HONEYCOMB (GEOMETRY)

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.
Space-filling tessellations of hyperbolic space are also called ''honeycombs''.

cubic honeycomb

Tetrahedral-octahedral honeycomb

Contents

General characteristics

Classification

Uniform honeycombs

Space-filling polyhedra

Non-convex honeycombs

Hyperbolic honeycombs

Duality of honeycombs

References

See also

External links

General characteristics

It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles:
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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.
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, 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

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 packing of cubes.

★ The uniform packing of truncated octahedra.

★ The rhombic dodecahedral honeycomb.

★ The rhombo-hexagonal dodecahedron honeycomb.

★ A packing of any cuboid, rhombic hexahedron or parallelepiped.

Truncated octahedra

Rhombic dodecahedra

rhombo-hexagonal dodecahedra

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

Hyperbolic space behaves rather differently from ordinary 'Euclidean' space, with cells fitting together according to rather different rules. Several hyperbolic honeycombs already have Wiki pages - TBA. See for example the List of regular polytopes.

dodecahedral honeycomb
in hyperbolic space.

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 (1997).

References

★ Grünbaum & Shepherd, Uniform tilings of 3-space.

★ 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''.

★ Inchbald, G.: The Archimedean Honeycomb duals, ''The Mathematical Gazette'' '81', July 1997, p.p. 213-219.

See also

★ List of uniform tilings

★ Regular honeycombs

External links

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★ [1] Uniform space-filling using only rhombo-hexagonal dodecahedra

★ [2] Uniform space-filling using only rhombic dodecahedra

★ [3] Uniform space-filling using only truncated octahedra

★ [4] Uniform space-filling using triangular, square, and hexagonal prisms

★ Five space-filling polyhedra, Guy Inchbald

★ The Archimedean honeycomb duals, Guy Inchbald, The Mathematical Gazette '80', November 1996, p.p. 466-475.