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 C4v 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 edit
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 C4v symmetry.[1]
Space-filling edit
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 edit
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 edit
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 D4h 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 edit
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