Editing Truncated octahedron (section)

=== As a space-filling polyhedron ===
{{multiple image
 | image1 = Symmetric group 4; permutohedron 3D; transpositions (1-based).png
 | caption1 = Truncated octahedron as a permutahedron of order 4
 | image2 = Truncated octahedra.png
 | caption2 = Truncated octahedra tiling space
 | total_width = 400
}}
The truncated octahedron can be described as a [[permutohedron]] of order 4 or '''4-permutohedron''', meaning it can be represented with even more symmetric coordinates in four dimensions: all permutations of <math> (1, 2, 3, 4) </math> form the vertices of a truncated octahedron in the three-dimensional subspace <math> x + y + z + w = 10 </math>.{{r|jj}} Therefore, each vertex corresponds to a permutation of <math> (1, 2, 3, 4) </math> and each edge represents a single pairwise swap of two elements.{{r|crisman}} With this labeling, the swaps are of elements whose values differ by one. If, instead, the truncated octahedron is labeled by the inverse permutations, the edges correspond to swaps of elements whose positions differ by one. With this alternative labeling, the edges and vertices of the truncated octahedron form the [[Cayley graph]] of the [[symmetric group]] <math> S_4 </math>, the group of four-element permutations, as generated by swaps of consecutive positions.{{r|budden}}

The truncated octahedron can tile space. It is classified as [[plesiohedron]], meaning it can be defined as the [[Voronoi cell]] of a symmetric [[Delone set]].{{r|erdahl}} Plesiohedra, [[Translation (geometry)|translated]] without rotating, can be repeated to fill space. There are five three-dimensional primary [[parallelohedron]]s, one of which is the truncated octahedron. This polyhedron is generated from six line segments with four triples of coplanar segments, with the most symmetric form being generated from six line segments parallel to the face diagonals of a cube;{{r|alexandrov}} an example of the honeycomb is the [[ bitruncated cubic honeycomb]].{{r|tz}} More generally, every permutohedron and parallelohedron is a [[zonohedron]], a polyhedron that is [[centrally symmetric]] and can be defined by a [[Minkowski sum]].{{r|jtdd}}