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 S = 37 4 a 2 cot ⁡ π 37 ≃ 108.67963 a 2 {\displaystyle S={\frac {37}{4}}a^{2}\cot {\frac {\pi }{37}}\simeq 108.67963a^{2}} cos ⁡ ( 2 π / 37 ) {\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。 λ 1 = 2 cos ⁡ 2 π 37 + 2 cos ⁡ 12 π 37 + 2 cos ⁡ 16 π 37 + 2 cos ⁡ 20 π 37 + 2 cos ⁡ 22 π 37 + 2 cos ⁡ 28 π 37 = − 1 3 + 37 3 − 11 + 3 3 i 2 37 3 ω 2 + 37 3 − 11 − 3 3 i 2 37 3 ω λ 2 = 2 cos ⁡ 4 π 37 + 2 cos ⁡ 18 π 37 + 2 cos ⁡ 24 π 37 + 2 cos ⁡ 30 π 37 + 2 cos ⁡ 32 π 37 + 2 cos ⁡ 34 π 37 = − 1 3 + 37 3 − 11 + 3 3 i 2 37 3 ω + 37 3 − 11 − 3 3 i 2 37 3 ω 2 λ 3 = 2 cos ⁡ 6 π 37 + 2 cos ⁡ 8 π 37 + 2 cos ⁡ 10 π 37 + 2 cos ⁡ 14 π 37 + 2 cos ⁡ 26 π 37 + 2 cos ⁡ 36 π 37 = − 1 3 + 37 3 − 11 + 3 3 i 2 37 3 + 37 3 − 11 − 3 3 i 2 37 3 {\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。 cos ⁡ k π 37 {\displaystyle \cos {\frac {k\pi }{37}}} と cos ⁡ 2 9 k π 37 ( = cos ⁡ − 6 k π 37 ) {\displaystyle \cos {\frac {2^{9}k\pi }{37}}\left(=\cos {\frac {-6k\pi }{37}}\right)} λ 1 = ( 2 cos ⁡ 2 π 37 + 2 cos ⁡ 12 π 37 ) + ( 2 cos ⁡ 20 π 37 + 2 cos ⁡ 28 π 37 ) + ( 2 cos ⁡ 16 π 37 + 2 cos ⁡ 22 π 37 ) = u 1 + u 2 + u 3 λ 2 = ( 2 cos ⁡ 4 π 37 + 2 cos ⁡ 24 π 37 ) + ( 2 cos ⁡ 30 π 37 + 2 cos ⁡ 32 π 37 ) + ( 2 cos ⁡ 18 π 37 + 2 cos ⁡ 34 π 37 ) = v 1 + v 2 + v 3 λ 3 = ( 2 cos ⁡ 10 π 37 + 2 cos ⁡ 14 π 37 ) + ( 2 cos ⁡ 6 π 37 + 2 cos ⁡ 36 π 37 ) + ( 2 cos ⁡ 8 π 37 + 2 cos ⁡ 26 π 37 ) = w 1 + w 2 + w 3 {\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、 cos ⁡ ( π − θ ) = − cos ⁡ θ {\displaystyle \cos(\pi -\theta )=-\cos \theta } の関係を使って変形する。 λ 1 = ( 2 cos ⁡ 30 π 37 ⋅ 2 cos ⁡ 32 π 37 ) + ( 2 cos ⁡ 4 π 37 ⋅ 2 cos ⁡ 24 π 37 ) + ( 2 cos ⁡ 18 π 37 ⋅ 2 cos ⁡ 34 π 37 ) = u 1 + u 2 + u 3 λ 2 = ( 2 cos ⁡ 10 π 37 ⋅ 2 cos ⁡ 14 π 37 ) + ( 2 cos ⁡ 6 π 37 ⋅ 2 cos ⁡ 36 π 37 ) + ( 2 cos ⁡ 8 π 37 ⋅ 2 cos ⁡ 26 π 37 ) = v 1 + v 2 + v 3 λ 3 = ( 2 cos ⁡ 2 π 37 ⋅ 2 cos ⁡ 12 π 37 ) + ( 2 cos ⁡ 16 π 37 ⋅ 2 cos ⁡ 22 π 37 ) + ( 2 cos ⁡ 20 π 37 ⋅ 2 cos ⁡ 28 π 37 ) = w 1 + w 2 + w 3 {\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 cos ⁡ 2 π 37 = u 1 + u 1 2 − 4 w 1 4 {\displaystyle \cos {\frac {2\pi }{37}}={\frac {u_{1}+{\sqrt {u_{1}^{2}-4w_{1}}}}{4}}} ここで、 u 1 , w 1 {\displaystyle u_{1},w_{1}} は以下の三次方程式の解である。 u 3 − λ 1 u 2 + ( λ 2 − 1 ) u + ( λ 1 − 2 ) = 0 {\displaystyle u^{3}-\lambda _{1}u^{2}+(\lambda _{2}-1)u+(\lambda _{1}-2)=0} w 3 − λ 3 w 2 + ( λ 1 − 1 ) w + ( λ 3 − 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 u 1 = λ 1 3 + 2 11 − 2 λ 1 − 2 λ 2 3 ⋅ cos ⁡ ( 1 3 arccos ⁡ 111 − 4 λ 1 − 9 λ 2 62 11 − 2 λ 1 − 2 λ 2 ) {\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)} w 1 = λ 3 3 + 2 11 − 2 λ 3 − 2 λ 1 3 ⋅ cos ⁡ ( 1 3 arccos ⁡ 111 − 4 λ 3 − 9 λ 1 62 11 − 2 λ 3 − 2 λ 1 ) {\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 u 1 = λ 1 3 + 2 11 − 2 λ 1 − 2 λ 2 3 111 − 4 λ 1 − 9 λ 2 62 11 − 2 λ 1 − 2 λ 2 + i 27 ( 1092 − 253 λ 1 − 205 λ 2 ) 62 11 − 