Rhombohedron

Rhombohedron
Type Prism
Faces 6 rhombi
Edges 12
Vertices 8
Symmetry group Ci, [2+,2+], (×), Order 2
Properties convex, zonohedron

In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but 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.

In general the rhombohedron can have three types of rhombic faces in congruent opposite pairs, Ci 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.[1]

Rhombohedral lattice system

The rhombohedral lattice system has rhombohedral cells, with 3 pairs of unique rhombic faces:

Special cases

Form Cube Trigonal trapezohedron Right rhombic prism General rhombic prism General rhombohedron
Symmetry Oh, [4,3], order 48 D3d, [2+,6], order 12 D2h, [2,2], order 8 C2h, [2], order 4 Ci, [2+,2+], order 2
Image
Faces 6 squares 6 identical rhombi Two rhombi and 4 squares 6 rhombic faces 6 rhombic faces
• Cube: with Oh symmetry, order 48. All faces are squares.
• Trigonal trapezohedron: with D3d symmetry, order 12. If all of the non-obtuse internal angles of the faces are equal (all faces are same). This can be see by stretching a cube on its body-digonal axis. For example a regular octahedron with two tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron:.
• Right rhombic prism: with D2h symmetry, order 8. It constructed by two rhombi and 4 squares. This can be see by stretching a cube on its face-digonal axis. For example two triangular prisms attached together makes a 60 degree rhombic prism.
• A general rhombic prism: With C2h symmetry, order 4. It has only one plane of symmetry through four vertices, and 6 rhombic faces.

References

1. ^ Court, N. A. (October 1934), "Notes on the orthocentric tetrahedron", American Mathematical Monthly: 499–502, JSTOR 2300415.