三十七角形

三十七角形（さんじゅうしちかくけい、さんじゅうななかっけい、triacontaheptagon）は、多角形の一つで、37本の辺と37個の頂点を持つ図形である。内角の和は6300°、対角線の本数は629本である。

正三十七角形

正三十七角形

正三十七角形においては、中心角と外角は9.729…°で、内角は170.27…°となる。一辺の長さが a の正三十七角形の面積 S は

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

${\displaystyle \cos(2\pi /37)}$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。

以下には、中間結果（三次方程式を1回解いた際の関係式）を示す。

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

各式を3つの組に分ける。${\displaystyle \cos {\frac {k\pi }{37}}}$と${\displaystyle \cos {\frac {2^{9}k\pi }{37}}\left(=\cos {\frac {-6k\pi }{37}}\right)}$

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

和積公式で変形する。また、${\displaystyle \cos(\pi -\theta )=-\cos \theta }$ の関係を使って変形する。

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

解と係数の関係を使って二次方程式を解くと

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

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

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

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

三角関数、逆三角関数を用いた解は

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

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

平方根、立方根で表すと

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

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

別解

${\displaystyle \cos(2\pi /37)}$を二次方程式→三次方程式→三次方程式の順で求めることもできる。 まず、以下のようにx1～x6を定める。

${\displaystyle {\begin{aligned}&x_{1}=2\cos {\frac {2\pi }{37}}+2\cos {\frac {20\pi }{37}}+2\cos {\frac {22\pi }{37}}\\&x_{2}=2\cos {\frac {4\pi }{37}}+2\cos {\frac {34\pi }{37}}+2\cos {\frac {30\pi }{37}}\\&x_{3}=2\cos {\frac {8\pi }{37}}+2\cos {\frac {6\pi }{37}}+2\cos {\frac {14\pi }{37}}\\&x_{4}=2\cos {\frac {16\pi }{37}}+2\cos {\frac {12\pi }{37}}+2\cos {\frac {28\pi }{37}}\\&x_{5}=2\cos {\frac {32\pi }{37}}+2\cos {\frac {24\pi }{37}}+2\cos {\frac {18\pi }{37}}\\&x_{6}=2\cos {\frac {10\pi }{37}}+2\cos {\frac {26\pi }{37}}+2\cos {\frac {36\pi }{37}}\\\end{aligned}}}$

α、βを以下のように置き

${\displaystyle {\begin{aligned}&\alpha =x_{1}+x_{3}+x_{5}\\&\beta =x_{2}+x_{4}+x_{6}\\\end{aligned}}}$

α、βの和と差の平方を求めると

${\displaystyle {\begin{aligned}&\alpha +\beta =-1\\&(\alpha -\beta )^{2}=37\\\end{aligned}}}$

となる。よって、

${\displaystyle {\begin{aligned}&x_{1}+x_{3}+x_{5}={\frac {-1+{\sqrt {37}}}{2}}\\&x_{2}+x_{4}+x_{6}={\frac {-1-{\sqrt {37}}}{2}}\\\end{aligned}}}$

さらに以下の値A,B,C,Dも三角関数の積和の公式から求まる。

${\displaystyle {\begin{aligned}\left(x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}\right)^{3}=&A={\frac {-37-8{\sqrt {37}}-3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\left(x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}\right)^{3}=&B={\frac {-37-8{\sqrt {37}}+3{\sqrt {3}}(37+6{\sqrt {37}})i}{2}}\\\left(x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}\right)^{3}=&C={\frac {-37+8{\sqrt {37}}-3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\left(x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}\right)^{3}=&D={\frac {-37+8{\sqrt {37}}+3{\sqrt {3}}(37-6{\sqrt {37}})i}{2}}\\\end{aligned}}}$

両辺の立方根を取ると

${\displaystyle {\begin{aligned}x_{3}+\omega \cdot x_{5}+\omega ^{2}\cdot x_{1}=&{\sqrt[{3}]{A}}\\x_{3}+\omega ^{2}\cdot x_{5}+\omega \cdot x_{1}=&{\sqrt[{3}]{B}}\\x_{4}+\omega \cdot x_{6}+\omega ^{2}\cdot x_{2}=&{\sqrt[{3}]{C}}\\x_{4}+\omega ^{2}\cdot x_{6}+\omega \cdot x_{2}=&{\sqrt[{3}]{D}}\\\end{aligned}}}$

以上より、x1～x6が求まる。

${\displaystyle {\begin{aligned}x_{1}=&{\frac {\alpha +\omega {\sqrt[{3}]{A}}+\omega ^{2}{\sqrt[{3}]{B}}}{3}}\\x_{3}=&{\frac {\alpha +{\sqrt[{3}]{A}}+{\sqrt[{3}]{B}}}{3}}\\x_{5}=&{\frac {\alpha +\omega ^{2}{\sqrt[{3}]{A}}+\omega {\sqrt[{3}]{B}}}{3}}\\x_{2}=&{\frac {\beta +\omega {\sqrt[{3}]{C}}+\omega ^{2}{\sqrt[{3}]{D}}}{3}}\\x_{4}=&{\frac {\beta +{\sqrt[{3}]{C}}+{\sqrt[{3}]{D}}}{3}}\\x_{6}=&{\frac {\beta +\omega ^{2}{\sqrt[{3}]{C}}+\omega {\sqrt[{3}]{D}}}{3}}\\\end{aligned}}}$

さらに以下のy11,y12の値をx1～x6を使って求める。

${\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{37}}+\omega \cdot 2\cos {\frac {20\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{37}}\right)^{3}=&y_{11}=3x_{1}+x_{3}+6(x_{2}+2)+3\omega (2x_{1}+x_{4}+x_{5})+3\omega ^{2}(2x_{1}+x_{5}+x_{6})\\\left(2\cos {\frac {2\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {20\pi }{37}}+\omega \cdot 2\cos {\frac {22\pi }{37}}\right)^{3}=&y_{12}=3x_{1}+x_{3}+6(x_{2}+2)+3\omega ^{2}(2x_{1}+x_{4}+x_{5})+3\omega (2x_{1}+x_{5}+x_{6})\\\end{aligned}}}$

両辺の立方根を取ると

${\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{37}}+\omega \cdot 2\cos {\frac {20\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{37}}=&{\sqrt[{3}]{y_{11}}}\\2\cos {\frac {2\pi }{37}}+\omega ^{2}\cdot 2\cos {\frac {20\pi }{37}}+\omega \cdot 2\cos {\frac {22\pi }{37}}=&{\sqrt[{3}]{y_{12}}}\\\end{aligned}}}$

以上より

${\displaystyle {\begin{aligned}\cos {\frac {2\pi }{37}}={\frac {x_{1}+{\sqrt[{3}]{y_{11}}}+{\sqrt[{3}]{y_{12}}}}{6}}\end{aligned}}}$

正三十七角形の作図

正三十七角形は定規とコンパスによる作図が不可能な図形である。

正三十七角形は折紙により作図可能である^[1]。