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Great dodecahemidodecahedron
Great dodecahemidodecahedron.png
Type Uniform star polyhedron
Elements F = 18, E = 60
V = 30 (χ = −12)
Faces by sides 12{5/2}+6{10/3}
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 Great dodecahemidodecahedron vertfig.png
5/2.10/3.5/3.10/3
Bowers acronym Gidhid

In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. Its vertex figure is a crossed quadrilateral.

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}, besides the regular small stellated dodecahedron {5/2,5} and great stellated dodecahedron {5/2,3}.

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

Contents

Related polyhedraEdit

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

 
Great icosidodecahedron
 
Great dodecahemidodecahedron
 
Great icosihemidodecahedron
 
Icosidodecahedron (convex hull)

FillingEdit

There is some controversy on how to colour the faces of this polyhedron. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in).[1]

 
Traditional filling
 
"Neo filling"

See alsoEdit

ReferencesEdit

External linksEdit

  • Weisstein, Eric W. "Great dodecahemidodecahedron". MathWorld.
  • Uniform polyhedra and duals