Every family of lattice planes can be described by a set of integer Miller indices that have no common divisors (i.e. are relative prime).
Conversely, every set of Miller indices without common divisors defines a family of lattice planes.
If, on the other hand, the Miller indices are not relative prime, the family of planes defined by them is not a family of lattice planes, because not every plane of the family then intersects lattice points.
[2] Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).
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