Icositruncated dodecadodecahedron

Icositruncated dodecadodecahedron
File:Icositruncated dodecadodecahedron.png
Type	Uniform star polyhedron
Elements	F = 44, E = 180 V = 120 (χ = −16)
Faces by sides	20{6}+12{10}+12{10/3}
Coxeter diagram	Template:CDD
Wythoff symbol
Symmetry group	Ih, [5,3], *532
Index references	U45, C57, W84
Dual polyhedron	Tridyakis icosahedron
Vertex figure	File:Icositruncated dodecadodecahedron vertfig.png 6.10.10/3
Bowers acronym	Idtid

File:Icositruncated dodecadodecahedron.stl

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.

File:Great rhombicosidodecahedron.png Truncated icosidodecahedron

File:Icositruncated dodecadodecahedron convex hull.png Convex hull

File:Icositruncated dodecadodecahedron.png Icositruncated dodecadodecahedron

Cartesian coordinates

Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of $$\begin{array}{crrlc}(& \pm [2-\frac{1}{\phi }],& \pm 1,& \pm [2+\phi ]& ),\\ (& \pm 1,& \pm \frac{1}{{\phi }^2},& \pm [3\phi -1]& ),\\ (& \pm 2,& \pm \frac{2}{\phi },& \pm 2\phi & ),\\ (& \pm 3,& \pm \frac{1}{{\phi }^2},& \pm {\phi }^2& ),\\ (& \pm {\phi }^2,& \pm 1,& \pm [3\phi -2]& ),\end{array}$$

where $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio.

Related polyhedra

Tridyakis icosahedron

Template:Uniform dual polyhedron stat table The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.

See also

• Catalan solid Duals to convex uniform polyhedra
• Uniform polyhedra
• List of uniform polyhedra

References

• Photo on page 96, Dorman Luke construction and stellation pattern on page 97.

External links

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