In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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

John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge. Some authors do not count it as a uniform polyhedron, because some pairs of edges coincide.

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 edges, this figure can be considered to have 360 total edges, and the Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 becomes -88.

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 has 4 square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 faces passing through the center of the model.

Related polyhedron

It shares the same edge arrangement as the great dirhombicosidodecahedron

Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...

, but has a different set of triangular faces. The vertices and edges are also shared with the uniform compounds of 20 octahedra

Compound of twenty octahedra

This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...

 or 20 tetrahemihexahedra

Compound of twenty tetrahemihexahedra

This uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry .John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by...

. 180 of the edges are shared with the great snub dodecicosidodecahedron

Great snub dodecicosidodecahedron

In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...

.

Convex hull Convex hull In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....	Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...	Great dirhombicosidodecahedron Great dirhombicosidodecahedron In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...
Great disnub dirhombidodecahedron	Compound of twenty octahedra Compound of twenty octahedra This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...	Compound of twenty tetrahemihexahedra Compound of twenty tetrahemihexahedra This uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry .John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by...

Dual polyhedron

The dual

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

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

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

.

Like the visually identical [great dirhombicosidodecacron]] in Magnus Wenninger

Magnus Wenninger

Father Magnus J. Wenninger OSB is a mathematician who works on constructing polyhedron models, and wrote the first book on their construction.-Early life and education:...

's Dual Models, it is represented with intersecting infinite prisms

Prism (geometry)

In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

 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

Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

 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.

External links

• http://www.orchidpalms.com/polyhedra/uniform/skilling.htm
• http://www.georgehart.com/virtual-polyhedra/great_disnub_dirhombidodecahedron.html