Tetrakis cuboctahedron
Introduction 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 angles 3D model of a tetrakis cuboctahedronIn geometry, the '''tetrakis cuboctahedron''' is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.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.File:Tetrakis cuboctahedron on octahedron.png|Octahedron with edges bisected and faces divided into subtriangles of the tetrakis cuboctahedronFile:Cuboctahedron.png|CuboctahedronFile:Tetrakishexahedron.jpg|Tetrakis hexahedronFile:Octahemioctahedron.png|The nonconvex octahemioctahedron looks like a concave tetrakis cuboctahedron with inverted square pyramids meeting at the polyhedron center.
See also * Pentakis icosidodecahedron
References * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, ** Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p284)
External links *  VTML polyhedral generator Try "k4aC" (Conway polyhedron notation)