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二十六角形 .
正二十六角形 二十六角形 (にじゅうろくかくけい、にじゅうろっかっけい、icosihexagon)は、多角形 の一つで、26本の辺 と26個の頂点 を持つ図形である。内角の和 は4320°、対角線 の本数は299本である。
正二十六角形においては、中心角と外角は13.846…°で、内角は166.153…°となる。一辺の長さが a の正二十六角形の面積 S は
S
=
26
4
a
2
cot
π
26
≃
53.53232
a
2
{\displaystyle S={\frac {26}{4))a^{2}\cot {\frac {\pi }{26))\simeq 53.53232a^{2))
cos
(
2
π
/
26
)
{\displaystyle \cos(2\pi /26)}
を平方根と立方根で表すと
cos
2
π
26
=
cos
π
13
=
1
12
72
+
72
⋅
cos
2
π
13
=
1
12
72
+
72
⋅
1
12
(
104
−
20
13
+
12
−
39
3
+
104
−
20
13
−
12
−
39
3
+
13
−
1
)
=
0.970941...
{\displaystyle \cos {\frac {2\pi }{26))=\cos {\frac {\pi }{13))={\frac {1}{12)){\sqrt {72+72\cdot \cos {\frac {2\pi }{13))))={\frac {1}{12)){\sqrt {72+72\cdot {\frac {1}{12))\left({\sqrt[{3}]{104-20{\sqrt {13))+12{\sqrt {-39))))+{\sqrt[{3}]{104-20{\sqrt {13))-12{\sqrt {-39))))+{\sqrt {13))-1\right)))=0.970941...}
関係式
α
=
2
cos
2
π
26
+
2
cos
6
π
26
+
2
cos
18
π
26
=
1
+
13
2
β
=
2
cos
14
π
26
+
2
cos
10
π
26
+
2
cos
22
π
26
=
1
−
13
2
{\displaystyle {\begin{aligned}&\alpha =2\cos {\frac {2\pi }{26))+2\cos {\frac {6\pi }{26))+2\cos {\frac {18\pi }{26))={\frac {1+{\sqrt {13))}{2))\\&\beta =2\cos {\frac {14\pi }{26))+2\cos {\frac {10\pi }{26))+2\cos {\frac {22\pi }{26))={\frac {1-{\sqrt {13))}{2))\\\end{aligned))}
三次方程式の係数を求めると
2
cos
2
π
26
⋅
2
cos
6
π
26
+
2
cos
6
π
26
⋅
2
cos
18
π
26
+
2
cos
18
π
26
⋅
2
cos
2
π
26
=
−
1
2
cos
2
π
26
⋅
2
cos
6
π
26
⋅
2
cos
18
π
26
=
β
−
2
{\displaystyle {\begin{aligned}&2\cos {\frac {2\pi }{26))\cdot 2\cos {\frac {6\pi }{26))+2\cos {\frac {6\pi }{26))\cdot 2\cos {\frac {18\pi }{26))+2\cos {\frac {18\pi }{26))\cdot 2\cos {\frac {2\pi }{26))=-1\\&2\cos {\frac {2\pi }{26))\cdot 2\cos {\frac {6\pi }{26))\cdot 2\cos {\frac {18\pi }{26))=\beta -2\end{aligned))}
解と係数の関係より
x
3
−
α
x
2
−
x
−
(
β
−
2
)
=
0
{\displaystyle x^{3}-\alpha x^{2}-x-(\beta -2)=0}
変数変換
x
=
y
+
α
/
3
{\displaystyle x=y+\alpha /3}
整理すると
y
3
−
13
+
13
6
y
+
26
+
5
13
27
=
0
{\displaystyle y^{3}-{\frac {13+{\sqrt {13))}{6))y+{\frac {26+5{\sqrt {13))}{27))=0}
三角関数、逆三角関数を用いて解は
x
=
1
+
13
6
+
2
3
13
+
13
2
cos
(
1
3
arccos
−
(
26
+
5
13
)
2
(
13
+
13
2
)
3
2
)
{\displaystyle x={\frac {1+{\sqrt {13))}{6))+{\frac {2}{3)){\sqrt {\frac {13+{\sqrt {13))}{2))}\cos \left({\frac {1}{3))\arccos {\frac {-(26+5{\sqrt {13)))}{2\left({\frac {13+{\sqrt {13))}{2))\right)^{\tfrac {3}{2))))\right)}
平方根、立方根を用いて
x
=
1
+
13
6
+
1
3
13
+
13
2
−
(
26
+
5
13
)
2
(
13
+
13
2
)
3
2
+
i
3
39
2
(
13
+
13
2
)
3
2
3
+
1
3
13
+
13
2
−
(
26
+
5
13
)
2
(
13
+
13
2
)
3
2
−
i
3
39
2
(
13
+
13
2
)
3
2
3
{\displaystyle x={\frac {1+{\sqrt {13))}{6))+{\frac {1}{3)){\sqrt {\frac {13+{\sqrt {13))}{2))}{\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2\left({\frac {13+{\sqrt {13))}{2))\right)^{\tfrac {3}{2))))+i{\frac {3{\sqrt {39))}{2\left({\frac {13+{\sqrt {13))}{2))\right)^{\tfrac {3}{2))))))+{\frac {1}{3)){\sqrt {\frac {13+{\sqrt {13))}{2))}{\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2\left({\frac {13+{\sqrt {13))}{2))\right)^{\tfrac {3}{2))))-i{\frac {3{\sqrt {39))}{2\left({\frac {13+{\sqrt {13))}{2))\right)^{\tfrac {3}{2))))))}
x
=
1
+
13
6
+
1
3
−
(
26
+
5
13
)
2
+
i
3
39
2
3
+
1
3
−
(
26
+
5
13
)
2
−
i
3
39
2
3
{\displaystyle x={\frac {1+{\sqrt {13))}{6))+{\frac {1}{3)){\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2))+i{\frac {3{\sqrt {39))}{2))))+{\frac {1}{3)){\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2))-i{\frac {3{\sqrt {39))}{2))))}
cos
(
2
π
/
26
)
{\displaystyle \cos(2\pi /26)}
を平方根と立方根で表すと
cos
2
π
26
=
1
+
13
12
+
1
6
−
(
26
+
5
13
)
2
+
i
3
39
2
3
+
1
6
−
(
26
+
5
13
)
2
−
i
3
39
2
3
{\displaystyle \cos {\frac {2\pi }{26))={\frac {1+{\sqrt {13))}{12))+{\frac {1}{6)){\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2))+i{\frac {3{\sqrt {39))}{2))))+{\frac {1}{6)){\sqrt[{3}]((\frac {-(26+5{\sqrt {13)))}{2))-i{\frac {3{\sqrt {39))}{2))))}
正二十六角形は定規 とコンパス による作図 が不可能な図形である。
正二十六角形は折紙 により作図可能である。
非古典的 (2辺以下) 辺の数: 3–10
辺の数: 11–20 辺の数: 21–30 辺の数: 31–40 辺の数: 41–50 辺の数: 51–70 (selected) 辺の数: 71–100 (selected) 辺の数: 101– (selected) 無限 星型多角形 (辺の数: 5–12)多角形のクラス
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