In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
'' 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 ("flat") space. They may also be constructed in
non-Euclidean spaces, such as
hyperbolic honeycombs. Any finite
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimen ...
can be projected to its
circumsphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcirc ...
to form a uniform honeycomb in spherical space.
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
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
based on some
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s of the plane. In particular, for every
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
, 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
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
, and having all vertices the same (i.e., the group of
sometries of 3-space that preserve the tilingis ''
transitive on vertices''). There are 28
convex examples in Euclidean 3-space, 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.
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''. A
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
for a polyhedron to be a space-filling polyhedron is that its
Dehn invariant must be zero, ruling out any of the
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s 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
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...
, rhombic
hexahedron or
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
)
#
Hexagonal prismatic honeycomb
#
Rhombic dodecahedral honeycomb
#
Elongated dodecahedral honeycomb
#
Bitruncated cubic honeycomb
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of ...
or
truncated octahedra
Other known examples of space-filling polyhedra include:
* The
triangular prismatic honeycomb
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms.
It is constructed from a triangular tiling extruded into p ...
* The
gyrated triangular prismatic honeycomb
* The
triakis truncated tetrahedral honeycomb
The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914.
Voronoi tessellation
It is the Voronoi tessellation of the carb ...
. The Voronoi cells of the carbon atoms in diamond are this shape.
* The
trapezo-rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equ ...
*
Isohedral tilings
Other honeycombs with two or more polyhedra
Sometimes, two 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
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
paracompact uniform hyperbolic honeycombs have been enumerated.
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.
[.]
Self-dual honeycombs
Honeycombs can also be
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
. All ''n''-dimensional
hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 - hypercu ...
s with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
s , are self-dual.
See also
*
List of uniform tilings
*
Regular honeycombs
*
Infinite skew polyhedron
*
Plesiohedron
References
Further reading
*
Coxeter, H. S. M.: ''
Regular Polytopes''.
* 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.
*
* Goldberg, Michael ''The Space-filling Pentahedra II'', Journal of Combinatorial Theory 17 (1974), 375–378.
*
*
* Goldberg, Michael ''Convex Polyhedral Space-Fillers of More than Twelve Faces.'' Geom. Dedicata 8, 491-500, 1979.
*
*
*
External links
*
Five space-filling polyhedra Guy Inchbald, The Mathematical Gazette 80, November 1996, p.p. 466-475.
Raumfueller (Space filling polyhedra) by T.E. Dorozinski*
{{DEFAULTSORT:Honeycomb (Geometry)
Polytopes