七十八角形 – Wikipedia

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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[編集]

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=784a2cot⁡π78≃483.88751a2{displaystyle S={frac {78}{4}}a^{2}cot {frac {pi }{78}}simeq 483.88751a^{2}}

関係式
2cos⁡2π78+2cos⁡46π78+2cos⁡34π78=14(−1−13+6(13+313))=x12cos⁡22π78+2cos⁡38π78+2cos⁡62π78=14(−1+13−6(13−313))=x22cos⁡70π78+2cos⁡50π78+2cos⁡58π78=14(−1−13−6(13+313))=x32cos⁡10π78+2cos⁡74π78+2cos⁡14π78=14(−1+13+6(13−313))=x4{displaystyle {begin{aligned}2cos {frac {2pi }{78}}+2cos {frac {46pi }{78}}+2cos {frac {34pi }{78}}={frac {1}{4}}left(-1-{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}right)=x_{1}2cos {frac {22pi }{78}}+2cos {frac {38pi }{78}}+2cos {frac {62pi }{78}}={frac {1}{4}}left(-1+{sqrt {13}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)=x_{2}2cos {frac {70pi }{78}}+2cos {frac {50pi }{78}}+2cos {frac {58pi }{78}}={frac {1}{4}}left(-1-{sqrt {13}}-{sqrt {6left(13+3{sqrt {13}}right)}}right)=x_{3}2cos {frac {10pi }{78}}+2cos {frac {74pi }{78}}+2cos {frac {14pi }{78}}={frac {1}{4}}left(-1+{sqrt {13}}+{sqrt {6left(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。

(2cos⁡2π78+ω⋅2cos⁡46π78+ω2⋅2cos⁡34π78)3=3x1+2cos⁡2π26+2cos⁡6π26+2cos⁡18π26+6(x4−2)+3ω(2x1+x3+2cos⁡14π26+2cos⁡10π26+2cos⁡22π26)+3ω2(2x1+x2+2cos⁡14π26+2cos⁡10π26+2cos⁡22π26)=3x1+1+132+6(x2−2)+3ω(2x1+x3+1−132)+3ω2(2x1+x2+1−132)=−104+3413−36(13+313)+156(13−313)−33(213+6(13+313)−6(13−313))i8(2cos⁡2π78+ω2⋅2cos⁡46π78+ω⋅2cos⁡34π78)3=3x1+2cos⁡2π26+2cos⁡6π26+2cos⁡18π26+6(x4−2)+3ω2(2x1+x3+2cos⁡14π26+2cos⁡10π26+2cos⁡22π26)+3ω(2x1+x4+2cos⁡14π26+2cos⁡10π26+2cos⁡22π26)=3x1+1+132+6(x2−2)+3ω2(2x1+x3+1−132)+3ω(2x1+x2+1−132)=−104+3413−36(13+313)+156(13−313)+33(213+6(13+313)−6(13−313))i8{displaystyle {begin{aligned}left(2cos {frac {2pi }{78}}+omega cdot 2cos {frac {46pi }{78}}+omega ^{2}cdot 2cos {frac {34pi }{78}}right)^{3}=&3x_{1}+2cos {frac {2pi }{26}}+2cos {frac {6pi }{26}}+2cos {frac {18pi }{26}}+6(x_{4}-2)+3omega left(2x_{1}+x_{3}+2cos {frac {14pi }{26}}+2cos {frac {10pi }{26}}+2cos {frac {22pi }{26}}right)+3omega ^{2}left(2x_{1}+x_{2}+2cos {frac {14pi }{26}}+2cos {frac {10pi }{26}}+2cos {frac {22pi }{26}}right)=&3x_{1}+{frac {1+{sqrt {13}}}{2}}+6(x_{2}-2)+3omega left(2x_{1}+x_{3}+{frac {1-{sqrt {13}}}{2}}right)+3omega ^{2}left(2x_{1}+x_{2}+{frac {1-{sqrt {13}}}{2}}right)=&{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}-3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}left(2cos {frac {2pi }{78}}+omega ^{2}cdot 2cos {frac {46pi }{78}}+omega cdot 2cos {frac {34pi }{78}}right)^{3}=&3x_{1}+2cos {frac {2pi }{26}}+2cos {frac {6pi }{26}}+2cos {frac {18pi }{26}}+6(x_{4}-2)+3omega ^{2}left(2x_{1}+x_{3}+2cos {frac {14pi }{26}}+2cos {frac {10pi }{26}}+2cos {frac {22pi }{26}}right)+3omega left(2x_{1}+x_{4}+2cos {frac {14pi }{26}}+2cos {frac {10pi }{26}}+2cos {frac {22pi }{26}}right)=&3x_{1}+{frac {1+{sqrt {13}}}{2}}+6(x_{2}-2)+3omega ^{2}left(2x_{1}+x_{3}+{frac {1-{sqrt {13}}}{2}}right)+3omega left(2x_{1}+x_{2}+{frac {1-{sqrt {13}}}{2}}right)=&{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}+3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}end{aligned}}}

両辺の立方根を取ると

2cos⁡2π78+ω⋅2cos⁡46π78+ω2⋅2cos⁡34π78=−104+3413−36(13+313)+156(13−313)−33(213+6(13+313)−6(13−313))i832cos⁡2π78+ω2⋅2cos⁡46π78+ω⋅2cos⁡34π78=−104+3413−36(13+313)+156(13−313)+33(213+6(13+313)−6(13−313))i83{displaystyle {begin{aligned}2cos {frac {2pi }{78}}+omega cdot 2cos {frac {46pi }{78}}+omega ^{2}cdot 2cos {frac {34pi }{78}}=&{sqrt[{3}]{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}-3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}}2cos {frac {2pi }{78}}+omega ^{2}cdot 2cos {frac {46pi }{78}}+omega cdot 2cos {frac {34pi }{78}}=&{sqrt[{3}]{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}+3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}}end{aligned}}}

よって

cos⁡2π78=16(−1−13+6(13+313)4+−104+3413−36(13+313)+156(13−313)−33(213+6(13+313)−6(13−313))i83+−104+3413−36(13+313)+156(13−313)+33(213+6(13+313)−6(13−313))i83){displaystyle {begin{aligned}cos {frac {2pi }{78}}=&{frac {1}{6}}left({tfrac {-1-{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}}{4}}+{sqrt[{3}]{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}-3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}}+{sqrt[{3}]{tfrac {-104+34{sqrt {13}}-3{sqrt {6left(13+3{sqrt {13}}right)}}+15{sqrt {6left(13-3{sqrt {13}}right)}}+3{sqrt {3}}left(2{sqrt {13}}+{sqrt {6left(13+3{sqrt {13}}right)}}-{sqrt {6left(13-3{sqrt {13}}right)}}right)i}{8}}}right)end{aligned}}}

正七十八角形の作図[編集]

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。

関連項目[編集]

外部リンク[編集]