The great dirhombicosidodecahedron is the uniform polyhedron with Maeder index 75 (Maeder 1997), Wenninger index 119 (Wenninger
1989), Coxeter index 82 (Coxeter et al. 1954), and Har'El index 80 (Har'El
1993). This polyhedron is exceptional because it cannot
be derived from Schwarz triangles and because
it is the only uniform polyhedron with more
than six polygons surrounding each polyhedron
vertex (four squares alternating with two triangles
and two pentagrams). It has pseudo-Wythoff
symbol 
and faces 
. This unique polyhedron has features in
common with both snub forms and hemipolyhedra, and its octagrammic faces pass through
the origin.
The great dirhombicosidodecahedron is implemented in the Wolfram Language as UniformPolyhedron[119],
UniformPolyhedron["GreatDirhombicosidodecahedron"],
UniformPolyhedron[
"Coxeter",
82
],
UniformPolyhedron[
"Kaleido",
80
],
UniformPolyhedron[
"Uniform",
75
],
or UniformPolyhedron[
"Wenninger",
119
].
It is also implemented in the Wolfram
Language as PolyhedronData["GreatDirhombicosidodecahedron"].
Its skeleton is the great dirhombicosidodecahedral graph, illustrated above in a couple embeddings.
Its circumradius for unit edge length is
 |
Its dual is the great dirhombicosidodecacron.
The great dirhombicosidodecahedron appears on the cover of issue 4, volume 3 of The Mathematica Journal.
