Rhombohedron

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, 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.

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

For a unit rhombohedron (side length = 1) whose rhombic acute angle is θ and has one vertex is at the origin (0, 0, 0) with one edge lying along the x-axis the three vectors are

e₁: ${\displaystyle {\biggl (}1,0,0{\biggr )}}$

e₂: ${\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )}}$

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