Honeycomb (geometry)

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For other uses, see Honeycomb (disambiguation).
cubic honeycomb

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 Euclidean ("flat") 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.

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.

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 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. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the 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., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples,[1] also called the Archimedean honeycombs. Of these, just one is regular and two are quasiregular (made from two types of regular cells):

Type Regular cubic honeycomb Quasiregular honeycombs
Cells Cubic Octahedra 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[2]

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


Truncated octahedra
Rhombic dodecahedra
Elongated dodecahedra

Sometimes, two [10] 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 [11]

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, as in the Yoshimoto Cube.
  • 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 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 regular hyperbolic honeycombs and many uniform hyperbolic honeycombs have been enumerated.

Duality of honeycombs

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

cells for vertices.
faces 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.[12]

Self-dual honeycombs

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

See also

Wikimedia Commons has media related to Tilings of 3-space.
  • List of uniform tilings
  • Regular honeycombs

References

  1. ↑ Grünbaum & Shephard, Uniform tilings of 3-space.
  2. ↑ Weisstein, Eric W., "Space-filling polyhedron", MathWorld.
  3. ↑ Uniform space-filling using triangular, square, and hexagonal prisms
  4. ↑ Uniform space-filling using only truncated octahedra
  5. ↑ Uniform space-filling using only rhombic dodecahedra
  6. ↑ John Conway (Mon, 22 Dec 2003 17:37:08 +0000 (UTC)). "Voronoi Polyhedron. geometry.puzzles". geometry.puzzles. Web link.
  7. ↑ X. Qian, D. Strahs and T. Schlick, J. Comp. Chem. 22(15) 1843–1850 (2001)
  8. ↑ Uniform space-filling using only rhombo-hexagonal dodecahedra
  9. ↑ O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Cryst. (2005) A61, 358-362
  10. ↑ Gabbrielli, Ruggero. A thirteen-sided polyhedron which fills space with its chiral copy.
  11. ↑ Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960
  12. ↑ Inchbald, G.: The Archimedean Honeycomb duals, The Mathematical Gazette 81, July 1997, p.p. 213–219.
  • Coxeter, H. S. M.: Regular Polytopes.
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. pp. 164–199. ISBN 0-486-23729-X.  Chapter 5: Polyhedra packing and space filling
  • Critchlow, K.: Order in space.
  • Pearce, P.: Structure in nature is a strategy for design.

External links

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