Small dodecicosidodecahedron

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.

Small dodecicosidodecahedron
TypeUniform star polyhedron
ElementsF = 44, E = 120
V = 60 (χ = 16)
Faces by sides20{3}+12{5}+12{10}
Coxeter diagram
Wythoff symbol3/2 5 | 5
3 5/4 | 5
Symmetry groupIh, [5,3], *532
Index referencesU33, C42, W72
Dual polyhedronSmall dodecacronic hexecontahedron
Vertex figure
5.10.3/2.10
Bowers acronymSaddid
3D model of a small dodecicosidodecahedron

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
TypeStar polyhedron
Face
ElementsF = 60, E = 120
V = 44 (χ = 16)
Symmetry groupIh, [5,3], *532
Index referencesDU33
dual polyhedronSmall 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

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

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


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