The trapezo-rhombic dodecahedron can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles.
It also has d₃ₕ symmetry and is space-filling.
It has 21 edges and 11 vertices.
With square faces it can be seen as a cube split across the 3-fold axis, separated with the two halves rotated 180 degrees, and filling the gaps with triangles.
When used as a space-filler, connecting dodecahedra on their triangles leaves two cubical step surfaces on the top and bottom which can connect with complementary steps.
The trapezo-rhombic dodecahedron can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles. It also has d₃ₕ symmetry and is space-filling. It has 21 edges and 11 vertices. With square faces it can be seen as a cube split across the 3-fold axis, separated with the two halves rotated 180 degrees, and filling the gaps with triangles. When used as a space-filler, connecting dodecahedra on their triangles leaves two cubical step surfaces on the top and bottom which can connect with complementary steps.
Trapezo-rhombic dodecahedron