In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary flat, or Euclidean space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a honeycomb in spherical space.

cubic honeycomb

Tetrahedral-octahedral honeycomb

General characteristics

This is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell.

Note that if we interpret each brick face as a hexagon having two interior angles of 180 degrees, we can now accept the pattern as a proper tiling. However, not all geometers accept such hexagons.

Just as a plane tiling is in some respects an infinite polyhedron or apeirohedron, so a honeycomb is in some respects an infinite four-dimensional polycell/polychoron.

Classification

There are infinitely many honeycombs, which have never been fully 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 tessellation of the plane. 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.

Uniform honeycombs

A uniform honeycomb is a honeycomb in Euclidean 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e. it is vertex-transitive or isogonal). There are 28 convex examples^[1], also called the Archimedean honeycombs. Of these, just one is regular and one quasiregular:

• Regular honeycomb: Cubes.
• Quasiregular honeycomb: Octahedra and tetrahedra.

Space-filling polyhedra^[2]

A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. A cell is said to be a space-filling polyhedron. Well-known examples include:

• The regular packings^[3] of cubes, hexagonal prisms, and triangular prisms.
• The uniform packing of truncated octahedra^[4].
• The rhombic dodecahedral honeycomb. ^[5]
• The squashed (rhombic) dodecahedron honeycomb ^[6].
• The rhombo-hexagonal dodecahedron honeycomb ^[7].
• A packing of any cuboid, rhombic hexahedron or parallelepiped.

Truncated octahedra

Rhombic dodecahedra

rhombo-hexagonal dodecahedra

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.

Weaire-Phelan structure
(With two types of cells)

Non-convex 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.
• Overlapping of cells whose positive and negative densities 'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane.

A dodecahedral honeycomb in hyperbolic space

Hyperbolic honeycombs

In 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 ?/2 and 2?/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.

Several hyperbolic honeycombs are already documented.

Duality of honeycombs

For every honeycomb there is a dual honeycomb, which may be obtained by exchanging:

cells for vertices.

walls for edges.

These are just the rules for dualising four-dimensional polychora, 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^[8].

References

• Coxeter; Regular polytopes.
• Williams, R.; The geometrical foundation of natural structure.
• Critchlow, K.; Order in space.
• Pearce, P.; Structure in nature is a strategy for design.

See also

• List of uniform tilings
• Regular honeycombs

External links

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

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