What are Miller indices in mineralogy ?

Miller indices : definition

Miller indices, denoted h, k, l (these 3 letters representing whole numbers, positive, negative or zero), are used in crystallography to index the faces, shapes and axes of crystals.

By convention, faces are indexed in brackets (hkl), shapes in braces {hkl} and axes in brackets [hkl]. The indices correspond to the inverse of the number of base vectors (for the crystalline system concerned) which define the plane or the axis. Clearly, a plane (111) is defined by the 3 basic vectors of the crystal system (vectors x, y and z on the diagram below). A plane (221) is defined by 1/2 x vector and 1/2 y vector as well as 1 full z vector.

Attention for different crystal systems two planes of the same indices are not parallel (because the base vectors have different lengths and angles between them according to each crystal system).

Attention for the hexagonal system there are 4 indices (hkil).