Diminished rhombic dodecahedron

In geometry, a diminished rhombic dodecahedron is a rhombic dodecahedron with one or more vertices removed. This article describes diminishing one 4-valence vertex. This diminishment creates one new square face while 4 rhombic faces are reduced to triangles. It has 13 vertices, 24 edges, and 13 faces. It has C_4v symmetry, order 8.

Diminished rhombic dodecahedron
Faces	13: 8 rhombi 4 triangles 1 square
Edges	24
Vertices	13
Symmetry group	C4v, order 8
Dual polyhedron	Self-dual
Properties	convex
Net

Like the rhombic dodecahedron, the long diagonal of each rhombic face is √2 times the length of the short diagonal, so that the acute angles on each face measure arccos(⁠1/3⁠), or approximately 70.53°.

Self-dual

Like the dihedral symmetry pyramids, and elongated pyramids, it is self-dual, with the dual geometry inverted across the axis of symmetry. It is one of three self-dual tridecahedra with C_4v symmetry.

Space-filling

This polyhedron along with the cube is space-filling, like the rhombic dodecahedral honeycomb. Six diminished points come together to form cubic holes.

Cartesian coordinate

It has 13 of 14 Cartesian coordinates of the rhombic dodecahedron are:

8: (±1, ±1, ±1)

1: (2, 0, 0)

2: (0, ±2, 0)

2: (0, 0, ±2)

Augmented cuboctahedron

The same topological polyhedron with different proportions can be constructed as an augmented cuboctahedron, with a square face augmented by a square pyramid. This construction requires merging of neighboring coparallel triangular faces into new 60° rhombic faces. This can also be seen as an asymmetric stellation of a cuboctahedron. It retains 5 square faces, and 4 equilateral triangle faces of the cuboctahedron.

Augmented cuboctahedron

Net

The 13 Cartesian coordinate can be positioned as:

1: ( 2, 0, 0)

4: (±1, ±1, 0)

4: (±1, 0, ±1)

4: ( 0, ±1, ±1)

Augmenting two opposite squares will create a dihedral rhombic dodecahedron, doubling the symmetry to D_4h symmetry, order 16. Augmenting pyramids on all six square faces, with merged faces will produce a regular octahedron, restoring full octahedral symmetry, order 48.

References

• Symmetric Canonical Self-Dual Tridecahedra #7