出典: フリー百科事典『ウィキペディア(Wikipedia)』
正三十七角形
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

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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaa509c42a49cdde6cdf34b45d3b0450b7ca6ef)
各式を3つの組に分ける。
と

和積公式で変形する。また、
の関係を使って変形する。

解と係数の関係を使って二次方程式を解くと

ここで、
は以下の三次方程式の解である。


三角関数、逆三角関数を用いた解は


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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d40aeafd8a448349e035f816f35ddacfec79af8)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4f273587ebc8a69b4adfd35729ea715712dde3)
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。

α、βを以下のように置き

α、βの和と差の平方を求めると

となる。よって、

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。

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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c94d7c3774e2e5b55949d89045c3bbb5ad0b74)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c14db6399a80af3218990d775fa5fcb1aab9713d)
さらに以下のy11,y12の値をx1~x6を使って求める。

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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0cd0298023d05125736c8af0a2083181fccc76)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/823ac72f1126b44dbf8ec7a5b39057ed1c0a784c)
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]。
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非古典的 (2辺以下) | |
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辺の数: 3–10 |
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辺の数: 11–20 | |
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辺の数: 21–30 | |
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辺の数: 31–40 | |
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辺の数: 41–50 | |
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辺の数: 51–70 (抜粋) | |
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辺の数: 71–100 (抜粋) | |
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辺の数: 101– (抜粋) | |
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無限 | |
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星型多角形 (辺の数: 5–12) | |
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多角形のクラス | |
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