Polyhedron with 44 faces
3D model of an icositruncated dodecadodecahedron In geometry , the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron , indexed as U45 .
Convex hull
Its convex hull is a nonuniform truncated icosidodecahedron .
Cartesian coordinates
Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of
(
± ± -->
[
2
− − -->
1
φ φ -->
]
,
± ± -->
1
,
± ± -->
[
2
+
φ φ -->
]
)
,
(
± ± -->
1
,
± ± -->
1
φ φ -->
2
,
± ± -->
[
3
φ φ -->
− − -->
1
]
)
,
(
± ± -->
2
,
± ± -->
2
φ φ -->
,
± ± -->
2
φ φ -->
)
,
(
± ± -->
3
,
± ± -->
1
φ φ -->
2
,
± ± -->
φ φ -->
2
)
,
(
± ± -->
φ φ -->
2
,
± ± -->
1
,
± ± -->
[
3
φ φ -->
− − -->
2
]
)
,
{\displaystyle {\begin{array}{crrlc}{\Bigl (}&\pm {\bigl [}2-{\frac {1}{\varphi }}{\bigr ]},&\pm \,1,&\pm {\bigl [}2+\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,1,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm {\bigl [}3\varphi -1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,2,&\pm \,{\frac {2}{\varphi }},&\pm \,2\varphi &{\Bigr )},\\{\Bigl (}&\pm \,3,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm \,\varphi ^{2}&{\Bigr )},\\{\Bigl (}&\pm \,\varphi ^{2},&\pm \,1,&\pm {\bigl [}3\varphi -2{\bigr ]}&{\Bigr )},\end{array}}}
where
φ φ -->
=
1
+
5
2
{\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}
golden ratio .
Tridyakis icosahedron
The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
See also
References
External links
Kepler-Poinsot (nonconvex Uniform truncations Nonconvex uniform hemipolyhedra Duals of nonconvex Duals of nonconvex
![{\displaystyle {\begin{array}{crrlc}{\Bigl (}&\pm {\bigl [}2-{\frac {1}{\varphi }}{\bigr ]},&\pm \,1,&\pm {\bigl [}2+\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,1,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm {\bigl [}3\varphi -1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm \,2,&\pm \,{\frac {2}{\varphi }},&\pm \,2\varphi &{\Bigr )},\\{\Bigl (}&\pm \,3,&\pm \,{\frac {1}{\varphi ^{2}}},&\pm \,\varphi ^{2}&{\Bigr )},\\{\Bigl (}&\pm \,\varphi ^{2},&\pm \,1,&\pm {\bigl [}3\varphi -2{\bigr ]}&{\Bigr )},\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc946dd2ad1cf9490fbc40f2b13dd9958049b453)
