MULTIREGULAR POINT SYSTEMS

This paper gives several conditions in geometric crystallography which force a structure X in R-n to be an ideal crystal. An ideal crystal i n R-n is a finite union of translates of a full-dimensional lattice. A n (r, R)-set is a discrete set X in R-n such that each open hall of ra dius r contains at most one point of X and each closed ball of radius R contains at least one point of X. A multiregular point system X is a n (r, R)-set whose points are partitioned into finitely many orbits un der the symmetry group Sym(X) of isometries of R-n that leave X invari ant. Every multiregular point system is an ideal crystal and vice vers a. We present two different types of geometric conditions on a set X t hat imply that it is a multiregular point system. The first is that if X ''looks the same'' when viewed from n + 2 points {y(i) : 1 less tha n or equal to i less than or equal to n + 2}, such that one of these p oints is in the interior of the convex hull of all the others, then X is a multiregular point system. The second is a ''local rules'' condit ion, which asserts that if X is an (r, R)-set and all neighborhoods of X within distance rho of each x is an element of X are isometric to o ne of k given point configurations, and rho exceeds C Rk for C = 2(n(2 ) + 1)log(2)(2R/r + 2), then X is a multiregular point system that has at most k orbits under the action of Sym(X) on R-n. In particular, id eal crystals have perfect local rules under isometries.