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 C4v symmetry, order 8. Like the rhombic dodecahedron, the long diagonal of each rhombic face is times the length of the short diagonal, so that the acute angles on each face measure arccos(), 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 C4v 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