Editing Cuboctahedron (section)

=== Radial equilateral symmetry ===
In a cuboctahedron, the long radius (center to vertex) is the same as the edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths.{{Sfn|Coxeter|1973|p=69|loc=§4.7 Other honeycombs}} Its center is like the apical vertex of a canonical pyramid: one edge length away from ''all'' the other vertices. (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform [[polytopes]], including the two-dimensional [[hexagon]], the three-dimensional cuboctahedron, and the four-dimensional [[24-cell]] and [[tesseract|8-cell (tesseract)]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I (ii): column ''<sub>0</sub>R/l''}} ''Radially equilateral'' polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. 

Each of these radially equilateral polytopes also occurs as cells of a characteristic space-filling [[tessellation]]: the tiling of regular hexagons, the [[rectified cubic honeycomb]] (of alternating cuboctahedra and octahedra), the [[24-cell honeycomb]] and the [[tesseractic honeycomb]], respectively.{{Sfn|Coxeter|1973|p=296|loc=Table II: Regular Honeycombs}} Each tessellation has a [[dual tessellation]]; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular [[sphere-packing]] in two, three and four dimensions uses the cell centers of one of these tessellations as sphere centers.

Because it is radially equilateral, the cuboctahedron's center is one edge length distant from the 12 vertices.