## The truncated octahedron

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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.

## The cuboctahedron

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Here we see the illustrious cuboctahedron, or vector equilibrium. Along with the truncated octahedron, it can be considered, in some sense, a three-dimensional analogue to the hexagon. 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. 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. (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.

## Hexagonal projection of the platonic solids

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In this diagram we see that four of the five platonic solids can be projected in two dimensions as hexagonally-symmetric figures. The hexahedron, octahedron, and icosahedron can all be orthographically projected as regular hexagons, and the dodecahedron can be so projected as a somewhat lopsided yet equal-angled hexagon (not shown), or as a hexagonally-symmetrical dodecagon (shown here). The tetrahedron—or "freak" polyhedron if you will—can of course be projected as an equilateral triangle, which although not truly hexagonal is still of the same general angular family.

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