四十二角形

正四十二角形においては、中心角と外角は8.571…°で、内角は171.428…°となる。一辺の長さが a の正四十二角形の面積 S は

${\displaystyle S={\frac {42}{4}}a^{2}\cot {\frac {\pi }{42}}\simeq 140.11276a^{2}}$

正三角形、正七角形、正四十二角形でのタイリング

${\displaystyle \cos(2\pi /42)}$を平方根と立方根で表すことが可能である。

${\displaystyle {\begin{aligned}\cos {\frac {2\pi }{42}}=&\cos {\frac {\pi }{21}}\\=&{\frac {1}{12}}{\sqrt {72+72\cos {\frac {2\pi }{21}}}}\\=&{\frac {1}{12}}{\sqrt {72+72\cdot {\frac {1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}{12}}}}\\=&{\frac {1}{12}}{\sqrt {72+6\left({1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(42{\sqrt {3}}-18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{154-30{\sqrt {21}}+\left(18{\sqrt {7}}-42{\sqrt {3}}\right)i}}}\right)}}\end{aligned}}}$

関係式

以下のように定義すると

${\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{42}}+2\cos {\frac {10\pi }{42}}+2\cos {\frac {34\pi }{42}}={\frac {-1+{\sqrt {21}}}{2}}\\&\beta =2\cos {\frac {22\pi }{42}}+2\cos {\frac {26\pi }{42}}+2\cos {\frac {38\pi }{42}}={\frac {-1-{\sqrt {21}}}{2}}\\\end{aligned}}}$

${\displaystyle \alpha ,\beta }$は以下の関係式より求められる。

${\displaystyle {\begin{aligned}&\alpha +\beta =-1\\&(\alpha -\beta )^{2}=21\\\end{aligned}}}$

三次方程式の係数を求めると

${\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{42}}\cdot 2\cos {\frac {10\pi }{42}}+2\cos {\frac {10\pi }{42}}\cdot 2\cos {\frac {34\pi }{42}}+2\cos {\frac {34\pi }{42}}\cdot 2\cos {\frac {2\pi }{42}}=-\alpha -1\\&2\cos {\frac {2\pi }{42}}\cdot 2\cos {\frac {10\pi }{42}}\cdot 2\cos {\frac {34\pi }{42}}=\beta -2\\\end{aligned}}}$

解と係数の関係より

${\displaystyle x^{3}-\alpha x^{2}+(-\alpha -1)x-(\beta -2)=0}$

変数変換、関係式より

${\displaystyle x=y+\alpha /3,\quad \beta =-1-\alpha ,\quad \alpha ^{2}=5-\alpha ,\quad \alpha ^{3}=6\alpha -5}$

整理すると

${\displaystyle y^{3}-{\frac {2\alpha +8}{3}}x+{\frac {15\alpha +46}{27}}=0}$

三角関数、逆三角関数を使用した解は

${\displaystyle x={\frac {\alpha }{3}}+{\frac {2{\sqrt {2\alpha +8}}}{3}}\cos \left({\frac {1}{3}}\arccos {\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}\right)}$

平方根と立方根で表すと

${\displaystyle {\begin{aligned}&x={\frac {\alpha }{3}}+{\frac {\sqrt {2\alpha +8}}{3}}{\sqrt[{3}]{{\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}+i{\frac {\sqrt {189(\alpha +3)}}{2({2\alpha +8})^{\frac {3}{2}}}}}}+{\frac {\sqrt {2\alpha +8}}{3}}{\sqrt[{3}]{{\frac {-(15\alpha +46)}{2({2\alpha +8})^{\frac {3}{2}}}}-i{\frac {\sqrt {189(\alpha +3)}}{2({2\alpha +8})^{\frac {3}{2}}}}}}\\&x={\frac {\alpha }{3}}+{\frac {1}{6}}{\sqrt[{3}]{{-4(15\alpha +46)}+i\cdot 4{\sqrt {189(\alpha +3)}}}}+{\frac {1}{6}}{\sqrt[{3}]{-4(15\alpha +46)-i\cdot 4{\sqrt {189(\alpha +3)}}}}\end{aligned}}}$

αの値((-1+√21)/2)を代入して、整理すると

${\displaystyle \cos {\frac {2\pi }{42}}={\frac {-1+{\sqrt {21}}+{\sqrt[{3}]{-154-30{\sqrt {21}}+\left(42{\sqrt {3}}+18{\sqrt {7}}\right)i}}+{\sqrt[{3}]{-154-30{\sqrt {21}}-\left(42{\sqrt {3}}+18{\sqrt {7}}\right)i}}}{12}}}$

• 七角形
• 二十一角形