Rhombic dodecahedral honeycomb

Rhombic dodecahedral honeycomb
Type	convex uniform honeycomb dual
Coxeter-Dynkin diagram	=
Cell type	Rhombic dodecahedron V3.4.3.4
Face types	Rhombus
Space group	Fm3m (225)
Coxeter notation	½${\displaystyle {\tilde {C}}_{3}}$, [1+,4,3,4] ${\displaystyle {\tilde {B}}_{3}}$, [4,31,1] ${\displaystyle {\tilde {A}}_{3}}$×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.

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.

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.

4-coloring	6-coloring
Alternate square layers of yellow/blue and red/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
Type	convex uniform honeycomb dual
Cell type	trapezo-rhombic dodecahedron VG3.4.3.4
Face types	rhombus, trapezoid
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.

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

Rhombic pyramidal honeycomb
(No image)
Type	Dual 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 group	Fm3m (225)
vertex figures	, ,
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.

rhombic dodecahedral honeycomb

Rhombohedral dissection

Within a cube

It is dual to the cantic cubic honeycomb:

• Architectonic and catoptric tessellation

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

Wikimedia Commons has media related to Rhombic dodecahedral honeycomb.

• Weisstein, Eric W. "Space-filling polyhedron". MathWorld.
• Examples of Housing Construction using this geometry