A(3) Orthotope

An "orthotope" is a union of finitely many boxes whose edges are all parallel to coordinate axes.  This represents a realization of the Coxeter complex for A(3) (corresponding to the symmetric group on 4 letters) by an orthotope.  The 24 balls correspond to the elements of the group.  The 2-element cosets are represented by the 48 edges.  Notice that every vertex has degree 3, corresponding to the rank of the Coxeter complex.  The rank-2 cosets are 6 rectangles and 8 non-convex hexagons.  Combinatorially, this is identical to the "usual" realization of the A(3) Coxeter complex by a truncated octahedron (also known as a permutahedron).