In geometry, a honeycomb is a space filling or close packing of polyhedral or higherdimensional 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 nhoneycomb for a honeycomb of ndimensional space.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in nonEuclidean 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. 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 (Euclidean) space. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.
Uniform 3honeycombs
A 3dimensional uniform honeycomb is a honeycomb in 3space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3space,[1] 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 3space, the cubic honeycomb. Two are quasiregular (made from two types of regular cells):
Type  Regular cubic honeycomb  Quasiregular honeycombs 

Cells  Cubic  Octahedra and tetrahedra 
Slab layer 
The tetrahedraloctahedral honeycomb and gyrated tetrahedraloctahedral 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.
Spacefilling polyhedra
See also: Stereohedron, Plesiohedron, and Parallelohedron
A honeycomb having all cells identical within its symmetries is said to be celltransitive or isochoric. In the 3dimensional euclidean space, a cell of such a honeycomb is said to be a spacefilling polyhedron.[2] A necessary condition for a polyhedron to be a spacefilling polyhedron is that its Dehn invariant must be zero,[3][4] ruling out any of the Platonic solids other than the cube.
Five spacefilling polyhedra can tessellate 3dimensional euclidean space using translations only. They are called parallelohedra:
Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped)
Hexagonal prismatic honeycomb[5]
Rhombic dodecahedral honeycomb
Elongated dodecahedral honeycomb.[6]
Bitruncated cubic honeycomb or truncated octahedra[7]
cubic honeycomb 
Hexagonal prismatic honeycomb 
Rhombic dodecahedra 
Elongated dodecahedra 
Truncated octahedra 
Cube (parallelepiped) 
Hexagonal prism  Rhombic dodecahedron  Elongated dodecahedron  Truncated octahedron 

3 edgelengths  3+1 edgelengths  4 edgelengths  4+1 edgelengths  6 edgelengths 
Other known examples of spacefilling polyhedra include:
The Triangular prismatic honeycomb.
The gyrated triangular prismatic honeycomb
The triakis truncated tetrahedral honeycomb. The Voronoi cells of the carbon atoms in diamond are this shape.[8]
The trapezorhombic dodecahedral honeycomb[9]
Isohedral tilings.[10]
Other honeycombs with two or more polyhedra
Sometimes, two [11] 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 [12]
1214hedral honeycomb.png
Weaire–Phelan structure (With two types of cells)
Nonconvex 3honeycombs
Documented examples are rare. Two classes can be distinguished:
Nonconvex 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.
Hyperbolic honeycombs
In 3dimensional 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.
{5,3,4} 
{4,3,5} 
{3,5,3} 
{5,3,5} 
11 paracompact regular honeycombs  

{6,3,3} 
{6,3,4} 
{6,3,5} 
{6,3,6} 
{4,4,3} 
{4,4,4} 

{3,3,6} 
{4,3,6} 
{5,3,6} 
{3,6,3} 
{3,4,4} 
Duality of 3honeycombs
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 fourdimensional 4polytopes, 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 selfdual.
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 celltransitive and have been described by Inchbald.[13]
Selfdual honeycombs
Honeycombs can also be selfdual. All ndimensional hypercubic honeycombs with Schläfli symbols {4,3n−2,4}, are selfdual.
See also
List of uniform tilings
Regular honeycombs
Infinite skew polyhedron
Plesiohedron
References
Grünbaum (1994). "Uniform tilings of 3space". Geombinatorics 4(2)
Weisstein, Eric W. "Spacefilling polyhedron". MathWorld.
Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln", Archiv der Mathematik (in German), 35 (6): 583–587, doi:10.1007/BF01235384, MR 0604258.
Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill R n {\displaystyle \mathbb {R} ^{n}} \mathbb {R} ^{n} and scissors congruence", Discrete and Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797.
[1] Uniform spacefilling using triangular, square, and hexagonal prisms
[2] Uniform spacefilling using only rhombohexagonal dodecahedra
[3] Uniform spacefilling using only truncated octahedra
John Conway (20031222). "Voronoi Polyhedron. geometry.puzzles". Newsgroup: geometry.puzzles. Usenet: Pine.LNX.4.44.0312221226380.25139100000@fine318a.math.Princeton.EDU.
X. Qian, D. Strahs and T. Schlick, J. Comput. Chem. 22(15) 1843–1850 (2001)
[4] O. DelgadoFriedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358362
[5] Archived 20150630 at the Wayback Machine Gabbrielli, Ruggero. A thirteensided polyhedron which fills space with its chiral copy.
Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960
Inchbald, Guy (July 1997), "The Archimedean honeycomb duals", The Mathematical Gazette, 81 (491): 213–219, doi:10.2307/3619198, JSTOR 3619198.
Further reading
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 048623729X. Chapter 5: Polyhedra packing and space filling
Critchlow, K.: Order in space.
Pearce, P.: Structure in nature is a strategy for design.
Goldberg, Michael Three Infinite Families of Tetrahedral SpaceFillers Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Goldberg, Michael (1972). "The spacefilling pentahedra". Journal of Combinatorial Theory, Series A. 13 (3): 437–443. doi:10.1016/00973165(72)900775.
Goldberg, Michael The Spacefilling Pentahedra II, Journal of Combinatorial Theory 17 (1974), 375–378.
Goldberg, Michael (1977). "On the spacefilling hexahedra". Geometriae Dedicata. 6. doi:10.1007/BF00181585.
Goldberg, Michael (1978). "On the spacefilling heptahedra". Geometriae Dedicata. 7 (2): 175–184. doi:10.1007/BF00181630.
Goldberg, Michael Convex Polyhedral SpaceFillers of More than Twelve Faces. Geometriae Dedicata 8, 491500, 1979.
Goldberg, Michael (1981). "On the spacefilling octahedra". Geometriae Dedicata. 10 (1–4): 323–335. doi:10.1007/BF01447431.
Goldberg, Michael (1982). "On the Spacefilling Decahedra".
Goldberg, Michael (1982). "On the spacefilling enneahedra". Geometriae Dedicata. 12 (3). doi:10.1007/BF00147314.
External links
Olshevsky, George. "Honeycomb". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Five spacefilling polyhedra, Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466475.
Raumfueller (Space filling polyhedra) by T.E. Dorozinski
Weisstein, Eric W. "SpaceFilling Polyhedron". MathWorld.
Fundamental convex regular and uniform honeycombs in dimensions 29



Space  Family  $\stackrel{}{\mathrm{\backslash (\; \{\backslash tilde\; \{A\}\}\_\{n1\}\; \backslash )}}}_{$  $\stackrel{}{\mathrm{\backslash (\; \{\backslash tilde\; \{C\}\}\_\{n1\}\; \backslash )}}}_{$  $\stackrel{}{\mathrm{\backslash (\; \{\backslash tilde\; \{B\}\}\_\{n1\}\; \backslash )}}}_{$  $\stackrel{}{\mathrm{\backslash (\; \{\backslash tilde\; \{D\}\}\_\{n1\}\; \backslash )}}}_{$  $\stackrel{}{\mathrm{\backslash (\; \{\backslash tilde\; \{G\}\}\_\{2\}\; /\; \{\backslash tilde\; \{F\}\}\_\{4\}\; /\; \{\backslash tilde\; \{E\}\}\_\{n1\}\; \backslash )}}}_{$ 
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 
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