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.

[2] 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.

[3] 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.

However, an alternative interpretation for the same skeletal form is that it depicts a third shape with a similar appearance, the polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.

The first stellation of the rhombic dodecahedron has 12 hexagonal faces, 36 edges, and 20 vertices, yielding an Euler characteristic of 20 − 36 + 12 = −4.

[1] Escher's solid instead has 48 triangular faces, 72 edges, and 26 vertices, yielding an Euler characteristic of 26 − 72 + 48 = 2.