Honeycomb (geometry)

Last updated
cubic honeycomb Cubic honeycomb.png
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. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. A novel honeycomb tessellation generator MATLAB code, HoneyMesher, can be found in Ref. [1]

Contents

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. Wallpaper group-cmm-1.jpg
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, [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):

Type Regular cubic honeycomb Quasiregular honeycombs
Cells Cubic Octahedra and tetrahedra
Slab layer Cubic semicheck.png Tetroctahedric semicheck.png

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 . [3] A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be zero, [4] [5] 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:

  1. Cubic honeycomb (or variations: cuboid, rhombic hexahedron or parallelepiped)
  2. Hexagonal prismatic honeycomb [6]
  3. Rhombic dodecahedral honeycomb
  4. Elongated dodecahedral honeycomb [7]
  5. Bitruncated cubic honeycomb or truncated octahedra [8]
Rhombohedral prism honeycomb.png
cubic honeycomb
Skew hexagonal prism honeycomb.png
Hexagonal prismatic honeycomb
Rhombic dodecahedra.png
Rhombic dodecahedra
Elongated rhombic dodecahedron honeycomb.png
Elongated dodecahedra
Truncated octahedra.png
Truncated octahedra
Cube
(parallelepiped)
Hexagonal prism Rhombic dodecahedron Elongated dodecahedron Truncated octahedron
Parallelohedron edges cube.png Parallelohedron edges hexagonal prism.png Parallelohedron edges rhombic dodecahedron.png Parallelohedron edges elongated rhombic dodecahedron.png Parallelohedron edge truncated octahedron.png
3 edge-lengths3+1 edge-lengths4 edge-lengths4+1 edge-lengths6 edge-lengths

Other known examples of space-filling polyhedra include:

Other honeycombs with two or more polyhedra

Sometimes, two [12] 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 [13]

12-14-hedral honeycomb.png
The periodic unit of the Weaire–Phelan structure.
P8-gabbrielli.gif
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:

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
H3 534 CC center.png
{5,3,4}
H3 435 CC center.png
{4,3,5}
H3 353 CC center.png
{3,5,3}
H3 535 CC center.png
{5,3,5}
11 paracompact regular honeycombs
H3 633 FC boundary.png
{6,3,3}
H3 634 FC boundary.png
{6,3,4}
H3 635 FC boundary.png
{6,3,5}
H3 636 FC boundary.png
{6,3,6}
H3 443 FC boundary.png
{4,4,3}
H3 444 FC boundary.png
{4,4,4}
H3 336 CC center.png
{3,3,6}
H3 436 CC center.png
{4,3,6}
H3 536 CC center.png
{5,3,6}
H3 363 FC boundary.png
{3,6,3}
H3 344 CC center.png
{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:

Self-dual honeycombs

Honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schläfli symbols {4,3n2,4}, are self-dual.

See also

Related Research Articles

Cuboctahedron Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

4-polytope Four-dimensional geometric object with flat sides

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

Schläfli symbol Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

Elongated square bipyramid

In geometry, the elongated square bipyramid is one of the Johnson solids (J15). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves.

Cubic honeycomb Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

Tetrahedral-octahedral honeycomb Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

Rhombic dodecahedral honeycomb Space-filling tesselation

The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space.

Bitruncated cubic honeycomb

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

Quarter cubic honeycomb

The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.

Isohedral figure ≥3-dimensional polytope with identical faces

In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Triangular prismatic honeycomb

The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

Order-6 dodecahedral honeycomb Regular geometrical object in hyperbolic space

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

In geometry and crystallography, a stereohedron is a convex polyhedron that fills space isohedrally, meaning that the symmetries of the tiling take any copy of the stereohedron to any other copy.

References

  1. Kumar, P (2022). "HoneyTop90: A 90-line MATLAB code for topology optimization using honeycomb tessellation". Optimization and Engineering. doi:10.1007/s11081-022-09715-6.
  2. Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics 4(2)
  3. Weisstein, Eric W. "Space-filling polyhedron". MathWorld .
  4. 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 .
  5. Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill and scissors congruence", Discrete and Computational Geometry , 13 (3–4): 573–583, doi: 10.1007/BF02574064 , MR   1318797 .
  6. Uniform space-filling using triangular, square, and hexagonal prisms
  7. Uniform space-filling using only rhombo-hexagonal dodecahedra
  8. Uniform space-filling using only truncated octahedra
  9. John Conway (2003-12-22). "Voronoi Polyhedron. geometry.puzzles". Newsgroup:  geometry.puzzles. Usenet:   Pine.LNX.4.44.0312221226380.25139-100000@fine318a.math.Princeton.EDU.
  10. X. Qian, D. Strahs and T. Schlick, J. Comput. Chem.22(15) 1843–1850 (2001)
  11. O. Delgado-Friedrichs and M. O'Keeffe. Isohedral simple tilings: binodal and by tiles with <16 faces. Acta Crystallogr. (2005) A61, 358-362
  12. Archived 2015-06-30 at the Wayback Machine Gabbrielli, Ruggero. A thirteen-sided polyhedron which fills space with its chiral copy.
  13. Pauling, Linus. The Nature of the Chemical Bond. Cornell University Press, 1960
  14. Inchbald, Guy (July 1997), "The Archimedean honeycomb duals", The Mathematical Gazette , 81 (491): 213–219, doi:10.2307/3619198, JSTOR   3619198 .

Further reading

Space Family / /
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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21