Honeycomb (geometry)#Uniform honeycombs

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.

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,Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics 4(2) 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):

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.

{{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.{{mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron}} A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero,{{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}}.{{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 $\backslash mathbb\{R\}^n$ and scissors congruence| volume = 13| year = 1995| doi-access = free}}. ruling out any of the Platonic solids other than the cube.

Five space-filling convex polyhedra can tessellate 3-dimensional euclidean space using translations only. They are called parallelohedra:

1. Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped)
2. Hexagonal prismatic honeycomb[http://www.nanomedicine.com/NMI/Figures/5.4.jpg] Uniform space-filling using triangular, square, and hexagonal prisms
3. Rhombic dodecahedral honeycomb
4. Elongated dodecahedral honeycomb[http://www.nanomedicine.com/NMI/Figures/5.7.jpg] Uniform space-filling using only rhombo-hexagonal dodecahedra
5. Bitruncated cubic honeycomb or truncated octahedra[http://www.nanomedicine.com/NMI/Figures/5.5.jpg] Uniform space-filling using only truncated octahedra

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.{{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}}
• The trapezo-rhombic dodecahedral honeycombX. Qian, D. Strahs and T. Schlick, J. Comput. Chem. 22(15) 1843–1850 (2001)
• Isohedral tilings[http://scripts.iucr.org/cgi-bin/paper?S0108767305009578] O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358-362

Sometimes, two [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. 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 Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960

{{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.

}}

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.

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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.

{{Regular compact H3 honeycombs}}

{{Regular_paracompact_H3_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.{{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}}.

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

{{Commons category|Honeycombs (geometry)}}

• List of uniform tilings
• Regular honeycombs
• Infinite skew polyhedron
• Plesiohedron

{{reflist}}

• Coxeter, H. S. M.: 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. 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}}

• {{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. 466-475.
• [http://www.3doro.de/space-filling/ Raumfueller (Space filling polyhedra) by T.E. Dorozinski]
• {{MathWorld|Space-FillingPolyhedron|Space-Filling Polyhedron}}

{{Honeycombs}}

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Category:Polytopes