There are four fully supported stellations of the rhombic dodecahedron including as
usual the original solid in the count (Wells 1991; Webb). The three nontrivial ones
(excluding the base solid) are illustrated above.
Applying Miller's rules gives one additional stellation,
bringing the total to 5, all of which are reflexible (Webb).
The original rhombic dodecahedron, its facial planes, and the intersections of those planes with the facial plane of the "top" face are illustrated above.
The first stellation is sometimes simply known as the stellated rhombic dodecahedron. It consists of 12 intersecting bowtie-shaped concave hexagons. Its outer boundary
(concave hull) corresponds to Escher's solid and
can be constructed by drawing diagonals across the square faces of a cuboctahedron
and connecting centers of these diagonals with the vertices of neighboring squares.
The outer edges of the third stellation correspond with those of the truncated
octahedron.
These stellations are implemented in the Wolfram Language as PolyhedronData["RhombicDodecahedronStellation",
n] for , 2, 3.
, 2, 3.
