![]() 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. ![]()
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