The truncated octahedron is the fourth order permutohedron, and forms constituent cells in higher-order permutohedra. Its vertices represent every combination of the coordinates 1, 2, 3, and 4 in 4-space, in same way that a hexagon can be embedded in 3-space with vertices at every permution of 1, 2, and 3 (i.e., bisecting a cube spanning coordinates 1, 1, 1 to 3, 3, 3). It is also, along with the lowly cube, one of only two space-filling uniform polyhedra, and is one of only five regular-faced convex polyhedra able to do so—along with the aforeementioned cube, the triangular and hexagonal prisms, and the gyrobifastigium, whatever the crap that is.
We seem to have fallen a bit behind in terms of keeping this site updated with RELEVANT MATHEMATICAL INFORMATION about hexagons. This is a deficit we're looking to correct as soon as possible. First up, let us consider the permutohedra. We at HEXNET.ORG have been meaning to write something about permutohedra for a couple of years now, but have never really found a good opportunity to so. WE WILL NOT EXPEND GREAT EFFORT DOING SO NOW. It will suffice to merely describe the concept in conjunction with some helpful imagery, which will hopefully serve as a useful foundation for further geometrical observations and investigations in the near future.