In geometria , il grande icosidodecaedro retrocamuso è un poliedro stellato uniforme avente 92 facce - 80 triangolari e 12 a forma di pentagramma - 150 spigoli e 60 vertici.[ 1]
Coordinate cartesiane
Le coordinate cartesiane per i vertici del grande icosidodecaedro retrocamuso, spesso indicato con il simbolo U74 e il cui inviluppo convesso è un dodecaedro camuso non uniforme, sono date da tutte le permutazioni pari di:
(
± ± -->
2
α α -->
,
± ± -->
2
,
± ± -->
2
β β -->
)
{\displaystyle \left(\,\pm 2\alpha ,\,\pm 2,\,\pm 2\beta \,\right)}
(
± ± -->
(
α α -->
− − -->
β β -->
φ φ -->
− − -->
φ φ -->
− − -->
1
)
,
± ± -->
(
α α -->
φ φ -->
− − -->
1
+
β β -->
− − -->
φ φ -->
)
,
± ± -->
(
− − -->
α α -->
φ φ -->
− − -->
β β -->
φ φ -->
− − -->
1
− − -->
1
)
)
{\displaystyle \left(\,\pm (\alpha -\beta \varphi -\varphi ^{-1}),\,\pm (\alpha \varphi ^{-1}+\beta -\varphi ),\,\pm (-\alpha \varphi -\beta \varphi ^{-1}-1)\,\right)}
(
± ± -->
(
α α -->
φ φ -->
− − -->
β β -->
φ φ -->
− − -->
1
+
1
)
,
± ± -->
(
− − -->
α α -->
− − -->
β β -->
φ φ -->
+
φ φ -->
− − -->
1
)
,
± ± -->
(
− − -->
α α -->
φ φ -->
− − -->
1
+
β β -->
+
φ φ -->
)
)
{\displaystyle \left(\,\pm (\alpha \varphi -\beta \varphi ^{-1}+1),\,\pm (-\alpha -\beta \varphi +\varphi ^{-1}),\,\pm (-\alpha \varphi ^{-1}+\beta +\varphi )\,\right)}
(
± ± -->
(
α α -->
φ φ -->
− − -->
β β -->
φ φ -->
− − -->
1
− − -->
1
)
,
± ± -->
(
α α -->
+
β β -->
φ φ -->
+
φ φ -->
− − -->
1
)
,
± ± -->
(
− − -->
α α -->
φ φ -->
− − -->
1
+
β β -->
− − -->
φ φ -->
)
)
{\displaystyle \left(\,\pm (\alpha \varphi -\beta \varphi ^{-1}-1),\,\pm (\alpha +\beta \varphi +\varphi ^{-1}),\,\pm (-\alpha \varphi ^{-1}+\beta -\varphi )\,\right)}
(
± ± -->
(
α α -->
− − -->
β β -->
φ φ -->
+
φ φ -->
− − -->
1
)
,
± ± -->
(
− − -->
α α -->
φ φ -->
− − -->
1
− − -->
β β -->
− − -->
φ φ -->
)
,
± ± -->
(
− − -->
α α -->
φ φ -->
− − -->
β β -->
φ φ -->
− − -->
1
+
1
)
)
{\displaystyle \left(\,\pm (\alpha -\beta \varphi +\varphi ^{-1}),\,\pm (-\alpha \varphi ^{-1}-\beta -\varphi ),\,\pm (-\alpha \varphi -\beta \varphi ^{-1}+1)\,\right)}
con un numero pari di segni più, dove
φ φ -->
=
1
+
5
2
{\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}
sezione aurea ,
ξ ξ -->
≈ ≈ -->
0
,
3264046
{\displaystyle \xi \approx 0,3264046}
radice positiva dell'equazione
ξ ξ -->
3
− − -->
2
ξ ξ -->
=
− − -->
1
φ φ -->
⟹ ⟹ -->
ξ ξ -->
=
(
1
+
i
3
)
1
2
φ φ -->
+
1
4
φ φ -->
2
− − -->
8
27
3
+
(
1
− − -->
i
3
)
1
2
φ φ -->
− − -->
1
4
φ φ -->
2
− − -->
8
27
3
2
{\displaystyle {\begin{aligned}\xi ^{3}-2\xi =-{\frac {1}{\varphi }}\quad \implies \quad \xi &={\frac {\left(1+i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}+{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}+\left(1-i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}-{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}}{2}}\\[4pt]\end{aligned}}}
e
α α -->
=
ξ ξ -->
− − -->
1
ξ ξ -->
,
β β -->
=
− − -->
ξ ξ -->
φ φ -->
+
1
φ φ -->
2
− − -->
1
ξ ξ -->
φ φ -->
.
