First stellation of rhombic dodecahedron

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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[edit]

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[edit]

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

See also[edit]

External links[edit]

  • Weisstein, Eric W. "First stellation of rhombic dodecahedron". MathWorld.
  • George Hart, Stellations