RHOMBIC DODECAHEDRON

The 'rhombic dodecahedron' is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.

Contents

Properties

Area and volume

Cartesian coordinates

See also

References

External links

Properties

It is the polyhedral dual of the cuboctahedron and a zonohedron. The long diagonal of each face is exactly √2 times the length of the short diagonal, so that the acute angles on each face measure cos⁻¹(1/3), or approximately 70.53°.
Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.
The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron.

The rhombic dodecahedron can be used to tessellate 3-dimensional space. It can be stacked to fill a space much like hexagons fill a plane.
This tessellation can be seen as the Voronoi tessellation of the face-centred cubic lattice. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.
The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.

Area and volume

The area ''A'' and the volume ''V'' of the rhombic dodecahedron of edge length ''a'' are:
:
:

Cartesian coordinates

The eight vertices where three faces meet at their obtuse angles have Cartesian coordinates
: (±1, ±1, ±1)
The six vertices where four faces meet at their acute angles are given by the permutations of
: (0, 0, ±2)

See also

★ Rhombic triacontahedron

★ Truncated rhombic dodecahedron

★ 24-cell - 4D analog of rhombic dodecahedron

References

★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X (Section 3-9)

External links

★

★ Virtual Reality Polyhedra – The Encyclopedia of Polyhedra

★ Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue