b and c have rational ratios, then the same family of planes can be written in terms of integer indices (hkℓ) by scaling a, b and c appropriately: divide by the largest of the three numbers, and then multiply by the least common denominator. Thus, integer Miller indices implicitly include indices with all rational ratios. The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the lattice planes: they are the only planes whose intersections with the crystal are 2d-periodic. For a plane (abc) where a, b and c have irrational ratios, on the other hand, the intersection of the plane with the crystal is not periodic. It forms an aperiodic pattern known as a quasicrystal. This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices. (Although many quasicrystals, such as the Penrose tiling, are formed by "cuts" of periodic lattices in more than three dimensions, involving the intersection of more than one such hyperplane.) See also Crystal structure Crystal habit Kikuchi line Zone axis References External links IUCr Online Dictionary of Crystallography Miller index description with diagrams Online tutorial about lattice planes and Miller indices. MTEX – Free MATLAB toolbox for Texture Analysis http://sourceforge.net/projects/orilib – A collection of routines for rotation / orientation manipulation, including special tools for crystal orientations. Geometry Crystallography