# 三十七角形

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## 正三十七角形

${\displaystyle S={\frac {37}{4}}a^{2}\cot {\frac {\pi }{37}}\simeq 108.67963a^{2}}$

${\displaystyle \cos(2\pi /37)}$を平方根と立方根で表すことが可能であるが、三次方程式三次方程式（2つ）→二次方程式と解く必要がある。

{\displaystyle {\begin{aligned}\lambda _{1}=&2\cos {\frac {2\pi }{37}}+2\cos {\frac {12\pi }{37}}+2\cos {\frac {16\pi }{37}}+2\cos {\frac {20\pi }{37}}+2\cos {\frac {22\pi }{37}}+2\cos {\frac {28\pi }{37}}=-{\frac {1}{3}}+{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}\omega ^{2}+{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\omega \\\lambda _{2}=&2\cos {\frac {4\pi }{37}}+2\cos {\frac {18\pi }{37}}+2\cos {\frac {24\pi }{37}}+2\cos {\frac {30\pi }{37}}+2\cos {\frac {32\pi }{37}}+2\cos {\frac {34\pi }{37}}=-{\frac {1}{3}}+{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}\omega +{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\omega ^{2}\\\lambda _{3}=&2\cos {\frac {6\pi }{37}}+2\cos {\frac {8\pi }{37}}+2\cos {\frac {10\pi }{37}}+2\cos {\frac {14\pi }{37}}+2\cos {\frac {26\pi }{37}}+2\cos {\frac {36\pi }{37}}=-{\frac {1}{3}}+{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11+3{\sqrt {3}}i}{2{\sqrt {37}}}}}+{\frac {\sqrt {37}}{3}}{\sqrt[{3}]{\frac {-11-3{\sqrt {3}}i}{2{\sqrt {37}}}}}\\\end{aligned}}}

{\displaystyle {\begin{aligned}\lambda _{1}=&\left(2\cos {\frac {2\pi }{37}}+2\cos {\frac {12\pi }{37}}\right)+\left(2\cos {\frac {20\pi }{37}}+2\cos {\frac {28\pi }{37}}\right)+\left(2\cos {\frac {16\pi }{37}}+2\cos {\frac {22\pi }{37}}\right)=u_{1}+u_{2}+u_{3}\\\lambda _{2}=&\left(2\cos {\frac {4\pi }{37}}+2\cos {\frac {24\pi }{37}}\right)+\left(2\cos {\frac {30\pi }{37}}+2\cos {\frac {32\pi }{37}}\right)+\left(2\cos {\frac {18\pi }{37}}+2\cos {\frac {34\pi }{37}}\right)=v_{1}+v_{2}+v_{3}\\\lambda _{3}=&\left(2\cos {\frac {10\pi }{37}}+2\cos {\frac {14\pi }{37}}\right)+\left(2\cos {\frac {6\pi }{37}}+2\cos {\frac {36\pi }{37}}\right)+\left(2\cos {\frac {8\pi }{37}}+2\cos {\frac {26\pi }{37}}\right)=w_{1}+w_{2}+w_{3}\\\end{aligned}}}

{\displaystyle {\begin{aligned}\lambda _{1}=&\left(2\cos {\frac {30\pi }{37}}\cdot 2\cos {\frac {32\pi }{37}}\right)+\left(2\cos {\frac {4\pi }{37}}\cdot 2\cos {\frac {24\pi }{37}}\right)+\left(2\cos {\frac {18\pi }{37}}\cdot 2\cos {\frac {34\pi }{37}}\right)=u_{1}+u_{2}+u_{3}\\\lambda _{2}=&\left(2\cos {\frac {10\pi }{37}}\cdot 2\cos {\frac {14\pi }{37}}\right)+\left(2\cos {\frac {6\pi }{37}}\cdot 2\cos {\frac {36\pi }{37}}\right)+\left(2\cos {\frac {8\pi }{37}}\cdot 2\cos {\frac {26\pi }{37}}\right)=v_{1}+v_{2}+v_{3}\\\lambda _{3}=&\left(2\cos {\frac {2\pi }{37}}\cdot 2\cos {\frac {12\pi }{37}}\right)+\left(2\cos {\frac {16\pi }{37}}\cdot 2\cos {\frac {22\pi }{37}}\right)+\left(2\cos {\frac {20\pi }{37}}\cdot 2\cos {\frac {28\pi }{37}}\right)=w_{1}+w_{2}+w_{3}\\\end{aligned}}}

${\displaystyle \cos {\frac {2\pi }{37}}={\frac {u_{1}+{\sqrt {u_{1}^{2}-4w_{1}}}}{4}}}$

ここで、${\displaystyle u_{1},w_{1}}$は以下の三次方程式の解である。

${\displaystyle u^{3}-\lambda _{1}u^{2}+(\lambda _{2}-1)u+(\lambda _{1}-2)=0}$
${\displaystyle w^{3}-\lambda _{3}w^{2}+(\lambda _{1}-1)w+(\lambda _{3}-2)=0}$

${\displaystyle u_{1}={\frac {\lambda _{1}}{3}}+{\frac {2{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}{3}}\cdot \cos \left({\frac {1}{3}}\arccos {\frac {111-4\lambda _{1}-9\lambda _{2}}{62{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}}\right)}$
${\displaystyle w_{1}={\frac {\lambda _{3}}{3}}+{\frac {2{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}{3}}\cdot \cos \left({\frac {1}{3}}\arccos {\frac {111-4\lambda _{3}-9\lambda _{1}}{62{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}}\right)}$

{\displaystyle {\begin{aligned}u_{1}={\frac {\lambda _{1}}{3}}+{\frac {2{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}{3}}{\sqrt[{3}]{{\frac {111-4\lambda _{1}-9\lambda _{2}}{62{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}}+i{\frac {\sqrt {27(1092-253\lambda _{1}-205\lambda _{2})}}{62{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}}}}\\+{\frac {2{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}{3}}{\sqrt[{3}]{{\frac {111-4\lambda _{1}-9\lambda _{2}}{62{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}}-i{\frac {\sqrt {27(1092-253\lambda _{1}-205\lambda _{2})}}{62{\sqrt {11-2\lambda _{1}-2\lambda _{2}}}}}}}\end{aligned}}}
{\displaystyle {\begin{aligned}w_{1}={\frac {\lambda _{3}}{3}}+{\frac {2{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}{3}}{\sqrt[{3}]{{\frac {111-4\lambda _{3}-9\lambda _{1}}{62{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}}+i{\frac {\sqrt {27(1092-253\lambda _{3}-205\lambda _{1})}}{62{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}}}}\\+{\frac {2{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}{3}}{\sqrt[{3}]{{\frac {111-4\lambda _{3}-9\lambda _{1}}{62{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}}-i{\frac {\sqrt {27(1092-253\lambda _{3}-205\lambda _{1})}}{62{\sqrt {11-2\lambda _{3}-2\lambda _{1}}}}}}}\end{aligned}}}

## 脚注

 [脚注の使い方]
1. ^ 西村保三, 山本一海「折り紙による正37角形の作図」『福井大学教育地域科学部紀要』第2巻、福井大学教育地域科学部、2012年、 63-70頁、 ISSN 2185-369XNAID 110008795238