First stellation of rhombic dodecahedron
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).
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.
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.
- George Hart, Stellations