Hexagonal and rhombohedral unit cells are fundamental structures in crystallography that exemplify the diversity of atomic arrangements in solid materials.
The simplest way to form a three-dimensional Bravais lattice with 6-fold rotational symmetry is to stack the lattices of figure V one above another (Va), giving the primitive hexagonal unit cell (demarcated by red lines in Vb) with IaI = IbI ≠ IcI and α = β = 90o, γ = 120o.
A three-dimensional Bravais lattice with 3-fold rotational symmetry is formed by a staggered stacking of layers of figure V, with equal separation distance between layers. The lateral position of the second layer is such that the lattice points of this layer are above the middle of equilateral triangles formed by the first layer (Vc). The third layer is also staggered in the similar way with its lattice points above the middle of equilateral triangles formed by the first as well as the second layer. This is known as the triple hexagonal lattice.
If we continue this manner of staggered-stacking, we have the lattice points of fourth layer lying directly above those of the first layer, giving an XYZXYZ stacking arrangement and producing the primitive rhombohedral unit cell (Vd) with IaI = IbI = IcI and α = β = γ ≠ 90o.
Note that the triple hexagonal lattice (Vc) does not have any 6-fold rotational symmetry, since a rotation of 600 using an axis perpendicular to the plane of the page and through the red lattice point maps lattice point 1 of the second layer to lattice point 2 of the third layer and not to lattice point 3.
