<P> Octahedra and tetrahedra can be alternated to form a vertex, edge, and face - uniform tessellation of space, called the octet truss by Buckminster Fuller . This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs . Another is a tessellation of octahedra and cuboctahedra . </P> <P> The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex . Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces . </P> <P> Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid . Truncation of two opposite vertices results in a square bifrustum . </P> <P> The octahedron is 4 - connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices . It is one of only four 4 - connected simplicial well - covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size . The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces . </P>

3d shape with 12 edges and 6 faces