First stellation of the rhombic dodecahedron

In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces. Escher's solid can tessellate space to form the stellated rhombic dodecahedral honeycomb. The first stellation of the rhombic dodecahedron has 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron, meaning that each of its faces lies in the same plane as one of the rhombus faces of the rhombic dodecahedron, with each face containing the rhombus in the same plane, and that it has the same symmetries as the rhombic dodecahedron. It is the first stellation, meaning that no other self-intersecting polyhedron with the same face planes and the same symmetries has smaller faces. Extending the faces outwards even farther in the same planes leads to two more stellations, if the faces are required to be simple polygons. For polyhedra formed only using faces in the same 12 planes and with the same symmetries, but with the faces allowed to become non-simple or with multiple faces in a single plane, additional possibilities arise. In particular, removing the inner rhombus from each hexagonal face of the stellation leaves four triangles, and the resulting system of 48 triangles forms a different non-convex polyhedron without self-intersections that forms the boundary of a solid shape, sometimes called Escher's solid. This shape appears in M. C. Escher's works Waterfall and in a study for Stars (although Stars itself features a different shape, the compound of three octahedra). As the stellation and the solid have the same visual appearance, it is not possible to determine which of the two Escher intended to depict in Waterfall.