out any of the Platonic solids other than the cube. Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra: Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped) Hexagonal prismatic honeycomb Rhombic dodecahedral honeycomb Elongated dodecahedral honeycomb Bitruncated cubic honeycomb or truncated octahedra Other known examples of space-filling polyhedra include: The triangular prismatic honeycomb The gyrated triangular prismatic honeycomb The triakis truncated tetrahedral honeycomb. The Voronoi cells of the carbon atoms in diamond are this shape. The trapezo-rhombic dodecahedral honeycomb Isohedral tilings Other honeycombs with two or more polyhedra Sometimes, two or more different polyhedra may be combined to fill space. Besides many of the uniform honeycombs, another well known example is the Weaire–Phelan structure, adopted from the structure of clathrate hydrate crystals Non-convex 3-honeycombs Documented examples are rare. Two classes can be distinguished: Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities 'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane. Hyperbolic honeycombs In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora. The 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated. Duality of 3-honeycombs For every honeycomb there is a dual honeycomb, which may be obtained by exchanging: cells for vertices. faces for edges. These are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems. The more regular honeycombs dualise neatly: The cubic honeycomb is self-dual. That of octahedra and tetrahedra is dual to that of rhombic dodecahedra. The slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are. The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald. Self-dual honeycombs Honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols