Honeycomb (geometry)

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.

General characteristics

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. 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 [Grünbaum & Shepherd, Uniform tilings of 3-space.] , 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 [mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron] =

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 [ [http://www.nanomedicine.com/NMI/Figures/5.4.jpg] Uniform space-filling using triangular, square, and hexagonal prisms] of cubes, hexagonal prisms, and triangular prisms.
* The uniform packing of truncated octahedra [ [http://www.nanomedicine.com/NMI/Figures/5.5.jpg] Uniform space-filling using only truncated octahedra] .
* The rhombic dodecahedral honeycomb. [ [http://www.nanomedicine.com/NMI/Figures/5.6.jpg] Uniform space-filling using only rhombic dodecahedra]
* The squashed (rhombic) dodecahedron honeycomb [X. Qian, D. Strahs and T. Schlick, J. Comp. Chem. 22(15) 1843-1850 (2001)] .
* The rhombo-hexagonal dodecahedron honeycomb [ [http://www.nanomedicine.com/NMI/Figures/5.7.jpg] Uniform space-filling using only rhombo-hexagonal dodecahedra] .
* A packing of any cuboid, rhombic hexahedron or parallelepiped.

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.

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.

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 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.

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 [Inchbald, G.: The Archimedean Honeycomb duals, "The Mathematical Gazette" 81, July 1997, p.p. 213-219.] .

Self-dual honeycombs

Honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schlafli symbols {4,3^"n"−2,4}, are self-dual.

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".

ee also

* List of uniform tilings
* Regular honeycombs

External links

*
* [http://www.steelpillow.com/polyhedra/five_sf/five.htm Five space-filling polyhedra] , Guy Inchbald
* [http://www.steelpillow.com/polyhedra/AHD/AHD.htm The Archimedean honeycomb duals] , Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.