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, [1+,4,3,4]
, [4,31,1]
×2, <[3[4]]>
Dualtetrahedral-octahedral honeycomb
Propertiesedge-transitive, face-transitive, cell-transitive


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.


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.

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,
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
rhombic pyramid
Faces Rhombus
Coxeter groups[4,31,1],
Symmetry groupFm3m (225)
vertex figures
, ,
DualCantic cubic honeycomb

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:

See also


  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 168. ISBN 0-486-23729-X.
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