Great dodecahemidodecahedron

In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U₇₀. It has 18 faces (12 pentagrams and 6 decagrams), 60 edges, and 30 vertices.^[1] Its vertex figure is a crossed quadrilateral.

Great dodecahemidodecahedron
Type	Uniform star polyhedron
Elements	F = 18, E = 60 V = 30 (χ = −12)
Faces by sides	12{5/2}+6{10/3}
Coxeter diagram
Wythoff symbol	5/3 5/2 | 5/3 (double covering)
Symmetry group	Ih, [5,3], *532
Index references	U70, C86, W107
Dual polyhedron	Great dodecahemidodecacron
Vertex figure	5/2.10/3.5/3.10/3
Bowers acronym	Gidhid

3D model of a great dodecahemidodecahedron

Aside from the regular small stellated dodecahedron {⁵/₂,5} and great stellated dodecahedron {⁵/₂,3}, it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons {⁵/₂} and {¹⁰/₃}.

It is a hemipolyhedron with 6 decagrammic faces passing through the model center.

Related polyhedra

Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the pentagrammic faces in common) and the great icosihemidodecahedron (having the decagrammic faces in common).

Great icosidodecahedron	Great dodecahemidodecahedron	Great icosihemidodecahedron
Icosidodecahedron (convex hull)

Gallery

Traditional filling

Modulo-2 filling

See also

• List of uniform polyhedra