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# Rhombic dodecahedral honeycomb

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 Title: Rhombic dodecahedral honeycomb Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 is the densest possible packing of equal spheres in ordinary space (see Kepler conjecture).

## Contents

• Geometry 1
• Related honeycombs 2
• Trapezo-rhombic dodecahedral honeycomb 2.1
• Related honeycombs 2.1.1
• Rhombic pyramidal honeycomb 2.2
• Related honeycombs 2.2.1
• References 3

## 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
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 a truncated tetraoctahedrille.

#### Related honeycombs

It is dual to the cantic cubic honeycomb: