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 dodecahedra 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 is believed to be 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.

Trapezo-rhombic dodecahedral 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.

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
, ,
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 an truncated tetraoctahedrille.

Related honeycombs

It is dual to the cantic cubic honeycomb: