Trapezo-rhombic dodecahedron
Type	Johnson solid Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...  dual
Faces	6 rhombus Rhombus In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every... , 6 trapezoid Trapezoid In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
Edges	24
Vertices	14
Vertex configuration Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...	(2) 4.4.4 (6) 4.4.4.4 (6) 4.4.4
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	Dihedral (D3h)
Dual polyhedron 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...	Triangular orthobicupola Triangular orthobicupola In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases...
Properties	convex Convex set In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

The trapezo-rhombic dodecahedron is a convex

Convex polygon

In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...

 polyhedron

Polyhedron

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

 with 6 rhombic

Rhombus

In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...

 and 6 trapezoidal

Trapezoid

In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

 faces.

This shape could be constructed by taking a tall uniform hexagonal prism

Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...

, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts.

A space-filling tessellation (or honeycomb

Honeycomb (geometry)

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....

) can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up.

In the special case that the long sides of the trapezoids equals twice the length of the short sides, the solid now represents the 3D

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

 Voronoi cell

Voronoi diagram

In mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...

 of a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 in a Hexagonal Close Packing (HCP), next to Face-Centered-Cubic, an ideal way to stack spheres. It is therefore similar to the rhombic dodecahedron

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

, which can be represented by turning the lower half of the picture at right over an angle of 60 degrees. The rhombic dodecahedron is a Voronoi cell of the other ideal way to stack spheres.

It is the polyhedral 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 triangular orthobicupola

Triangular orthobicupola

In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases...

.

External links

• VRML
  VRML
  VRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
  
  model http://www.georgehart.com/virtual-polyhedra/vrml/twisted_rhombic_dodecahedron.wrl