Polyhedron with 92 faces
3D model of a great snub icosidodecahedron
In geometry , the great snub icosidodecahedron is a nonconvex uniform polyhedron , indexed as U57 . It has 92 faces (80 triangles and 12 pentagrams ), 150 edges, and 60 vertices.[1] It can be represented by a Schläfli symbol sr{5 ⁄2 ,3}, and Coxeter-Dynkin diagram .
This polyhedron is the snub member of a family that includes the great icosahedron , the great stellated dodecahedron and the great icosidodecahedron .
In the book Polyhedron Models by Magnus Wenninger , the polyhedron is misnamed great inverted snub icosidodecahedron , and vice versa.
Cartesian coordinates [ edit ]
Cartesian coordinates for the vertices of a great retrosnub 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 negative real root of:
ξ
3
−
2
ξ
=
−
1
φ
⟹
ξ
≈
−
1.5488772.
{\displaystyle \xi ^{3}-2\xi =-{\frac {1}{\varphi }}\quad \implies \quad \xi \approx -1.5488772.}
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
=
0.64502
…
{\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.64502\dots }
where
x
=
−
0.505561
{\displaystyle x=-0.505561}
is the second largest real root of the polynomial
[2]
x
3
+
2
x
2
=
φ
−
2
=
(
1
+
5
2
)
−
2
=
(
1
−
5
2
)
2
.
{\displaystyle x^{3}+2x^{2}=\varphi ^{-2}=\left({\tfrac {1+{\sqrt {5}}}{2}}\right)^{-2}=\left({\tfrac {1-{\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, in order, the circumradii of the
great retrosnub icosidodecahedron (U
74 ), great snub icosidodecahedron (U
57 ),
great inverted snub icosidodecahedron (U
69 ) and
snub dodecahedron (U
29 ).
Related polyhedra [ edit ]
Great pentagonal hexecontahedron [ edit ]
3D model of a great pentagonal hexecontahedron
The great pentagonal hexecontahedron (or great petaloid ditriacontahedron ) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron . It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.
Proportions [ edit ]
Denote the golden ratio by
ϕ
{\displaystyle \phi }
. Let
ξ
≈
−
0.199
510
322
83
{\displaystyle \xi \approx -0.199\,510\,322\,83}
be the negative 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
)
≈
101.508
325
512
64
∘
{\displaystyle \arccos(\xi )\approx 101.508\,325\,512\,64^{\circ }}
and one angle of
arccos
(
−
ϕ
−
1
+
ϕ
−
2
ξ
)
≈
133.966
697
949
42
∘
{\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 133.966\,697\,949\,42^{\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
≈
1.315
765
089
00
{\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.315\,765\,089\,00}
.
The dihedral angle equals
arccos
(
ξ
/
(
ξ
+
1
)
)
≈
104.432
268
611
86
∘
{\displaystyle \arccos(\xi /(\xi +1))\approx 104.432\,268\,611\,86^{\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 inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron .
See also [ edit ]
References [ edit ]
External links [ edit ]
Kepler-Poinsot polyhedra (nonconvex regular polyhedra)Uniform truncations of Kepler-Poinsot polyhedra Nonconvex uniform hemipolyhedra Duals of nonconvex uniform polyhedra Duals of nonconvex uniform polyhedra with infinite stellations