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The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling
Examples of Housing Construction using this geometry
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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 ...
(or honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) in Euclidean 3-space. It is the Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...
of the face-centered cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties o ...
sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see Kepler conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling s ...
).
Geometry
It consists of copies of a single cell, therhombic dodecahedron
In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...
. All faces are rhombi
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
, with diagonals in the ratio 1:. Three cells meet at each edge. The honeycomb is thus 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 congruen ...
, face-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 Face (geometry), faces are the same. More specifically, all faces must be not ...
, and edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea ...
; but it is not vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...
, as it has two kinds of vertex. Each vertex with the obtuse rhombic face angles is shared by 4 cells; each vertex with the acute rhombic face angles is shared by 6 cells.
The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...
of hexagonal 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 occ ...
.
Colorings
The tiling's cells can be 4-colored in square layers of 2 colors each, such that two cells of the same color touch only at vertices; or they can be 6-colored in hexagonal layers of 3 colors each, such that same-colored cells have no contact at all.Related honeycombs
The ''rhombic dodecahedral honeycomb'' can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4trigonal trapezohedron
In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron.
An alternative name for the same shape is the ''trig ...
s. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb.
Trapezo-rhombic dodecahedral honeycomb
The trapezo-rhombic dodecahedral honeycomb is a space-fillingtessellation
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 ...
(or honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) in Euclidean 3-space. It consists of copies of a single cell, the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi.
:
Related honeycombs
It is a dual to thevertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...
gyrated tetrahedral-octahedral honeycomb.
:
Rhombic pyramidal honeycomb
The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-fillingtessellation
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 ...
(or honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) in Euclidean 3-space.
This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 12 rhombic pyramids.
Related honeycombs
It is dual to the cantic cubic honeycomb: :
See also
*Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the Uniform convex honeycomb, uniform tessellations (or Honeycomb (geometry), honeycombs) of Euclidean 3-space with prime space groups and their Dual polytope, ...
References
*External links
* {{Mathworld , urlname = Space-FillingPolyhedron , title = Space-filling polyhedronExamples of Housing Construction using this geometry
3-honeycombs