View source for Great disnub dirhombidodecahedron

{{Short description|Polyhedron with 204 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Skilling}}
[[File:Great disnub dirhombidodecahedron.stl|thumb|3D model of a great disnub dirhombidodecahedron]]
In [[Geometry|geometry]], the '''great disnub dirhombidodecahedron''', also called ''Skilling's figure'', is a degenerate [[Uniform star polyhedron|uniform star polyhedron]].

It was proven in 1970 that there are only 75 [[Uniform polyhedron|uniform polyhedra]] other than the infinite families of [[Prism (geometry)|prisms]] and [[Antiprism|antiprism]]s.  John Skilling discovered another degenerate example, the great disnub dirhombidodecahedron, by relaxing the condition that edges must be single. More precisely, he allowed any even number of faces to meet at each edge, as long as the set of faces couldn't be separated into two connected sets (Skilling, 1975). Due to its geometric realization  having some double edges where 4 faces meet, it is considered a [[Degeneracy (mathematics)|degenerate]] uniform polyhedron but not strictly a uniform polyhedron.

The number of edges is ambiguous, because the underlying abstract polyhedron has 360 edges, but 120 pairs of these have the same image in the geometric realization, so that the geometric realization has 120 single edges and 120 double edges where 4 faces meet, for a total of 240 edges. The Euler characteristic of the abstract polyhedron is −96. If the pairs of coinciding edges in the geometric realization are considered to be single edges, then it has only 240 edges and Euler characteristic 24.

The [[Vertex figure|vertex figure]] has 4 square faces passing through the center of the model.

It may be constructed as the [[Exclusive or|exclusive or]] (blend) of the [[Great dirhombicosidodecahedron|great dirhombicosidodecahedron]] and [[Compound of twenty octahedra|compound of twenty octahedra]].

== Related polyhedra ==

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

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[File:Nonuniform2-rhombicosidodecahedron.png|100px]]<BR>[[Convex hull]]
| [[Image:Great snub dodecicosidodecahedron.png|100px]]<BR>[[Great snub dodecicosidodecahedron]]
| [[Image:Great dirhombicosidodecahedron.png|100px]]<BR>[[Great dirhombicosidodecahedron]]
|-
| [[Image:Great disnub dirhombidodecahedron.png|100px]]<BR>'''Great disnub dirhombidodecahedron'''
| [[Image:UC14-20 octahedra.png|100px]]<BR>[[Compound of twenty octahedra]]
| [[Image:UC19-20 tetrahemihexahedron.png|100px]]<BR>[[Compound of twenty tetrahemihexahedra]]
|}

=== Dual polyhedron ===
[[File:Great_dirhombicosidodecacron.png|thumb|The great disnub dirhombidodecacron]]

The [[Dual polyhedron|dual]] of the '''great disnub dirhombidodecahedron''' is called the ''great disnub dirhombidodecacron''. It is a nonconvex infinite [[Isohedral figure|isohedral]] [[Polyhedron|polyhedron]].

Like the visually identical great dirhombicosidodecacron in Magnus Wenninger's ''Dual Models'', it is represented with intersecting infinite [[Prism (geometry)|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|stellation]] polyhedra, called [[Stellation#Stellation to infinity|stellation to infinity]]. However, he also acknowledged that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
== Gallery ==

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
|align=center|[[Image:Great disnub dirhombidodecahedron.png|150px]]<BR>Traditional filling
|align=center|[[Image:Great disnub dirhombidodecahedron 2.png|150px]]<BR>Modulo-2 filling
|}

== See also ==
* [[List of uniform polyhedra]]

== References ==
* {{citation|first=John|last=Skilling|doi=10.1098/rsta.1975.0022|title=The complete set of uniform polyhedra|journal=Philosophical Transactions of the Royal Society A |volume=278|issue=1278|pages=111–135|year=1975}}.
* {{mathworld | urlname = GreatDirhombicosidodecahedron| title =Great dirhombicosidodecahedron}}
* http://www.software3d.com/MillersMonster.php
{{reflist}}

== External links ==
* http://www.orchidpalms.com/polyhedra/uniform/skilling.htm
* http://www.georgehart.com/virtual-polyhedra/great_disnub_dirhombidodecahedron.html

[[Category:Uniform polyhedra]]

{{Sourceattribution|Great disnub dirhombidodecahedron}}