direct-lattice basis-vectors a1, a2 and a3 as Hence zone indices of the direction perpendicular to plane (hkℓ) are, in suitably normalized triplet form, simply . When four indices are used for the zone normal to plane (hkℓ), however, the literature often uses instead. Thus as you can see, four-index zone indices in square or angle brackets sometimes mix a single direct-lattice index on the right with reciprocal-lattice indices (normally in round or curly brackets) on the left. And, note that for hexagonal interplanar distances, they take the form Crystallographic planes and directions Crystallographic directions are lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, crystallographic planes are planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal: optical properties: in condensed matter, light "jumps" from one atom to the other with the Rayleigh scattering; the velocity of light thus varies according to the directions, whether the atoms are close or far; this gives the birefringence adsorption and reactivity: adsorption and chemical reactions can occur at atoms or molecules on crystal surfaces, these phenomena are thus sensitive to the density of nodes; surface tension: the condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface Pores and crystallites tend to have straight grain boundaries following dense planes cleavage dislocations (plastic deformation) the dislocation core tends to spread on dense planes (the elastic perturbation is "diluted"); this reduces the friction (Peierls–Nabarro force), the sliding occurs more frequently on dense planes; the perturbation carried by the dislocation (Burgers vector) is along a dense direction: the shift of one node in a dense direction is a lesser distortion; the dislocation line tends to follow a dense direction, the dislocation line is often a straight line, a dislocation loop is often a polygon. For all these reasons, it is important to determine the planes and thus to have a notation system. Integer vs. irrational Miller indices: Lattice planes and quasicrystals Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane (abc) where the Miller "indices" a, b and c (defined as above) are not necessarily integers. If a,