Small dodecicosidodecahedron

Small dodecicosidodecahedron
Type Uniform star polyhedron
Elements F = 44, E = 120
V = 60 (χ = −16)
Faces by sides 20{3}+12{5}+12{10}
Coxeter diagram
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
Bowers acronym Saddid

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.

3D model of a small dodecicosidodecahedron
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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

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Small dodecacronic hexecontahedron
 
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
 
3D model of a small dodecacronic hexecontahedron

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

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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

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  1. ^ Maeder, Roman. "33: small dodecicosidodecahedron". MathConsult.
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