The octahemioctahedron, also called the octatetrahedron, is the uniform polyhedron with Maeder index 3 (Maeder 1997), Wenninger index 68 (Wenninger 1989), Coxeter index 37 (Coxeter et al. 1954), and Har'El index 8 (Har'El 1993). It has Wythoff symbol and its faces are . It is a dodecahedron with intersecting faces, and its hull is a faceted cuboctahedron.
The octahemioctahedron is implemented in the Wolfram Language as UniformPolyhedron[68], UniformPolyhedron["SmallDitrigonalIcosidodecahedron"], UniformPolyhedron["Coxeter", 37], UniformPolyhedron["Kaleido", 8], UniformPolyhedron["Uniform", 3], or UniformPolyhedron["Wenninger", 68]. It is also implemented in the Wolfram Language as PolyhedronData["Octahemioctahedron"].
Its skeleton is the cuboctahedral graph, illustrated above in a number of embeddings.
The convex hull of the octahemioctahedron is the cuboctahedron.
For unit edge lengths, its circumradius is
Its dual polyhedron is the octahemioctacron.