七十八角形

七十八角形（ななじゅうはちかくけい、ななじゅうはちかっけい、heptacontaoctagon）は、多角形の一つで、78本の辺と78個の頂点を持つ図形である。内角の和は13680°、対角線の本数は2925本である。 正七十八角形においては、中心角と外角は4.615…°で、内角は175.384…°となる。一辺の長さが a の正七十八角形の面積 S は S = 78 4 a 2 cot ⁡ π 78 ≃ 483.88751 a 2 {\displaystyle S={\frac {78}{4}}a^{2}\cot {\frac {\pi }{78}}\simeq 483.88751a^{2}} 関係式 2 cos ⁡ 2 π 78 + 2 cos ⁡ 46 π 78 + 2 cos ⁡ 34 π 78 = 1 4 ( − 1 − 13 + 6 ( 13 + 3 13 ) ) = x 1 2 cos ⁡ 22 π 78 + 2 cos ⁡ 38 π 78 + 2 cos ⁡ 62 π 78 = 1 4 ( − 1 + 13 − 6 ( 13 − 3 13 ) ) = x 2 2 cos ⁡ 70 π 78 + 2 cos ⁡ 50 π 78 + 2 cos ⁡ 58 π 78 = 1 4 ( − 1 − 13 − 6 ( 13 + 3 13 ) ) = x 3 2 cos ⁡ 10 π 78 + 2 cos ⁡ 74 π 78 + 2 cos ⁡ 14 π 78 = 1 4 ( − 1 + 13 + 6 ( 13 − 3 13 ) ) = x 4 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{78}}+2\cos {\frac {46\pi }{78}}+2\cos {\frac {34\pi }{78}}={\frac {1}{4}}\left(-1-{\sqrt {13}}+{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{1}\\2\cos {\frac {22\pi }{78}}+2\cos {\frac {38\pi }{78}}+2\cos {\frac {62\pi }{78}}={\frac {1}{4}}\left(-1+{\sqrt {13}}-{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{2}\\2\cos {\frac {70\pi }{78}}+2\cos {\frac {50\pi }{78}}+2\cos {\frac {58\pi }{78}}={\frac {1}{4}}\left(-1-{\sqrt {13}}-{\sqrt {6\left(13+3{\sqrt {13}}\right)}}\right)=x_{3}\\2\cos {\frac {10\pi }{78}}+2\cos {\frac {74\pi }{78}}+2\cos {\frac {14\pi }{78}}={\frac {1}{4}}\left(-1+{\sqrt {13}}+{\sqrt {6\left(13-3{\sqrt {13}}\right)}}\right)=x_{4}\\\end{aligned}}} さらに、以下のような関係式が得られる。 ( 2 cos ⁡ 2 π 78 + ω ⋅ 2 cos ⁡ 46 π 78 + ω 2 ⋅ 2 cos ⁡ 34 π 78 ) 3 = 3 x 1 + 2 cos ⁡ 2 π 26 + 2 cos ⁡ 6 π 26 + 2 cos ⁡ 18 π 26 + 6 ( x 4 − 2 ) + 3 ω ( 2 x 1 + x 3 + 2 cos ⁡ 14 π 26 + 2 cos ⁡ 10 π 26 + 2 cos ⁡ 22 π 26 ) + 3 ω 2 ( 2 x 1 + x 2 + 2 cos ⁡ 14 π 26 + 2 cos ⁡ 10 π 26 + 2 cos ⁡ 22 π 26 ) = 3 x 1 + 1 + 13 2 + 6 ( x 2 − 2 ) + 3 ω ( 2 x 1 + x 3 + 1 − 13 2 ) + 3 ω 2 ( 2 x 1 + x 2 + 1 − 13 2 ) = − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) − 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 ( 2 cos ⁡ 2 π 78 + ω 2 ⋅ 2 cos ⁡ 46 π 78 + ω ⋅ 2 cos ⁡ 34 π 78 ) 3 = 3 x 1 + 2 cos ⁡ 2 π 26 + 2 cos ⁡ 6 π 26 + 2 cos ⁡ 18 π 26 + 6 ( x 4 − 2 ) + 3 ω 2 ( 2 x 1 + x 3 + 2 cos ⁡ 14 π 26 + 2 cos ⁡ 10 π 26 + 2 cos ⁡ 22 π 26 ) + 3 ω ( 2 x 1 + x 4 + 2 cos ⁡ 14 π 26 + 2 cos ⁡ 10 π 26 + 2 cos ⁡ 22 π 26 ) = 3 x 1 + 1 + 13 2 + 6 ( x 2 − 2 ) + 3 ω 2 ( 2 x 1 + x 3 + 1 − 13 2 ) + 3 ω ( 2 x 1 + x 2 + 1 − 13 2 ) = − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) + 