Rhombohedron | |
---|---|
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Type | prism |
Faces | 6 rhombi |
Edges | 12 |
Vertices | 8 |
Symmetry group | Ci , [2+,2+], (×), order 2 |
Properties | convex, equilateral, zonohedron, parallelohedron |
In geometry, a rhombohedron (also called a rhombic hexahedron [1] [2] or, inaccurately, a rhomboid [a]) is a special case of a parallelepiped in which all six faces are congruent rhombi. [3] It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square.
The common angle at the two apices is here given as . There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.
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Oblate rhombohedron | Prolate rhombohedron |
In the oblate case and in the prolate case . For the figure is a cube.
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
Form | Cube | √2 Rhombohedron | Golden Rhombohedron |
---|---|---|---|
Angle constraints |
|||
Ratio of diagonals | 1 | √2 | Golden ratio |
Occurrence | Regular solid | Dissection of the rhombic dodecahedron | Dissection of the rhombic triacontahedron |
For a unit (i.e.: with side length 1) rhombohedron, [4] 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 [5] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .
The volume of a 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 a 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.
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. [6]
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron[ citation needed]:
Rhombohedron | |
---|---|
![]() | |
Type | prism |
Faces | 6 rhombi |
Edges | 12 |
Vertices | 8 |
Symmetry group | Ci , [2+,2+], (×), order 2 |
Properties | convex, equilateral, zonohedron, parallelohedron |
In geometry, a rhombohedron (also called a rhombic hexahedron [1] [2] or, inaccurately, a rhomboid [a]) is a special case of a parallelepiped in which all six faces are congruent rhombi. [3] It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square.
The common angle at the two apices is here given as . There are two general forms of the rhombohedron, oblate (flattened) and prolate (stretched.
![]() |
![]() |
Oblate rhombohedron | Prolate rhombohedron |
In the oblate case and in the prolate case . For the figure is a cube.
Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms.
Form | Cube | √2 Rhombohedron | Golden Rhombohedron |
---|---|---|---|
Angle constraints |
|||
Ratio of diagonals | 1 | √2 | Golden ratio |
Occurrence | Regular solid | Dissection of the rhombic dodecahedron | Dissection of the rhombic triacontahedron |
For a unit (i.e.: with side length 1) rhombohedron, [4] 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 [5] of the 3 direction vectors: e1 + e2 , e1 + e3 , e2 + e3 , and e1 + e2 + e3 .
The volume of a 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 a 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.
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. [6]
The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron[ citation needed]: