A cat toy in the shape of a tetrakis cuboctahedron projected onto a sphere Tetrakis cuboctahedrons usefully represent carbon atoms in a 3D ball-and-stick model of a diamond lattice as the normals to alternate yellow-shaded faces in the top image correspond exactly to the tetrahedral bond angles3D model of a tetrakis cuboctahedron
Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12.
Related polyhedra
It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral.
This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices.
Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedron