# 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 spheres in ordinary space (see Kepler conjecture).

Rhombic dodecahedral honeycomb
Typeconvex uniform honeycomb dual
Coxeter-Dynkin diagram =
Cell type
Rhombic dodecahedron V3.4.3.4
Face typesRhombus
Space groupFm3m (225)
Coxeter notation½${\displaystyle {\tilde {C}}_{3}}$, [1+,4,3,4]
${\displaystyle {\tilde {B}}_{3}}$, [4,31,1]
${\displaystyle {\tilde {A}}_{3}}$×2, <[3[4]]>
Dualtetrahedral-octahedral honeycomb
Propertiesedge-transitive, face-transitive, cell-transitive

## Geometry

It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:2. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive, and edge-transitive; but it is not vertex-transitive, 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 of hexagonal close-packing.

 The honeycomb can be derived from an alternate cube tessellation by augmenting each face of each cube with a pyramid. The view from inside the rhombic dodecahedral honeycomb.

### Colorings

Cells can be given 4 colors in square layers of 2-colors where neighboring faces have different colors, and 6 colors in hexagonal layers of 3 colors where same-colored cells have no contact at all.

4-colors6-colors
Alternate square layers yellow, blue with red and green Alternate hexagonal layers of red,green,blue and magenta, yellow, cyan.

The rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb.

### Trapezo-rhombic dodecahedral honeycomb

Trapezo-rhombic dodecahedral honeycomb
Typeconvex uniform honeycomb dual
Cell typetrapezo-rhombic dodecahedron VG3.4.3.4
Face typesrhombus,
trapezoid
Symmetry groupP63/mmc
Dualgyrated tetrahedral-octahedral honeycomb
Propertiesedge-uniform, face-uniform, cell-uniform

The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation (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.

It is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb.

### Rhombic pyramidal honeycomb

Rhombic pyramidal honeycomb
(No image)
TypeDual uniform honeycomb
Coxeter-Dynkin diagrams
Cell
rhombic pyramid
Faces Rhombus
Triangle
Coxeter groups[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
[3[4]], ${\displaystyle {\tilde {A}}_{3}}$
Symmetry groupFm3m (225)
vertex figures
, ,
DualCantic cubic honeycomb
PropertiesCell-transitive

The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-filling tessellation (or honeycomb) 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.

 rhombic dodecahedral honeycomb Rhombohedral dissection Within a cube

It is dual to the cantic cubic honeycomb: