In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula, a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's De Divina Proportione, 1509.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound.
There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron.
In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
This uniform polyhedron compound is a composition of 2 great dodecahedra, in the same arrangement as in the compound of 2 icosahedra.
This uniform polyhedron compound is a composition of 2 small stellated dodecahedra, in the same arrangement as in the compound of 2 icosahedra.
This uniform polyhedron compound is a composition of 5 great dodecahedra, in the same arrangement as in the compound of 5 icosahedra.
This uniform polyhedron compound is a composition of 5 small stellated dodecahedra, in the same arrangement as in the compound of 5 icosahedra.
This uniform polyhedron compound is a composition of two truncated tetrahedra, formed by truncating each of the tetrahedra in the stellated octahedron. It is related to the cantic cube construction of the truncated tetrahedron, as , which is one of the two dual positions represented in this compound.
The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron.
