Small rhombidodecahedron

In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U₃₉. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices.^[1] Its vertex figure is a crossed quadrilateral.

Small rhombidodecahedron
Type	Uniform star polyhedron
Elements	F = 42, E = 120 V = 60 (χ = −18)
Faces by sides	30{4}+12{10}
Coxeter diagram	(with extra double-covered triangles) (with extra double-covered pentagons)
Wythoff symbol	2 5 (3/2 5/2) |
Symmetry group	Ih, [5,3], *532
Index references	U39, C46, W74
Dual polyhedron	Small rhombidodecacron
Vertex figure	4.10.4/3.10/9
Bowers acronym	Sird

3D model of a small rhombidodecahedron

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 square faces in common), and with the small dodecicosidodecahedron (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

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Small rhombidodecacron

Small rhombidodecacron
Type	Star polyhedron
Face
Elements	F = 60, E = 120 V = 42 (χ = −18)
Symmetry group	Ih, [5,3], *532
Index references	DU39
dual polyhedron	Small rhombidodecahedron

3D model of a small rhombidodecacron

The small rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.