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Five space-filling polyhedra, Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.

Archived from the original on 4 February 2007. All n-dimensional hypercubic honeycombs with Schläfli symbols : Cite journal requires |journal= ( help)

The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald.

The slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are.

That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.

The more regular honeycombs dualise neatly: 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 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been enumerated.įor every honeycomb there is a dual honeycomb, which may be obtained by exchanging: Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora. 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. In 3-dimensional hyperbolic space, the dihedral angle of a polyhedron depends on its size. Besides many of the uniform honeycombs, another well known example is the Weaire–Phelan structure, adopted from the structure of clathrate hydrate crystals Sometimes, two or more different polyhedra may be combined to fill space.

Other honeycombs with two or more polyhedra The trapezo-rhombic dodecahedral honeycomb.The Voronoi cells of the carbon atoms in diamond are this shape. The triakis truncated tetrahedral honeycomb.The gyrated triangular prismatic honeycomb.Other known examples of space-filling polyhedra include: Bitruncated cubic honeycomb or truncated octahedra.Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped).A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of the Platonic solids other than the cube.įive space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a space-filling polyhedron. See also: Stereohedron, Plesiohedron, and ParallelohedronĪ honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers. The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. Two are quasiregular (made from two types of regular cells):

However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Every regular honeycomb is automatically uniform. There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs.Ī honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.Ī 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of is transitive on vertices). In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. There are infinitely many honeycombs, which have only been partially classified. However, not all geometers accept such hexagons.

Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon.