View source for Honeycomb (geometry)

{{short description|Tiling of euclidean or hyperbolic space of three or more dimensions}}
{{other uses|Honeycomb (disambiguation)}}
[[Image:cubic honeycomb.png|thumb|150px|[[Cubic honeycomb]]]]
In [[geometry]], a '''honeycomb''' is a ''space filling'' or ''[[close packing]]'' of [[polyhedron|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 geometry|Euclidean]] ("flat") space. They may also be constructed in [[non-Euclidean geometry|non-Euclidean spaces]], such as [[#Hyperbolic honeycombs|hyperbolic honeycombs]].  Any finite [[uniform polytope]] can be projected to its [[circumsphere]] to form a uniform honeycomb in spherical space. 

[[Image:Wallpaper group-cmm-1.jpg|200px|right|thumb|It is possible to fill the plane with [[polygon]]s which do not meet at their corners, for example using [[rectangle]]s, 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 [[prism (geometry)|prisms]] based on some [[tessellation]]s 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 tetrahedron|Hill tetrahedra]] and their generalizations, which can also tile the space.

===Uniform 3-honeycombs===
A 3-dimensional '''uniform honeycomb''' is a honeycomb in [[Three-dimensional space|3-space]] composed of [[uniform polyhedron|uniform polyhedral]] [[Cell (geometry)|cells]], and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is ''[[vertex-transitive|transitive on vertices]]''). There are 28 [[Convex polytope|convex]] examples in Euclidean 3-space,<ref>Grünbaum (1994). "Uniform tilings of 3-space". ''Geombinatorics'' 4(2)</ref> also called the '''[[convex uniform honeycomb|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):

{| class=wikitable
!Type
![[Regular cubic honeycomb]]
!Quasiregular honeycombs
|-
!Cells
![[Cube|Cubic]]
![[Octahedron|Octahedra]] and [[tetrahedron|tetrahedra]]
|-
!Slab layer
|[[File:Cubic semicheck.png|240px]]
|[[File:Tetroctahedric semicheck.png|240px]]
|}
The [[tetrahedral-octahedral honeycomb]] and [[gyrated tetrahedral-octahedral honeycomb]]s 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===
{{See also|Stereohedron|Plesiohedron|Parallelohedron}}
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]]''.<ref>{{mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron}}</ref> A [[necessary condition]] for a polyhedron to be a space-filling polyhedron is that its [[Dehn invariant]] must be zero,<ref>{{citation| last = Debrunner | first = Hans E.| doi = 10.1007/BF01235384| issue = 6| journal = [[Archiv der Mathematik]]| language = German| mr = 604258| pages = 583–587 | title = Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln| volume = 35| year = 1980| s2cid = 121301319}}.</ref><ref>{{citation| last1 = Lagarias | first1 = J. C. | author1-link = Jeffrey Lagarias| last2 = Moews | first2 = D.| doi = 10.1007/BF02574064| issue = 3–4| journal = [[Discrete and Computational Geometry]]| mr = 1318797| pages = 573–583| title = Polytopes that fill <math>\mathbb{R}^n</math> and scissors congruence| volume = 13| year = 1995| doi-access = free}}.</ref> ruling out any of the [[Platonic solid]]s other than the cube.

Five space-filling convex polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called [[Parallelohedron|parallelohedra]]:
# [[Cubic honeycomb]] (or variations: [[cuboid]],  rhombic [[hexahedron]] or [[parallelepiped]])
# [[Hexagonal prismatic honeycomb]]<ref>[http://www.nanomedicine.com/NMI/Figures/5.4.jpg] Uniform space-filling using triangular, square, and hexagonal prisms</ref>
# [[Rhombic dodecahedral honeycomb]]
# [[Elongated dodecahedral honeycomb]]<ref>[http://www.nanomedicine.com/NMI/Figures/5.7.jpg] Uniform space-filling using only rhombo-hexagonal dodecahedra</ref>
# [[Bitruncated cubic honeycomb]] or [[truncated octahedra]]<ref>[http://www.nanomedicine.com/NMI/Figures/5.5.jpg] Uniform space-filling using only truncated octahedra</ref>

{| class=wikitable width=600
|- align=center valign=top
|[[File:Rhombohedral prism honeycomb.png|120px]]<BR>[[cubic honeycomb]]
|[[File:Skew hexagonal prism honeycomb.png|120px]]<BR>[[Hexagonal prismatic honeycomb]]
|[[File:Rhombic dodecahedra.png|120px]]<BR>[[rhombic dodecahedral honeycomb|Rhombic dodecahedra]]
|[[File:Elongated rhombic dodecahedron honeycomb.png|120px]]<BR>[[Elongated rhombic dodecahedral honeycomb|Elongated dodecahedra]]
||[[File:Truncated octahedra.png|120px]]<BR>[[Bitruncated cubic honeycomb|Truncated octahedra]]
|-
![[Cube]]<BR>(parallelepiped)
![[Hexagonal prism]]
![[Rhombic dodecahedron]]
![[Elongated dodecahedron]]
![[Truncated octahedron]]
|-
|[[File:Parallelohedron_edges_cube.png|120px]]
|[[File:Parallelohedron_edges_hexagonal prism.png|120px]]
|[[File:Parallelohedron_edges_rhombic dodecahedron.png|120px]]
|[[File:Parallelohedron_edges_elongated_rhombic_dodecahedron.png|120px]]
|[[File:Parallelohedron_edge_truncated_octahedron.png|140px]]
|-
!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.<ref>{{cite newsgroup|newsgroup=geometry.puzzles|message-id=Pine.LNX.4.44.0312221226380.25139-100000@fine318a.math.Princeton.EDU|author=John Conway|title=Voronoi Polyhedron. geometry.puzzles|date=2003-12-22 |url=https://groups.google.com/d/msg/geometry.puzzles/pkL3avbWPoc/ABSaqdQaqu4J}}</ref>
* The [[trapezo-rhombic dodecahedral honeycomb]]<ref>X. Qian, D. Strahs and T. Schlick, ''J. Comput. Chem.'' '''22'''(15) 1843–1850 (2001)</ref>
* [[Isohedral]] tilings<ref>[http://scripts.iucr.org/cgi-bin/paper?S0108767305009578] O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with &lt;16 faces. Acta Crystallogr. (2005) A61, 358-362</ref>

=== Other honeycombs with two or more polyhedra===
Sometimes, two <ref>[http://science.unitn.it/~gabbrielli/p8_gallery.html] {{Webarchive|url=https://web.archive.org/web/20150630072207/http://science.unitn.it/~gabbrielli/p8_gallery.html |date=2015-06-30 }} Gabbrielli, Ruggero. A thirteen-sided polyhedron which fills space with its chiral copy.</ref> 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 <ref>Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960</ref>

{{multiple image
 | align = right
 | total_width = 320

 | image1 = 12-14-hedral honeycomb.png
 | alt1 = [[Weaire–Phelan structure]] (with two types of cells)
 | caption1 = The periodic unit of the [[Weaire–Phelan structure]].

 | image2 = P8-gabbrielli.gif
 | alt2 = P8 tiling (with left and right-handed cells)
 | caption2 = 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 [[stellated rhombic dodecahedral honeycomb|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 [[dodecahedron|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 [[Convex uniform honeycombs in hyperbolic space|compact]] and [[Paracompact uniform honeycombs|paracompact]] uniform hyperbolic honeycombs have been enumerated.

{{Regular compact H3 honeycombs}}
{{Regular_paracompact_H3_honeycombs}}

==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-polytope]]s, 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.<ref>{{citation| last = Inchbald | first = Guy| date = July 1997| doi = 10.2307/3619198| issue = 491| pages = 213–219| journal = [[The Mathematical Gazette]]| title = The Archimedean honeycomb duals| url = http://www.steelpillow.com/polyhedra/AHD/AHD.html| volume = 81| jstor = 3619198}}.</ref>

== Self-dual honeycombs ==
Honeycombs can also be [[duality (mathematics)|self-dual]]. All ''n''-dimensional [[hypercubic honeycomb]]s with [[Schläfli symbol]]s {4,3<sup>''n''&minus;2</sup>,4}, are self-dual.

==See also==
{{Commons category|Honeycombs (geometry)}}
* [[List of uniform tilings]]
* [[List of regular polytopes#Tessellations|Regular honeycombs]]
* [[Infinite skew polyhedron]]
* [[Plesiohedron]]

==References==
{{reflist}}

==Further reading==
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]]: ''[[Regular Polytopes (book)|Regular Polytopes]]''.
* {{The Geometrical Foundation of Natural Structure (book)|pages=164–199}} 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 Space-Fillers'' Journal of Combinatorial Theory A, 16, pp.&nbsp;348–354, 1974.
* {{cite journal|doi=10.1016/0097-3165(72)90077-5|title=The space-filling pentahedra|journal=Journal of Combinatorial Theory, Series A|volume=13|issue=3|pages=437–443|year=1972|last1=Goldberg|first1=Michael|doi-access=}}
* Goldberg, Michael ''The Space-filling Pentahedra II'', Journal of Combinatorial Theory 17 (1974), 375–378.
* {{cite journal|doi=10.1007/BF00181585|title=On the space-filling hexahedra|journal=Geometriae Dedicata|volume=6|year=1977|last1=Goldberg|first1=Michael|s2cid=189889869}}
* {{cite journal|doi=10.1007/BF00181630|title=On the space-filling heptahedra|journal=Geometriae Dedicata|volume=7|issue=2|pages=175–184|year=1978|last1=Goldberg|first1=Michael|s2cid=120562040}}
* Goldberg, Michael ''Convex Polyhedral Space-Fillers of More than Twelve Faces.'' Geom. Dedicata 8, 491-500, 1979. 
* {{cite journal|url=http://documents.mx/documents/on-the-space-filling-octahedra.html |doi=10.1007/BF01447431|title=On the space-filling octahedra|journal=Geometriae Dedicata|volume=10|issue=1–4|pages=323–335|year=1981|last1=Goldberg|first1=Michael|s2cid=189876836}}
* {{cite journal|hdl=2099/990|title=On the Space-filling Decahedra|journal=Structural Topology|issue=7|year=1982|pages=39–44|last1=Goldberg|first1=Michael}}
* {{cite journal|doi=10.1007/BF00147314|title=On the space-filling enneahedra|journal=Geometriae Dedicata|volume=12|issue=3|year=1982|last1=Goldberg|first1=Michael|s2cid=120914105}}

==External links==
* {{GlossaryForHyperspace | anchor=Honeycomb | title=Honeycomb }}
* [http://www.steelpillow.com/polyhedra/five_sf/five.htm Five space-filling polyhedra], Guy Inchbald, The Mathematical Gazette '''80''', November 1996, p.p.&nbsp;466-475.
* [http://www.3doro.de/space-filling/ Raumfueller (Space filling polyhedra) by T.E. Dorozinski]
* {{MathWorld|Space-FillingPolyhedron|Space-Filling Polyhedron}}

{{Honeycombs}}

{{DEFAULTSORT:Honeycomb (Geometry)}}
[[Category:Honeycombs (geometry)| ]]
[[Category:Space-filling polyhedra| ]]
[[Category:Polytopes]]