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. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
Honeycombs are usually constructed in ordinary 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 uniform honeycomb in spherical space.
s 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. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. However, not all geometers accept such hexagons.

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 the cubic honeycomb being special because it is the only regular honeycomb in ordinary space. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.

Uniform 3-honeycombs

A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same. 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, the cubic honeycomb. Two are quasiregular :
TypeRegular cubic honeycombQuasiregular honeycombs
CellsCubicOctahedra and tetrahedra
Slab layer

The tetrahedral-octahedral honeycomb and gyrated tetrahedral-octahedral honeycombs are generated by 3 or 2 positions of slab layer of cells, each alternating tetrahedra and octahedra. An infinite number of unique honeycombs can be created by higher order of patterns of repeating these slab layers.

Space-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 the Platonic solids other than the cube.
Five space-filling polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra:
  1. Cubic honeycomb
  2. Hexagonal prismatic honeycomb
  3. Rhombic dodecahedral honeycomb
  4. Elongated dodecahedral honeycomb.
  5. Bitruncated cubic honeycomb or truncated octahedra

cubic honeycomb
Hexagonal prismatic honeycomb
Rhombic dodecahedra
Elongated dodecahedra
Truncated octahedra
Cube
Hexagonal prismRhombic dodecahedronElongated dodecahedronTruncated octahedron
3 edge-lengths3+1 edge-lengths4 edge-lengths4+1 edge-lengths6 edge-lengths

Other known examples of space-filling polyhedra include:
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 crystals

Weaire–Phelan structure

Non-convex 3-honeycombs

Documented examples are rare. Two classes can be distinguished:
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:
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:
Honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols, are self-dual.