Honeycomb (geometry)

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

Honeycomb is also sometimes used for higher dimensional tessellations as well. For clarity, George Olshevsky advocates limiting the term honeycomb to 3-space tessellations and expanding a systematic terminology for higher dimensions: tetracomb as tessellations of 4-space, and pentacomb as tessellations of 5-space, and so on.

Honeycombs may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs.

cubic honeycomb

Tetrahedral-octahedral honeycomb

Contents 1 General characteristics 2 Classification 2.1 Uniform honeycombs 2.2 Space-filling polyhedra[2] 2.3 Non-convex honeycombs 2.4 Hyperbolic honeycombs 3 Duality of honeycombs 4 References 5 See also 6 External links

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] Hyperbolic honeycombs

Hyperbolic space behaves rather differently from ordinary Euclidean space, with cells fitting together according to rather different rules. Several hyperbolic honeycombs are already documented.

[edit] 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].

[edit] 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.

[edit] See also

• List of uniform tilings
• Regular honeycombs

[edit] External links

• Olshevsky, George, Honeycomb at Glossary for Hyperspace.
• Five space-filling polyhedra, Guy Inchbald
• The Archimedean honeycomb duals, Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.