Great dodecahemicosahedron

In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices.[1] Its vertex figure is a crossed quadrilateral.

Great dodecahemicosahedron
TypeUniform star polyhedron
ElementsF = 22, E = 60
V = 30 (χ = −8)
Faces by sides12{5}+10{6}
Wythoff symbol5/4 5 | 3 (double covering)
Symmetry groupIh, [5,3], *532
Index referencesU65, C81, W102
Dual polyhedronGreat dodecahemicosacron
Vertex figure
5.6.5/4.6
Bowers acronymGidhei
3D model of a great dodecahemicosahedron

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

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
TypeStar polyhedron
Face—
ElementsF = 30, E = 60
V = 22 (χ = −8)
Symmetry groupIh, [5,3], *532
Index referencesDU65
dual polyhedronGreat 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.[2] 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.

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See also

See also
  • Hemi-icosahedron - The ten vertices at infinitity correspond directionally to the 10 vertices of this abstract polyhedron.

References
  1. Maeder, Roman. "65: great dodecahemicosahedron". MathConsult.
  2. (Wenninger 2003, p. 101)


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