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.

Cubic honeycomb

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. 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 3-honeycombs

A 3-dimensional uniform honeycomb is a honeycomb in 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 in Euclidean 3-space,[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 3-space, 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 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.[2] A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero,[3][4] 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:

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]

Cube (parallelepiped)	Hexagonal prism	Rhombic dodecahedron	Elongated dodecahedron	Truncated octahedron
cubic honeycomb	Hexagonal prismatic honeycomb	Rhombic dodecahedra	Elongated dodecahedra	Truncated octahedra

3 edge-lengths	3+1 edge-lengths	4 edge-lengths	4+1 edge-lengths	6 edge-lengths

Other known examples of space-filling 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 trapezo-rhombic 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]

Weaire–Phelan structure (with two types of cells)

The periodic unit of the Weaire–Phelan structure.

P8 tiling (with left and right-handed cells)

A honeycomb by left and right-handed versions of the same polyhedron.

Non-convex 3-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.

Hyperbolic honeycombs

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.

Four regular compact honeycombs in H³
{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 3-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 4-polytopes, 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.[13]

Self-dual honeycombs

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

References

Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics 4(2)
Weisstein, Eric W. "Space-filling 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, S2CID 121301319.
Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill $\mathbb {R} ^{n}$ and scissors congruence", Discrete and Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797.
Uniform space-filling using triangular, square, and hexagonal prisms
Uniform space-filling using only rhombo-hexagonal dodecahedra
Uniform space-filling using only truncated octahedra
John Conway (2003-12-22). "Voronoi Polyhedron. geometry.puzzles". Newsgroup: geometry.puzzles. Usenet: Pine.LNX.4.44.0312221226380.25139-100000@fine318a.math.Princeton.EDU.
X. Qian, D. Strahs and T. Schlick, J. Comput. Chem. 22(15) 1843–1850 (2001)
O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358-362
Archived 2015-06-30 at the Wayback Machine Gabbrielli, Ruggero. A thirteen-sided 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.

External links

Olshevsky, George. "Honeycomb". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Five space-filling polyhedra, Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.
Raumfueller (Space filling polyhedra) by T.E. Dorozinski
Weisstein, Eric W. "Space-Filling Polyhedron". MathWorld.

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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[1] Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics 4(2)

[2] Weisstein, Eric W. "Space-filling polyhedron". MathWorld.

[3] 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, S2CID 121301319.

[4] Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill $\mathbb {R} ^{n}$ and scissors congruence", Discrete and Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797.

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

[6] Uniform space-filling using only rhombo-hexagonal dodecahedra

[7] Uniform space-filling using only truncated octahedra

[8] John Conway (2003-12-22). "Voronoi Polyhedron. geometry.puzzles". Newsgroup: geometry.puzzles. Usenet: Pine.LNX.4.44.0312221226380.25139-100000@fine318a.math.Princeton.EDU.

[9] X. Qian, D. Strahs and T. Schlick, J. Comput. Chem. 22(15) 1843–1850 (2001)

[10] O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358-362

[11] Archived 2015-06-30 at the Wayback Machine Gabbrielli, Ruggero. A thirteen-sided polyhedron which fills space with its chiral copy.

[12] Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960

[13] Inchbald, Guy (July 1997), "The Archimedean honeycomb duals", The Mathematical Gazette, 81 (491): 213–219, doi:10.2307/3619198, JSTOR 3619198.