# Great dodecahemicosahedron

Great dodecahemicosahedron
Type Uniform star polyhedron
Elements F = 22, E = 60
V = 30 (χ = −8)
Faces by sides 12{5}+10{6}
Wythoff symbol 5/4 5 | 3 (double covering)
Symmetry group Ih, [5,3], *532
Index references U65, C81, W102
Dual polyhedron Great dodecahemicosacron
Vertex figure
5.6.5/4.6
Bowers acronym Gidhei

In geometry, the great dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U65. Its vertex figure is a crossed quadrilateral.

It is a hemipolyhedron with ten hexagonal faces passing through the model center.

## Related polyhedra

Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the dodecadodecahedron (having the pentagonal faces in common), and with the small dodecahemicosahedron (having the hexagonal faces in common).

 Dodecadodecahedron Small dodecahemicosahedron Great dodecahemicosahedron Icosidodecahedron (convex hull)

### Great dodecahemicosacron

Great dodecahemicosacron
Type Star polyhedron
Face
Elements F = 30, E = 60
V = 22 (χ = −8)
Symmetry group Ih, [5,3], *532
Index references DU65
dual polyhedron Great dodecahemicosahedron

The great dodecahemicosacron is the dual of the great dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemicosacron.

Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[1] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.

The great dodecahemicosahedron can be seen as having ten vertices at infinity.