Rhombohedron | |
---|---|
Type | prism |
Faces | 6 rhombi |
Edges | 12 |
Vertices | 8 |
Symmetry group | C_{i} , [2^{+},2^{+}], (×), order 2 |
Properties | convex, zonohedron |
In geometry, a rhombohedron (also called a rhombic hexahedron^{[1]} or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.
In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, C_{i} symmetry, order 2.
Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.^{[2]}
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron:
Form | Cube | Trigonal trapezohedron | Right rhombic prism | Oblique rhombic prism |
---|---|---|---|---|
Angle constraints |
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Symmetry | O_{h} order 48 |
D_{3d} order 12 |
D_{2h} order 8 |
C_{2h} order 4 |
Faces | 6 squares | 6 congruent rhombi | 2 rhombi, 4 squares | 6 rhombi |
For a unit (i.e.: with side length 1) isohedral rhombohedron,^{[3]} with rhombic acute angle , with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are
The other coordinates can be obtained from vector addition^{[4]} of the 3 direction vectors: e_{1} + e_{2} , e_{1} + e_{3} , e_{2} + e_{3} , and e_{1} + e_{2} + e_{3} .
The volume of an isohedral rhombohedron, in terms of its side length and its rhombic acute angle , is a simplification of the volume of a parallelepiped, and is given by
We can express the volume another way :
As the area of the (rhombic) base is given by , and as the height of a rhombohedron is given by its volume divided by the area of its base, the height of an isohedral rhombohedron in terms of its side length and its rhombic acute angle is given by
Note:
The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.