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 {\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{78}}+\omega \cdot 2\cos {\frac {46\pi }{78}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{78}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{26}}+2\cos {\frac {6\pi }{26}}+2\cos {\frac {18\pi }{26}}+6(x_{4}-2)+3\omega \left(2x_{1}+x_{3}+2\cos {\frac {14\pi }{26}}+2\cos {\frac {10\pi }{26}}+2\cos {\frac {22\pi }{26}}\right)+3\omega ^{2}\left(2x_{1}+x_{2}+2\cos {\frac {14\pi }{26}}+2\cos {\frac {10\pi }{26}}+2\cos {\frac {22\pi }{26}}\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_{2}+{\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 }{78}}+\omega ^{2}\cdot 2\cos {\frac {46\pi }{78}}+\omega \cdot 2\cos {\frac {34\pi }{78}}\right)^{3}=&3x_{1}+2\cos {\frac {2\pi }{26}}+2\cos {\frac {6\pi }{26}}+2\cos {\frac {18\pi }{26}}+6(x_{4}-2)+3\omega ^{2}\left(2x_{1}+x_{3}+2\cos {\frac {14\pi }{26}}+2\cos {\frac {10\pi }{26}}+2\cos {\frac {22\pi }{26}}\right)+3\omega \left(2x_{1}+x_{4}+2\cos {\frac {14\pi }{26}}+2\cos {\frac {10\pi }{26}}+2\cos {\frac {22\pi }{26}}\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_{2}+{\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}}} 両辺の立方根を取ると 2 cos ⁡ 2 π 78 + ω ⋅ 2 cos ⁡ 46 π 78 + ω 2 ⋅ 2 cos ⁡ 34 π 78 = − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) − 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 3 2 cos ⁡ 2 π 78 + ω 2 ⋅ 2 cos ⁡ 46 π 78 + ω ⋅ 2 cos ⁡ 34 π 78 = − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) + 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 3 {\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{78}}+\omega \cdot 2\cos {\frac {46\pi }{78}}+\omega ^{2}\cdot 2\cos {\frac {34\pi }{78}}=&{\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 }{78}}+\omega ^{2}\cdot 2\cos {\frac {46\pi }{78}}+\omega \cdot 2\cos {\frac {34\pi }{78}}=&{\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}}} よって cos ⁡ 2 π 78 = 1 6 ( − 1 − 13 + 6 ( 13 + 3 13 ) 4 + − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) − 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 3 + − 104 + 34 13 − 3 6 ( 13 + 3 13 ) + 15 6 ( 13 − 3 13 ) + 3 3 ( 2 13 + 6 ( 13 + 3 13 ) − 6 ( 13 − 3 13 ) ) i 8 3 ) {\displaystyle {\begin{aligned}\cos {\frac {2\pi }{78}}=&{\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}}} 正七十八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。 正七十八角形は折紙により作図可能である。 [脚注の使い方] 十三角形 二十六角形 三十九角形 五十二角形 ポータル 数学 表示 編集