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The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling
Examples of Housing Construction using this geometry
Honeycombs (geometry)
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
(or honeycomb) 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. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed th ...
of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see Kepler conjecture).
Geometry
It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive, and edge-transitive; but it is notvertex-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 in ...
, as it has two kinds of vertex. The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 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. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed th ...
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 occu ...
.
Colorings
Cells can be given 4 colors in square layers of 2-colors where neighboring cells have different colors, and 6 colors in hexagonal layers of 3 colors where 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 rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...
s. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal 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 ...
.
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 (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
(or honeycomb) 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 in ...
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 (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
(or honeycomb) in Euclidean 3-space.
This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra
Rhombic may refer to:
*Rhombus, a quadrilateral whose four sides all have the same length (often called a diamond)
*Rhombic antenna, a broadband directional antenna most commonly used on shortwave frequencies
* polyhedra formed from rhombuses, such ...
dissected
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...
with its center into 12 rhombic pyramids.
Related honeycombs
It is dual to thecantic cubic 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 incl ...
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:
See also
*Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space with prime space groups and their duals, as three-dimensional analogue of the Platonic, Archime ...
References
*External links
* {{Mathworld , urlname = Space-FillingPolyhedron , title = Space-filling polyhedronExamples of Housing Construction using this geometry
Honeycombs (geometry)