Rhombic dodecahedral honeycomb

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

Rhombic dodecahedral honeycomb
Rhombic dodecahedra.png
Type convex uniform honeycomb dual
Coxeter-Dynkin diagram CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node f1.pngCDel 3.pngCDel node.pngCDel split1-43.pngCDel nodes.png
CDel node f1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
Cell type Dodecahedrille cell.png
Rhombic dodecahedron V3.4.3.4
Face types Rhombus
Space group Fm3m (225)
Coxeter notation ½, [1+,4,3,4]
, [4,31,1]
×2, <[3[4]]>
Dual tetrahedral-octahedral honeycomb
Properties edge-transitive, face-transitive, cell-transitive

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).


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.

HC R1.png Cubes-R1 ani.gif
The honeycomb can be derived from an alternate cube tessellation by augmenting each face of each cube with a pyramid.
Rhombic dodecahedral honeycomb.png
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.

4-colors 6-colors
Rhombic dodecahedral honeycomb 4-color.gif Rhombic dodecahedral honeycomb 6-color.gif
Alternate square layers yellow, blue with red and green Alternate hexagonal layers of red,green,blue and magenta, yellow, cyan.

Related honeycombs[edit]

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[edit]

Trapezo-rhombic dodecahedral honeycomb
Trapez rhombic dodeca hb.png
Type convex uniform honeycomb dual
Cell type trapezo-rhombic dodecahedron VG3.4.3.4
Trapezo-rhombic dodecahedron.png
Face types rhombus,
Symmetry group P63/mmc
Dual gyrated tetrahedral-octahedral honeycomb
Properties edge-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.

Trapezo-rhombic dodecahedron honeycomb.png

Related honeycombs[edit]

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

Gyrated alternated cubic honeycomb.png

Rhombic pyramidal honeycomb[edit]

Rhombic pyramidal honeycomb
(No image)
Type Dual uniform honeycomb
Coxeter-Dynkin diagrams CDel node f1.pngCDel 3.pngCDel node f1.pngCDel split1-43.pngCDel nodes.png
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Cell Half oblate octahedrille cell.png
rhombic pyramid
Faces Rhombus
Coxeter groups [4,31,1],
Symmetry group Fm3m (225)
vertex figures Tetrakis cube.pngRhombic dodecahedron.jpgTriakis tetrahedron.png
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
Dual Cantic cubic honeycomb
Properties Cell-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.

HC R1.png
rhombic dodecahedral honeycomb
Half oblate octahedrille cell.png
Rhombohedral dissection
Half oblate octahedrille cell-cube.png
Within a cube

Related honeycombs[edit]

It is dual to the cantic cubic honeycomb:

Truncated Alternated Cubic Honeycomb.svg

See also[edit]


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

External links[edit]