The Crystal Lattice in Three Dimensions
A crystal's orientation specifies the rotation that aligns the crystal lattice frame with the external (sample or laboratory) reference frame. In practice this means specifying a rotation matrix g ∈ SO(3) such that any crystal direction dc is expressed in the sample frame as:
d_s = g · d_c
The lattice is fully characterised by its unit cell — the parallelepiped defined by lattice vectors a, b, c and interaxial angles α, β, γ. Each of the seven crystal systems (cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, triclinic) imposes different constraints on these parameters and carries a different set of point-group symmetry operations.
Symmetry Operations
Point-group symmetry operations (rotations, reflections, and roto-inversions) map the crystal lattice onto itself. For a crystal with point group G, any orientation g is physically equivalent to all orientations in its coset G·g. Proper rotations in G form a subgroup of SO(3) and define the symmetry-equivalent variants of an orientation — a key concept in texture and orientation mapping.
|G| variants: {h·g | h ∈ G}
For a cubic crystal the 24 proper rotations in Oh produce up to 24 symmetry-equivalent representations of the same physical orientation. Visualising all variants simultaneously reveals the multiplicity structure of each crystal system.
In the Simulation
The 3D viewer renders a wire-frame unit cell and a set of crystallographic reference directions, rotated live as the orientation parameters change. Key features:
Crystal system selector — choose from cubic, tetragonal, hexagonal, orthorhombic, monoclinic, and triclinic ·
Orientation controls — rotate via Euler angles, quaternion, or axis-angle, synchronised with the Rotations submodule ·
Symmetry overlay — display all symmetry-equivalent variants simultaneously as transparent copies ·
Axis labelling — a, b, c lattice vectors and x, y, z sample axes always visible.
The simulation makes it easy to see why different crystal systems have different numbers of indistinguishable orientations.