Editing Miller index (section)

==Integer versus 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'', ''b'' and ''c'' have [[rational number|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 plane]]s: they are the only planes whose intersections with the crystal are 2d-periodic.

For a plane (abc) where ''a'', ''b'' and ''c'' have [[irrational number|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]].)