Rhombic dodecahedron
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| Rhombic dodecahedron | |
|---|---|
 (Click here for rotating model) |
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| Type | Catalan solid |
| Face type | rhombus |
| Faces | 12 |
| Edges | 24 |
| Vertices | 14 |
| Vertices by type | 8{3}+6{4} |
| Face configuration | V3.4.3.4 |
| Symmetry group | Oh |
| Dihedral angle | 120° |
| Dual | Cuboctahedron |
| Properties | convex, face-transitive edge-transitive, zonohedron |
 Net |
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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.
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[edit] 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.
[edit] Area and volume
The area A and the volume V of the rhombic dodecahedron of edge length a are:
[edit] 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)
[edit] See also
- Dodecahedron
- Rhombic triacontahedron
- Truncated rhombic dodecahedron
- 24-cell - 4D analog of rhombic dodecahedron
- Rhombic dodecahedral honeycomb
[edit] References
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
[edit] External links
- Eric W. Weisstein, Rhombic dodecahedron (Catalan solid) at MathWorld.
- Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
[edit] Computer models
- Rhombic Dodecahedron -- interactive 3-d model
- Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron, Rhombic Dodecahedron 5-Compound and Rhombic Dodecahedron 5-Compound by Sándor Kabai, The Wolfram Demonstrations Project.
[edit] Paper projects
- Rhombic Dodecahedron Calendar – make a rhombic dodecahedron calendar without glue
- Another Rhombic Dodecahedron Calendar – made by plaiting paper strips
[edit] Practical applications
- Archimede Institute Examples of actual housing construction projects using this geometry
