正三十九角形 三十九角形(さんじゅうきゅうかくけい、さんじゅうきゅうかっけい、triacontaenneagon)は、多角形の一つで、39本の辺と39個の頂点を持つ図形である。内角の和は6660°、対角線の本数は702本である。
正三十九角形
正三十九角形においては、中心角と外角は9.23…°で、内角は170.769…°となる。一辺の長さが a の正三十九角形の面積 S は
![{\displaystyle S={\frac {39}{4}}a^{2}\cot {\frac {\pi }{39}}\simeq 120.77542a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4f536b50fe20b19d69996601beaca892d2abff)
を平方根と立方根で表すと
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{39}}=&\cos \left({\frac {2\pi }{3}}-{\frac {8\pi }{13}}\right)\\=&\cos {\frac {2\pi }{3}}\cos {\frac {8\pi }{13}}+\sin {\frac {2\pi }{3}}\sin {\frac {8\pi }{13}}\\=&-{\frac {1}{2}}\cos {\frac {8\pi }{13}}+{\frac {\sqrt {3}}{2}}\sin {\frac {8\pi }{13}}\\=&-{\frac {1}{2}}\cos {\frac {8\pi }{13}}+{\frac {\sqrt {3}}{2}}{\sqrt {\frac {1+\cos {\frac {16\pi }{13}}}{2}}}\\=&-{\frac {1}{24}}\cdot \left(12\cos {\frac {8\pi }{13}}\right)+{\frac {\sqrt {3}}{24}}{\sqrt {72+72\cos {\frac {10\pi }{13}}}}\\=&-{\frac {1}{24}}\left({\sqrt {13}}-1+\omega {\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega ^{2}{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}\right)\\&+{\frac {\sqrt {3}}{24}}{\sqrt {72+6\cdot (12\cos {\frac {10\pi }{13}})}}\\=&-{\frac {1}{24}}\left({\sqrt {13}}-1+\omega {\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega ^{2}{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}\right)\\&+{\frac {\sqrt {3}}{24}}{\sqrt {72+6\left(-{\sqrt {13}}-1+\omega ^{2}{\sqrt[{3}]{104+20{\sqrt {13}}+12i{\sqrt {39}}}}+\omega {\sqrt[{3}]{104+20{\sqrt {13}}-12i{\sqrt {39}}}}\right)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da6a97be2bcbdf3dd70d5f97a40f920ae699f0f)
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{39}}=&\cos {\frac {2\pi }{3\cdot 13}}\\=&{\frac {1}{2}}\cdot \left({\sqrt[{3}]{\cos {\frac {2\pi }{13}}+i\cdot \sin {\frac {2\pi }{13}}}}+{\sqrt[{3}]{\cos {\frac {2\pi }{13}}-i\cdot \sin {\frac {2\pi }{13}}}}\right)\\=&{\frac {1}{2}}\cdot {\sqrt[{3}]{\cos {\frac {2\pi }{13}}+i\cdot \sin {\frac {2\pi }{13}}}}+{\frac {1}{2}}\cdot {\sqrt[{3}]{\cos {\frac {2\pi }{13}}-i\cdot \sin {\frac {2\pi }{13}}}}\\=&{\frac {1}{2}}\cdot {\sqrt[{3}]{{\frac {1}{12}}({\sqrt {13}}-1+{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}})+i\cdot \sin {\frac {2\pi }{13}}}}\\&+{\frac {1}{2}}\cdot {\sqrt[{3}]{{\frac {1}{12}}({\sqrt {13}}-1+{\sqrt[{3}]{104-20{\sqrt {13}}-12i{\sqrt {39}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12i{\sqrt {39}}}})-i\cdot \sin {\frac {2\pi }{13}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/456bc6e32d72bcb0053aa4e8a36c56935d8d5f63)
- 関係式
![{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{39}}+2\cos {\frac {32\pi }{39}}+2\cos {\frac {34\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{1}\\2\cos {\frac {4\pi }{39}}+2\cos {\frac {14\pi }{39}}+2\cos {\frac {10\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{2}\\2\cos {\frac {8\pi }{39}}+2\cos {\frac {28\pi }{39}}+2\cos {\frac {20\pi }{39}}={\frac {1}{4}}\left(1-{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{3}\\2\cos {\frac {16\pi }{39}}+2\cos {\frac {22\pi }{39}}+2\cos {\frac {38\pi }{39}}={\frac {1}{4}}\left(1+{\sqrt {13}}-{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{4}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce12c46fe3103d37036e14da7c0a85eead7dec5)
さらに、以下のような関係式が得られる。
![{\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{39}}+\omega \cdot 2\cos {\frac {32\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{39}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {8\pi }{13}}+2\cos {\frac {6\pi }{13}}+6(x_{2}+2)+3\omega \left(2x_{1}+x_{3}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)+3\omega ^{2}\left(2x_{1}+x_{4}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)\\=&3x_{1}+{\frac {-1+{\sqrt {13}}}{2}}+6(x_{2}+2)+3\omega \left(2x_{1}+x_{3}+{\frac {-1-{\sqrt {13}}}{2}}\right)+3\omega ^{2}\left(2x_{1}+x_{4}+{\frac {-1-{\sqrt {13}}}{2}}\right)\\=&{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}+3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}\\\left(2\cos {\frac {2\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{39}}+\omega \cdot 2\cos {\frac {34\pi }{39}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{13}}+2\cos {\frac {8\pi }{13}}+2\cos {\frac {6\pi }{13}}+6(x_{2}+2)+3\omega ^{2}\left(2x_{1}+x_{3}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)+3\omega \left(2x_{1}+x_{4}+2\cos {\frac {4\pi }{13}}+2\cos {\frac {10\pi }{13}}+2\cos {\frac {12\pi }{13}}\right)\\=&3x_{1}+{\frac {-1+{\sqrt {13}}}{2}}+6(x_{2}+2)+3\omega ^{2}\left(2x_{1}+x_{3}+{\frac {-1-{\sqrt {13}}}{2}}\right)+3\omega \left(2x_{1}+x_{4}+{\frac {-1-{\sqrt {13}}}{2}}\right)\\=&{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}-3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad7e3c68c49497f938dfd41a2056d72dc21401a0)
両辺の立方根を取ると
![{\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{39}}+\omega \cdot 2\cos {\frac {32\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{39}}=&{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}+3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}\\2\cos {\frac {2\pi }{39}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{39}}+\omega \cdot 2\cos {\frac {34\pi }{39}}=&{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}-3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7374309a209e501884edecb7d45b78db803bedc2)
よって
![{\displaystyle {\begin{aligned}\cos {\frac {2\pi }{39}}=&{\frac {1}{6}}\left({\tfrac {1-{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}}{4}}+{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}+3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}+{\sqrt[{3}]{\tfrac {104+34{\sqrt {13}}+3{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+15{\sqrt {6\left(13+3{\sqrt {13}}\right)}}-3{\sqrt {3}}\left(-2{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)i}{8}}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3af3051621b9d776869e9785841606f6324793d8)
正三十九角形の作図
正三十九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。
正三十九角形は折紙により作図可能である。
脚注
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関連項目
外部リンク
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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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