Polyhedron with 92 faces
3D model of a great inverted snub icosidodecahedron
In geometry , the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron ) is a uniform star polyhedron , indexed as U69 . It is given a Schläfli symbol sr{5 ⁄3 ,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger , the polyhedron is misnamed great snub icosidodecahedron , and vice versa.
Cartesian coordinates [ edit ]
Cartesian coordinates for the vertices of a great inverted snub icosidodecahedron are all the even permutations of
(
±
2
α
,
±
2
,
±
2
β
)
,
(
±
[
α
−
β
φ
−
1
φ
]
,
±
[
α
φ
+
β
−
φ
]
,
±
[
−
α
φ
−
β
φ
−
1
]
)
,
(
±
[
α
φ
−
β
φ
+
1
]
,
±
[
−
α
−
β
φ
+
1
φ
]
,
±
[
−
α
φ
+
β
+
φ
]
)
,
(
±
[
α
φ
−
β
φ
−
1
]
,
±
[
α
+
β
φ
+
1
φ
]
,
±
[
−
α
φ
+
β
−
φ
]
)
,
(
±
[
α
−
β
φ
+
1
φ
]
,
±
[
−
α
φ
−
β
−
φ
]
,
±
[
−
α
φ
−
β
φ
+
1
]
)
,
{\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi -{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]},&\pm {\bigl [}-\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta +\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]},&\pm {\bigl [}\alpha +\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}-\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]}&{\Bigr )},\\\end{array}}}
with an even number of plus signs, where
α
=
ξ
−
1
ξ
β
=
−
ξ
φ
+
1
φ
2
−
1
ξ
φ
,
{\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }}\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}}
where
φ
=
1
+
5
2
{\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}
is the golden ratio and
ξ is the greater positive real solution to:
ξ
3
−
2
ξ
=
−
1
φ
⟹
ξ
≈
1.2224727.
{\displaystyle \xi ^{3}-2\xi =-{\tfrac {1}{\varphi }}\quad \implies \quad \xi \approx 1.2224727.}
Taking the
odd permutations of the above coordinates with an odd number of plus signs gives another form, the
enantiomorph of the other one.
The circumradius for unit edge length is
R
=
1
2
2
−
x
1
−
x
=
0.816081
…
{\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.816081\dots }
where
x is the appropriate root of
x
3
+
2
x
2
=
(
1
±
5
2
)
2
.
{\displaystyle x^{3}+2x^{2}=\left({\frac {1\pm {\sqrt {5}}}{2}}\right)^{2}.}
The four positive real roots of the
sextic in
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}
are the circumradii of the
snub dodecahedron (U
29 ),
great snub icosidodecahedron (U
57 ), great inverted snub icosidodecahedron (U
69 ), and
great retrosnub icosidodecahedron (U
74 ).
Related polyhedra [ edit ]
Great inverted pentagonal hexecontahedron [ edit ]
3D model of a great inverted pentagonal hexecontahedron
The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron ) is a nonconvex isohedral polyhedron . It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.
It is the dual of the uniform great inverted snub icosidodecahedron.
Proportions [ edit ]
Denote the golden ratio by
ϕ
{\displaystyle \phi }
. Let
ξ
≈
0.252
780
289
27
{\displaystyle \xi \approx 0.252\,780\,289\,27}
be the smallest positive zero of the polynomial
P
=
8
x
3
−
8
x
2
+
ϕ
−
2
{\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}
. Then each pentagonal face has four equal angles of
arccos
(
ξ
)
≈
75.357
903
417
42
∘
{\displaystyle \arccos(\xi )\approx 75.357\,903\,417\,42^{\circ }}
and one angle of
360
∘
−
arccos
(
−
ϕ
−
1
+
ϕ
−
2
ξ
)
≈
238.568
386
330
33
∘
{\displaystyle 360^{\circ }-\arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 238.568\,386\,330\,33^{\circ }}
. Each face has three long and two short edges. The ratio
l
{\displaystyle l}
between the lengths of the long and the short edges is given by
l
=
2
−
4
ξ
2
1
−
2
ξ
≈
3.528
053
034
81
{\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 3.528\,053\,034\,81}
.
The dihedral angle equals
arccos
(
ξ
/
(
ξ
+
1
)
)
≈
78.359
199
060
62
∘
{\displaystyle \arccos(\xi /(\xi +1))\approx 78.359\,199\,060\,62^{\circ }}
. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial
P
{\displaystyle P}
play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron .
See also [ edit ]
References [ edit ]
External links [ edit ]