三十五角形

正三十五角形においては、中心角と外角は10.285…°で、内角は169.714…°となる。一辺の長さが a の正三十五角形の面積 S は

${\displaystyle S={\frac {35}{4}}a^{2}\cot {\frac {\pi }{35}}\simeq 97.22046a^{2}}$

${\displaystyle \cos(2\pi /35)}$を平方根と立方根で表すことが可能である。

${\displaystyle {\begin{aligned}\cos {\frac {2\pi }{35}}=&\cos \left({\frac {\pi }{5}}-{\frac {\pi }{7}}\right)\\=&\cos {\frac {\pi }{5}}\cos {\frac {\pi }{7}}+\sin {\frac {\pi }{5}}\sin {\frac {\pi }{7}}\\=&{\frac {1}{4}}\left({\sqrt {5}}+1\right)\cdot \cos {\frac {\pi }{7}}+{\frac {1}{4}}\left({\sqrt {10-2{\sqrt {5}}}}\right)\cdot \sin {\frac {\pi }{7}}\\=&{\frac {{\sqrt {5}}+1}{4}}\cdot {\frac {\sqrt {3\left(20+2{\sqrt[{3}]{28-84i{\sqrt {3}}}}+2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}+{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\cdot {\frac {\sqrt {3\left(28-2{\sqrt[{3}]{28-84i{\sqrt {3}}}}-2{\sqrt[{3}]{28+84i{\sqrt {3}}}}\right)}}{12}}\end{aligned}}}$

関係式

${\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{35}}+2\cos {\frac {32\pi }{35}}+2\cos {\frac {22\pi }{35}}={\frac {1}{4}}\left(1+{\sqrt {5}}-{\sqrt {14\left(5-{\sqrt {5}}\right)}}\right)=x_{1}\\2\cos {\frac {4\pi }{35}}+2\cos {\frac {6\pi }{35}}+2\cos {\frac {26\pi }{35}}={\frac {1}{4}}\left(1-{\sqrt {5}}+{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)=x_{2}\\2\cos {\frac {8\pi }{35}}+2\cos {\frac {12\pi }{35}}+2\cos {\frac {18\pi }{35}}={\frac {1}{4}}\left(1+{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}\right)=x_{3}\\2\cos {\frac {16\pi }{35}}+2\cos {\frac {24\pi }{35}}+2\cos {\frac {34\pi }{35}}={\frac {1}{4}}\left(1-{\sqrt {5}}-{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)=x_{4}\\\end{aligned}}}$

さらに、以下のような関係式が得られる。

${\displaystyle {\begin{aligned}\left(2\cos {\frac {2\pi }{35}}+\omega \cdot 2\cos {\frac {32\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{35}}\right)^{3}=&3x_{1}+x_{2}+6x_{3}+12\cos {\frac {2\pi }{5}}+3\omega (2x_{1}+x_{4}+6\cos {\frac {4\pi }{5}})+3\omega ^{2}(2x_{1}+x_{2}+x_{3})\\=&{\tfrac {{-2+20{\sqrt {5}}+3{\sqrt {14\left(5-{\sqrt {5}}\right)}}+{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3\omega \left(-3-5{\sqrt {5}}-2{\sqrt {14\left(5-{\sqrt {5}}\right)}}-{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)+3\omega ^{2}\left(4+2{\sqrt {5}}-{\sqrt {14\left(5-{\sqrt {5}}\right)}}+{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)}{4}}\\=&{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}\\\left(2\cos {\frac {2\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{35}}+\omega \cdot 2\cos {\frac {22\pi }{35}}\right)^{3}=&3x_{1}+x_{2}+6x_{3}+12\cos {\frac {2\pi }{5}}+3\omega ^{2}(2x_{1}+x_{4}+6\cos {\frac {4\pi }{5}})+3\omega (2x_{1}+x_{2}+x_{3})\\=&{\tfrac {{-2+20{\sqrt {5}}+3{\sqrt {14\left(5-{\sqrt {5}}\right)}}+{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3\omega ^{2}\left(-3-5{\sqrt {5}}-2{\sqrt {14\left(5-{\sqrt {5}}\right)}}-{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)+3\omega \left(4+2{\sqrt {5}}-{\sqrt {14\left(5-{\sqrt {5}}\right)}}+{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)}{4}}\\=&{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}\\\end{aligned}}}$

両辺の立方根を取ると

${\displaystyle {\begin{aligned}2\cos {\frac {2\pi }{35}}+\omega \cdot 2\cos {\frac {32\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}-3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\2\cos {\frac {2\pi }{35}}+\omega ^{2}\cdot 2\cos {\frac {32\pi }{35}}+\omega \cdot 2\cos {\frac {22\pi }{35}}=&{\sqrt[{3}]{\tfrac {{-7+49{\sqrt {5}}+15{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}}+3{\sqrt {3}}\left(7+7{\sqrt {5}}+{\sqrt {14\left(5-{\sqrt {5}}\right)}}+2{\sqrt {14\left(5+{\sqrt {5}}\right)}}\right)i}{8}}}\\\end{aligned}}}$

よって

• 五角形
• 七角形
• 十角形
• 十四角形