2 λ 1 − 2 λ 2 3 + 2 11 − 2 λ 1 − 2 λ 2 3 111 − 4 λ 1 − 9 λ 2 62 11 − 2 λ 1 − 2 λ 2 − i 27 ( 1092 − 253 λ 1 − 205 λ 2 ) 62 11 − 2 λ 1 − 2 λ 2 3 {\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}}} w 1 = λ 3 3 + 2 11 − 2 λ 3 − 2 λ 1 3 111 − 4 λ 3 − 9 λ 1 62 11 − 2 λ 3 − 2 λ 1 + i 27 ( 1092 − 253 λ 3 − 205 λ 1 ) 62 11 − 2 λ 3 − 2 λ 1 3 + 2 11 − 2 λ 3 − 2 λ 1 3 111 − 4 λ 3 − 9 λ 1 62 11 − 2 λ 3 − 2 λ 1 − i 27 ( 1092 − 253 λ 3 − 205 λ 1 ) 62 11 − 2 λ 3 − 2 λ 1 3 {\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}}} cos ⁡ ( 2 π / 37 ) {\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。 x 1 = 2 cos ⁡ 2 π 37 + 2 cos ⁡ 20 π 37 + 2 cos ⁡ 22 π 37 x 2 = 2 cos ⁡ 4 π 37 + 2 cos ⁡ 34 π 37 + 2 cos ⁡ 30 π 37 x 3 = 2 cos ⁡ 8 π 37 + 2 cos ⁡ 6 π 37 + 2 cos ⁡ 14 π 37 x 4 = 2 cos ⁡ 16 π 37 + 2 cos ⁡ 12 π 37 + 2 cos ⁡ 28 π 37 x 5 = 2 cos ⁡ 32 π 37 + 2 cos ⁡ 24 π 37 + 2 cos ⁡ 18 π 37 x 6 = 2 cos ⁡ 10 π 37 + 2 cos ⁡ 26 π 37 + 2 cos ⁡ 36 π 37 {\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 α = x 1 + x 3 + x 5 β = x 2 + x 4 + x 6 {\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 α + β = − 1 ( α − β ) 2 = 37 {\displaystyle {\begin{aligned}&\alpha +\beta =-1\\&(\alpha -\beta )^{2}=37\\\end{aligned}}} となる。よって、 x 1 + x 3 + x 5 = − 1 + 37 2 x 2 + x 4 + x 6 = − 1 − 37 2 {\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。 ( x 3 + ω ⋅ x 5 + ω 2 ⋅ x 1 ) 3 = A = − 37 − 8 37 − 3 3 ( 37 + 6 37 ) i 2 ( x 3 + ω 2 ⋅ x 5 + ω ⋅ x 1 ) 3 = B = − 37 − 8 37 + 3 3 ( 37 + 6 37 ) i 2 ( x 4 + ω ⋅ x 6 + ω 2 ⋅ x 2 ) 3 = C = − 37 + 8 37 − 3 3 ( 37 − 6 37 ) i 2 ( x 4 + ω 2 ⋅ x 6 + ω ⋅ x 2 ) 3 = D = − 37 + 8 37 + 3 3 ( 37 − 6 37 ) i 2 {\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 x 3 + ω ⋅ x 5 + ω 2 ⋅ x 1 = A 3 x 3 + ω 2 ⋅ x 5 + ω ⋅ x 1 = B 3 x 4 + ω ⋅ x 6 + ω 2 ⋅ x 2 = C 3 x 4 + ω 2 ⋅ x 6 + ω ⋅ x 2 = D 3 {\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。 x 1 = α + ω A 3 + ω 2 B 3 3 x 3 = α + A 3 + B 3 3 x 5 = α + ω 2 A 3 + ω B 3 3 x 2 = β + ω C 3 + ω 2 D 3 3 x 4 = β + C 3 + D 3 3 x 6 = β + ω 2 C 3 + ω D 3 3 {\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。 ( 2 cos ⁡ 2 π 37 + ω ⋅ 2 cos ⁡ 20 π 37 + ω 2 ⋅ 2 cos ⁡ 22 π 37 ) 3 = y 11 = 3 x 1 + x 3 + 6 ( x 2 + 2 ) + 3 ω ( 2 x 1 + x 4 + x 5 ) + 3 ω 2 ( 2 x 1 + x 5 + x 6 ) ( 2 cos ⁡ 2 π 37 + ω 2 ⋅ 2 cos ⁡ 20 π 37 + ω ⋅ 2 cos ⁡ 22 π 37 ) 3 = y 12 = 3 x 1 + x 3 + 6 ( x 2 + 2 ) + 3 ω 2 ( 2 x 1 + x 4 + x 5 ) + 3 ω ( 2 x 1 + x 5 + x 6 ) {\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 2 cos ⁡ 2 π 37 + ω ⋅ 2 cos ⁡ 20 π 37 + ω 2 ⋅ 2 cos ⁡ 22 π 37 = y 11 3 2 cos ⁡ 2 π 37 + ω 2 ⋅ 2 cos ⁡ 20 π 37 + ω ⋅ 2 cos ⁡ 22 π 37 = y 12 3 {\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 cos ⁡ 2 π 37 = x 1 + y 11 3 + y 12 3 6 {\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。 [脚注の使い方] ^ 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 2185-369X、NAID 110008795238。  ポータル 数学 z^37=1 の解法 | てっぃちMarshの数学（Mathematics)教室 表示 編集