Small dodecicosidodecahedron
Small dodecicosidodecahedron | |
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Type | Uniform star polyhedron |
Elements | F = 44, E = 120 V = 60 (χ = −16) |
Faces by sides | 20{3}+12{5}+12{10} |
Wythoff symbol | 3/2 5 | 5 3 5/4 | 5 |
Symmetry group | Ih, [5,3], *532 |
Index references | U33, C42, W72 |
Dual polyhedron | Small dodecacronic hexecontahedron |
Vertex figure | ![]() 5.10.3/2.10 |
Saddid |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Small_dodecicosidodecahedron.stl/220px-Small_dodecicosidodecahedron.stl.png)
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices.[1] Its vertex figure is a crossed quadrilateral.
Related polyhedra[]
It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common).
![]() Rhombicosidodecahedron |
![]() Small dodecicosidodecahedron |
![]() Small rhombidodecahedron |
![]() Small stellated truncated dodecahedron |
![]() Compound of six pentagrammic prisms |
![]() Compound of twelve pentagrammic prisms |
Dual[]
Small dodecacronic hexecontahedron | |
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![]() | |
Type | Star polyhedron |
Face | ![]() |
Elements | F = 60, E = 120 V = 44 (χ = −16) |
Symmetry group | Ih, [5,3], *532 |
Index references | DU33 |
dual polyhedron | Small dodecicosidodecahedron |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Small_dodecacronic_hexecontahedron.stl/220px-Small_dodecacronic_hexecontahedron.stl.png)
The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.
Proportions[]
Faces have two angles of , one of and one of . Its dihedral angles equal . The ratio between the lengths of the long and short edges is .
References[]
- ^ Maeder, Roman. "33: small dodecicosidodecahedron". MathConsult.
{{cite web}}
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- Coxeter, H. S. M. (May 13, 1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246 (916): 401–450. doi:10.1098/rsta.1954.0003.
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208
External links[]
- Weisstein, Eric W. "Uniform Polyhedron". MathWorld.
- Weisstein, Eric W. "Small dodecicosidodecahedron". MathWorld.
- Weisstein, Eric W. "Small dodecacronic hexecontahedron". MathWorld.
- Uniform polyhedra
- Polyhedron stubs