Classification
There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or ''slabs'' of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with theUniform 3-honeycombs
A 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 sometries of 3-space that preserve the tilingis '' transitive on vertices''). There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs. A 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. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, theSpace-filling polyhedra
A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a '' space-filling polyhedron''. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, ruling out any of theOther 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 crystalsNon-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 withSee also
* List of uniform tilings * Regular honeycombs * Infinite skew polyhedron * PlesiohedronReferences
Further reading
* Coxeter, H. S. M.: '' Regular Polytopes''. * Chapter 5: Polyhedra packing and space filling * Critchlow, K.: ''Order in space''. * Pearce, P.: ''Structure in nature is a strategy for design''. * Goldberg, Michael ''Three Infinite Families of Tetrahedral Space-Fillers'' Journal of Combinatorial Theory A, 16, pp. 348–354, 1974. * * Goldberg, Michael ''The Space-filling Pentahedra II'', Journal of Combinatorial Theory 17 (1974), 375–378. * * * Goldberg, Michael ''Convex Polyhedral Space-Fillers of More than Twelve Faces.'' Geom. Dedicata 8, 491-500, 1979. * * *External links
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