{\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }}.\end{aligned}}}
Poliedri correlati
Dato un grande icosidodecaedro retrocamuso di spigolo pari a 1, il suo circumraggio è pari a
R
=
1
2
2
− − -->
x
1
− − -->
x
=
0
,
580002
… … -->
{\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0,580002\dots }
x
{\displaystyle x}
x
3
+
2
x
2
=
(
1
± ± -->
5
2
)
2
.
{\displaystyle x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}.}
Le quattro radici positive dell'equazione in
R
2
{\displaystyle \mathbb {R} ^{2}}
4096
R
12
− − -->
27648
R
10
+
47104
R
8
− − -->
35776
R
6
+
13872
R
4
− − -->
2696
R
2
+
209
=
0
{\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}
74 ), del grande icosidodecaedro camuso (U57 ), del grande icosidodecaedro camuso invertito (U69 ) e del dodecaedro camuso (U29 ).
Grande esacontaedro pentagrammico
Il grande esacontaedro pentagrammico è un poliedro stellato isoedro, nonché il duale del grande icosidodecaedro retrocamuso, avente per facce 60 pentagrammi irregolari.[ 2]
Dato un grande icosidodecaedro retrocamuso di spigolo pari a 1, immaginando il grande esacontaedro pentagrammico come composto da 60 facce intersecanti a forma di pentagramma irregolare, come riportato nella figura sottostante, di cui solo una parte visibile all'esterno del solido, e considerando la già citata sezione aurea e il numero
ξ ξ -->
≈ ≈ -->
0
,
946
730
033
56
{\displaystyle \xi \approx 0,946\,730\,033\,56}
P
=
8
x
3
− − -->
8
x
2
+
ϕ ϕ -->
− − -->
2
{\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}
arccos
-->
(
ξ ξ -->
)
≈ ≈ -->
18
,
785
633
958
24
∘ ∘ -->
{\displaystyle \arccos(\xi )\approx 18,785\,633\,958\,24^{\circ }}
arccos
-->
(
− − -->
ϕ ϕ -->
− − -->
1
+
ϕ ϕ -->
− − -->
2
ξ ξ -->
)
≈ ≈ -->
104
,
857
464
167
03
∘ ∘ -->
{\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 104,857\,464\,167\,03^{\circ }}
2
− − -->
4
ξ ξ -->
2
1
− − -->
2
ξ ξ -->
≈ ≈ -->
1
,
774
215
864
94
{\displaystyle {\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1,774\,215\,864\,94}
Note
Collegamenti esterni
![{\displaystyle {\begin{aligned}\xi ^{3}-2\xi =-{\frac {1}{\varphi }}\quad \implies \quad \xi &={\frac {\left(1+i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}+{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}+\left(1-i{\sqrt {3}}\right){\sqrt[{3}]{{\frac {1}{2\varphi }}-{\sqrt {{\frac {1}{4\varphi ^{2}}}-{\frac {8}{27}}}}}}}{2}}\\[4pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c43a5adc0a811e2954bcc9180672138c294649)
![{\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec51fa6dbea90b09896dbf0e7d6b128720208df6)
