Great disnub dirhombidodecahedron

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Great disnub dirhombidodecahedron
Great disnub dirhombidodecahedron.png
Type Uniform star polyhedron
Elements F = 204, E = 240
V = 60 (χ = 24)
Faces by sides 120{3}+60{4}+24{5/2}
Wythoff symbol | (3/2) 5/3 (3) 5/2
Symmetry group Ih, [5,3], *532
Index references U-, C-, W-
Dual polyhedron Great disnub dirhombidodecacron
Vertex figure Great disnub dirhombidodecahedron vertfig.png
(5/ 5/3.4.3/2.3/2.3/2.4)/2
Bowers acronym Gidisdrid

In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.

It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms with only two faces meeting at each edge. John Skilling discovered one more, the great disnub dirhombidodecahedron, by relaxing the condition that only two faces may meet at an edge. Due to it having four faces at some edges, it is not always counted as a uniform polyhedron.

It has 120 edges with 2 faces and 120 edges with 4 faces. If the 4-face edges are counted twice, as two topologically disjoint but coincident edges (one with 2 triangles meeting, and the other with a square meeting a pentagram), this figure can be considered to have 360 total edges, and the Euler characteristic becomes −96.

The vertex figure has 4 square faces passing through the center of the model.

It may be constructed as the exclusive or (blend) of the great dirhombicosidodecahedron and compound of twenty octahedra.

Related polyhedra[edit]

It shares the same edge arrangement as the great dirhombicosidodecahedron, but has a different set of triangular faces. The vertices and edges are also shared with the uniform compounds of 20 octahedra or 20 tetrahemihexahedra. 180 of the edges are shared with the great snub dodecicosidodecahedron.

Convex hull
Great snub dodecicosidodecahedron.png
Great snub dodecicosidodecahedron
Great dirhombicosidodecahedron.png
Great dirhombicosidodecahedron
Great disnub dirhombidodecahedron.png
Great disnub dirhombidodecahedron
UC14-20 octahedra.png
Compound of twenty octahedra
UC19-20 tetrahemihexahedron.png
Compound of twenty tetrahemihexahedra

Dual polyhedron[edit]

The great disnub dirhombidodecacron

The dual of the great disnub dirhombidodecahedron is called a great disnub dirhombidodecacron. It is a nonconvex infinite isohedral polyhedron.

Like the visually identical great dirhombicosidodecacron in Magnus Wenninger's Dual Models, it is represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation polyhedra, called stellation to infinity. However, he also acknowledged that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.


There is some controversy on how to colour the faces of this polyhedron. Although the common way to fill in a polygon is to colour its whole interior, this can result in some filled regions hanging as membranes over empty space. That problem is avoided by another filling rule, in which 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). In addition, overlapping regions of coplanar faces can cancel each other out.[1]

Great disnub dirhombidodecahedron.png
Traditional filling
Great disnub dirhombidodecahedron 2.png
"Neo filling"

See also[edit]


External links[edit]