Multiplicative congruential generators, their lattice structure, its relation to lattice-sublattice transformations and applications in crystallography
Abstract
An analysis of certain types of multiplicative congruential generators - otherwise known for their application to the sequential generation of pseudo-random numbers - reveals their relation to the coordinate description of lattice points in two-dimensional primitive sublattices. Taking the index of the lattice-sublattice transformation as the modulus of the multiplicative congruential generator, there are special choices for its multiplier which induce a symmetry-preserving permutation of lattice-point coordinates. From an analysis of similar sublattices with hexagonal and square symmetry it is conjectured that the cycle structure of the permutation has its crystallographic counterpart in the description of crystallographic orbits. Some applications of multiplicative congruential generators in structural chemistry and biology are discussed.