Rhombic dodecahedral honeycomb

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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 Rhombicdodecahedron.jpg
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[edit]

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


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,
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.

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
Cell Square pyramid.png
rhombic pyramid
Faces Rhombus
Triangle
Coxeter groups [4,31,1], {\tilde{B}}_3
[3[4]], {\tilde{A}}_3
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 is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an truncated tetraoctahedrille.

Related honeycombs[edit]

It is dual to the cantic cubic honeycomb:

Truncated Alternated Cubic Honeycomb.svg

References[edit]

External links[edit]