Rhombidodecadodecahedron
{{Short description|Polyhedron with 54 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|rDD}}
File:Rhombidodecadodecahedron.stl
In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices.{{Cite web|url=https://www.mathconsult.ch/static/unipoly/38.html|title=38: rhombidodecadodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}} It is given a Schläfli symbol t0,2{{mset|{{frac|5|2}},5}}, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of
: (±1/τ2, 0, ±τ2)
: (±1, ±1, ±{{radic|5}})
: (±2, ±1/τ, ±τ)
where τ = (1+{{radic|5}})/2 is the golden ratio (sometimes written φ).
Related polyhedra
It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the icosidodecadodecahedron (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron (having the square faces in common).
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= Medial deltoidal hexecontahedron=
{{Uniform polyhedra db|Uniform dual polyhedron stat table|rDD}}
File:Medial deltoidal hexecontahedron.stl
The medial deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.
See also
References
{{Reflist}}
- {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}}
External links
- {{mathworld | urlname = Rhombidodecadodecahedron| title = Rhombidodecadodecahedron}}
- {{mathworld | urlname = MedialDeltoidalHexecontahedron| title =Medial deltoidal hexecontahedron}}
- [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
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