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 ...
, a honeycomb is a ''space filling'' or ''
close packing
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
'' 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 o ...
'' 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 facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
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 ''circumcircle' ...
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 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 o ...
s of the plane. In particular, for every
parallelepiped, copies can fill space, with the
cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) 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 r ...
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
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
composed of
uniform polyhedral cells, 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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
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
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) 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 r ...
. Two are ''quasiregular'' (made from two types of regular cells):
The
tetrahedral-octahedral honeycomb
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
Other names i ...
and
gyrated tetrahedral-octahedral honeycomb
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
Other names in ...
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
A honeycomb having all cells identical within its symmetries is said to be
cell-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...
or isochoric. In the 3-dimensional euclidean space, a cell of such a honeycomb is said to be a ''
space-filling polyhedron
In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that, taken together, all the instances of the polyhedron const ...
''. 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 of ...
for a polyhedron to be a space-filling polyhedron is that its
Dehn invariant
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled (" dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who us ...
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
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) 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 r ...
(or variations:
cuboid, rhombic
hexahedron
A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.
Ther ...
or
parallelepiped)
#
Hexagonal 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 pri ...
#
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 equal s ...
#
Elongated dodecahedral honeycomb
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...
#
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 t ...
or
truncated octahedra
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
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 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 pri ...
* 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 carbo ...
. 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 equal ...
*
Isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
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
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
, 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
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon.
It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual ...
, as in the
Yoshimoto Cube
The Yoshimoto Cube is a polyhedral mechanical puzzle toy invented in 1971 by , who discovered that two stellated rhombic dodecahedra could be pieced together into a cube when he was finding different ways he could split a cube equally in half. Y ...
.
*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
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
, the
dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
of a polyhedron depends on its size. The regular hyperbolic honeycombs thus include two with four or five
dodecahedra
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
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 British ...
and
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
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-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
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.
[.]
Self-dual honeycombs
Honeycombs can also be
self-dual. All ''n''-dimensional
hypercubic honeycombs with
Schläfli symbols , are self-dual.
See also
*
List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
*
Regular honeycombs
*
Infinite skew polyhedron
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.
Skew apeirohedra have also been ...
*
Plesiohedron
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set.
Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. Th ...
References
Further reading
*
Coxeter, H. S. M.: ''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
''.
* 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