五十六角形

正五十六角形においては、中心角と外角は6.428571…°で、内角は173.571428…°となる。一辺の長さが a の正五十六角形の面積 S は

${\displaystyle S=14a^{2}\cot {\frac {\pi }{56}}a^{2}}$

関係式

${\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{56}}+2\cos {\frac {50\pi }{56}}+2\cos {\frac {18\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{2}}\\&x_{2}=2\cos {\frac {10\pi }{56}}+2\cos {\frac {26\pi }{56}}+2\cos {\frac {22\pi }{56}}={\frac {{\sqrt {14}}+{\sqrt {2}}}{2}}\\&x_{3}=2\cos {\frac {54\pi }{56}}+2\cos {\frac {6\pi }{56}}+2\cos {\frac {38\pi }{56}}={\frac {-{\sqrt {14}}+{\sqrt {2}}}{2}}\\&x_{4}=2\cos {\frac {46\pi }{56}}+2\cos {\frac {30\pi }{56}}+2\cos {\frac {34\pi }{56}}={\frac {-{\sqrt {14}}-{\sqrt {2}}}{2}}\\\end{aligned}}}$

三次方程式の係数を求めると

${\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}+2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}+2\cos {\frac {18\pi }{56}}\cdot 2\cos {\frac {2\pi }{56}}\\&=2\cos {\frac {26\pi }{28}}+2\cos {\frac {22\pi }{28}}+2\cos {\frac {10\pi }{28}}+2\cos {\frac {6\pi }{7}}+2\cos {\frac {4\pi }{7}}+2\cos {\frac {2\pi }{7}}=-{\sqrt {7}}-1={\sqrt {2}}x_{4}\\&2\cos {\frac {2\pi }{56}}\cdot 2\cos {\frac {50\pi }{56}}\cdot 2\cos {\frac {18\pi }{56}}=x_{4}-{\sqrt {2}}=x_{4}+x_{1}+x_{4}=x_{1}+2x_{4}\\\end{aligned}}}$

解と係数の関係より

${\displaystyle u^{3}-x_{1}u^{2}+{\sqrt {2}}x_{4}u-(x_{1}+2x_{4})=0}$

変数変換

${\displaystyle u=v+x_{1}/3}$

整理すると

${\displaystyle v^{3}-{\frac {7+2{\sqrt {7}}}{3}}v+{\frac {(7+2{\sqrt {7}})(5{\sqrt {2}}+{\sqrt {14}})}{54}}=0}$

三角関数、逆三角関数を用いた解は

${\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {2{\sqrt {7+2{\sqrt {7}}}}}{3}}\cos \left({\frac {1}{3}}\arccos \left(-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}\right)\right)}$

平方根、立方根で表すと

${\displaystyle u_{1}={\frac {{\sqrt {14}}-{\sqrt {2}}}{6}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{3}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}}$

${\displaystyle \cos(2\pi /56)}$を平方根と立方根で表すと

${\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}+i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}+{\frac {\sqrt {7+2{\sqrt {7}}}}{6}}{\sqrt[{3}]{-{\frac {5{\sqrt {2}}+{\sqrt {14}}}{4{\sqrt {7+2{\sqrt {7}}}}}}-i{\frac {\sqrt {48+12{\sqrt {7}}}}{4{\sqrt {7+2{\sqrt {7}}}}}}}}}$

${\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {1}{6}}{\sqrt[{3}]{-{\frac {49{\sqrt {2}}+17{\sqrt {14}}}{4}}+i{\frac {\sqrt {6048+2268{\sqrt {7}}}}{4}}}}+{\frac {1}{6}}{\sqrt[{3}]{-{\frac {49{\sqrt {2}}+17{\sqrt {14}}}{4}}-i{\frac {\sqrt {6048+2268{\sqrt {7}}}}{4}}}}}$

${\displaystyle \cos {\frac {2\pi }{56}}={\frac {{\sqrt {14}}-{\sqrt {2}}}{12}}+{\frac {1}{12}}{\sqrt[{3}]{-98{\sqrt {2}}-34{\sqrt {14}}+i\cdot 12{\sqrt {168+63{\sqrt {7}}}}}}+{\frac {1}{12}}{\sqrt[{3}]{-98{\sqrt {2}}-34{\sqrt {14}}-i\cdot 12{\sqrt {168+63{\sqrt {7}}}}}}}$

• 七角形
• 十四角形
• 二十八角形