(I can only assume 'hexagonal close packing' was named by someone with only the most superficial appreciation of hexagons, and who presumably didn't think very highly of them.) A moment's reflection on these shapes and their obvious contrast with the cuboctahedron and its extended family of tetrahedral-octahedral symmetries should make it somewhat self-evident why I consider FCC, and not HCP, to be the true successor to hexagonal symmetry in three dimensions. Is there a cuboctahedron-like polyhedron that represents their packing arrangement?' And of course there is: the somewhat absurd triangular orthobicupola, and its dual, the quite possibly more absurd trapezo-rhombi dodecahedron. That is, if spheres in a face-centered cubic packing were expanded until they filled all available space, or the vertices of a tetrahedral-octahedral tessellation were themselves turned into spheres and so expanded, they would become rhombic dodecahedra.Īt this point you may be wondering 'But wait, what about hexagonal close packed spheres? They, too, have twelve neighboring elements. The dual of the cuboctahedron is the delightfully quirky rhombic dodecahedron, which can, in fact, tessellate space by itself, in the manner of FCC-packed spheres.

This, again, can be expanded outward indefinitely, though the particular unit of spheres in this configuration will not fill such a packing arrangement by itself. Thus, a sphere placed at the center of a cuboctahedron with a diameter equal to the edge length of the latter can be surrounded by twelve neighboring spheres of equal size, each arranged at one of the vertices of the cuboctahedron.

The face-centered cubic arrangement of close-packed spheres (which, as we've described elsewhere, is arguably more hexagonal than the so-called 'hexagonal close packing' arrangement) is essentially identical to this tetrahedral-octahedral tessellation. Despite this, however-and unlike the truncated octahedron-the cuboctahedron cannot by itself tessellate space when only cuboctahedral units are fit together, one will always be left with either space left over, or overlapping tetrahedral/half-octahedral cells. Pairing the half-octahedra with their remaining halves, and continuing the same arrangement of edge-sharing tetrahedra around these octahedral 'cavities,' this pattern of tetrahedral-octahedral tessellation can be continued outward indefinitely, with identical cuboctahedral symmetry radiating from every vertex in the resulting tessellation. Likewise, as hexagons can be subdivided into equilateral triangles, so the cuboctahedron can be seen to consist of six half-octahedra and eight tetrahedra clustered around the center, forming the vertex figure of the tetrahedral-octahedral tessellation. As with the hexagon in 2-space, the cuboctahedron is the only uniform convex polyhedron with this quality. (Likewise, of course, the cuboctahedron can simply be seen as a sort of 'expanded' tetrahedron, with four of its eight triangular faces representing the original four faces of an inner tetrahedron, the remaining four triangles representing its four vertices, and the square faces representing its edges.) Tetrahedral symmetry being the simplest type of polyhedral symmetry, and the only one suited to this sort of fitting together of hexagonal planes, the cuboctahedron represents a unique extension of and analogue to hexagonal symmetry in three dimensions.įrom the idea of the cuboctahedron as four intersecting hexagonal planes, it follows that, like the hexagon, the radius between the center of the cuboctahedron and its vertices is equal to the length of its edges. That is, if one took a tetrahedron, replaced its four faces with hexagons (as for instance with a truncated tetrahedron), and collapsed all four hexagonal sides so that they all shared a common center, the vertices of the hexagons would describe a cuboctahedron, with each vertex shared between two intersecting hexagons, collapsing the original 24 vertices of the four hexagons into the 12 vertices of the cuboctahedron. Though it has no hexagonal faces, the cuboctahedron can be seen to consist of four hexagonal rings or planes arranged in the manner of tetrahedral symmetry. Along with the truncated octahedron, it can be considered, in some sense, a three-dimensional analogue to the hexagon. Here we see the illustrious cuboctahedron, or vector equilibrium.

Ipicture of a cuboctahedron flattened