaystyle {\begin{aligned}&((((x_{1}+x_{16})+(x_{13}+x_{4}))+((x_{25}+x_{8})+(x_{19}+x_{2})))+(((x_{5}+x_{22})+(x_{14}+x_{20}))+((x_{23}+x_{11})+(x_{7}+x_{10}))))=1\\&((((x_{1}+x_{16})+(x_{13}+x_{4}))+((x_{25}+x_{8})+(x_{19}+x_{2})))-(((x_{5}+x_{22})+(x_{14}+x_{20}))+((x_{23}+x_{11})+(x_{7}+x_{10}))))^{2}=17\\&\quad \vdots \\&((((x_{1}-x_{16})^{2}-(x_{13}-x_{4})^{2})^{2}-((x_{25}-x_{8})^{2}-(x_{19}-x_{2})^{2})^{2})^{2}+(((x_{5}-x_{22})^{2}-(x_{14}-x_{20})^{2})^{2}-((x_{23}-x_{11})^{2}-(x_{7}-x_{10})^{2})^{2})^{2})=6885\\&((((x_{1}-x_{16})^{2}-(x_{13}-x_{4})^{2})^{2}-((x_{25}-x_{8})^{2}-(x_{19}-x_{2})^{2})^{2})^{2}-(((x_{5}-x_{22})^{2}-(x_{14}-x_{20})^{2})^{2}-((x_{23}-x_{11})^{2}-(x_{7}-x_{10})^{2})^{2})^{2})^{2}=40264857\\\end{aligned}}}
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