# First stellation of rhombic dodecahedron

(Redirected from Stellated rhombic dodecahedron)
The stellation of the rhombic dodecahedron
STL model of the first stellation of the rhombic dodecahedron decomposed into 12 pyramids and 4 half-cubes
It is topologically equivalent to the disdyakis dodecahedron, a Catalan solid, which can be seen as a rhombic dodecahedron with shorter rhombic pyramids augumented to each face.

In geometry, the first stellation of the rhombic dodecahedron is a stellation of the rhombic dodecahedron. This polyhedron is sometimes called Escher's solid; it 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).

## Geometry

It has 48 faces, all triangles. It has 26 vertices in total: 6 with degree 8, 8 with degree 6, and 12 with degree 4. It has 72 edges in total, giving an Euler characteristic of 26 + 48 − 72 = +2. Its vertices are identical to those of the cuboctahedron. It can be constructed by taking a square pyramid of base length 2 and height 1, with the base centered at the origin and the apex on the Z-axis, and then rotating it so that the apex lies on each of the 6 half-axes in turn. The union of the six square pyramids gives the stellated rhombic dodecahedron.

## Space-filling

It can tessellate space in the stellated rhombic dodecahedral honeycomb. Six stellated rhombic dodecahedra meet at each vertex. This honeycomb is cell-transitive, edge-transitive and vertex-transitive.

Tesselation of space with stellated rhombic dodecahedra