A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell.
Each "layer" is a hexagonal tiling, or a rhombille tiling, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up.

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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 Voronoi cell of a sphere in a Hexagonal Close Packing (HCP), next to Face-Centered-Cubic an optimal way to stack spheres in a lattice.
It is therefore similar to the rhombic dodecahedron, 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 optimal way to stack spheres.

As the Voronoi cell of a regular space pattern, it is a plesiohedron.

It is the polyhedral dual of the triangular orthobicupola.
A space-filling tessellation, the trapezo-rhombic dodecahedral honeycomb, can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, or a rhombille 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 Voronoi cell of a sphere in a Hexagonal Close Packing (HCP), next to Face-Centered-Cubic an optimal way to stack spheres in a lattice. It is therefore similar to the rhombic dodecahedron, 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 optimal way to stack spheres.
As the Voronoi cell of a regular space pattern, it is a plesiohedron.
It is the polyhedral dual of the triangular orthobicupola.
Trapezo-rhombic dodecahedron