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${\tilde{C}}_3$, [1+,4,3,4] ${\tilde{B}}_3$, [4,31,1] ${\tilde{A}}_3$×2, <[3[4]]>
Dual	tetrahedral-octahedral honeycomb
Properties	edge-transitive, face-transitive, cell-transitive

The rhombic dodecahedral honeycomb 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).

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.

Related honeycombs

Trapezo-rhombic dodecahedral honeycomb

Trapezo-rhombic dodecahedral honeycomb
250px
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.

320px

Related honeycombs

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

Rhombic pyramidal honeycomb

Rhombic pyramidal honeycomb
(No image)
Type	Dual uniform honeycomb
Coxeter-Dynkin diagrams
Cell	rhombic pyramid
Faces	Rhombus Triangle
Coxeter groups	[4,31,1], ${\tilde{B}}_3$ [3[4]], ${\tilde{A}}_3$
Symmetry group	Fm3m (225)
vertex figures	30px , ,
Dual	Cantic cubic honeycomb
Properties	Cell-transitive

The rhombic pyramidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it a truncated tetraoctahedrille.

Related honeycombs

It is dual to the cantic cubic honeycomb:

External links

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