Multiregular Point Systems

Abstract. This paper gives several conditions in geometric crystallography which force a structure X in Rn to be an ideal crystal. An ideal crystal in Rn is a finite union of translates of a full-dimensional lattice. An (r,R) -set is a discrete set X in Rn such that each open ball of radius 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 an (r,R) -set whose points are partitioned into finitely many orbits under the symmetry group Sym(X) of isometries of Rn that leave X invariant. Every multiregular point system is an ideal crystal and vice versa. We present two different types of geometric conditions on a set X that imply that it is a multiregular point system. The first is that if X ``looks the same'' when viewed from n+2 points {yi : 1 \leq i \leq n + 2 } , such that one of these points 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'' condition, which asserts that if X is an (r,R) -set and all neighborhoods of X within distance ρ of each x∈X are isometric to one of k given point configurations, and ρ exceeds CRk for C = 2(n2 +1) log2 (2R/r+2) , then X is a multiregular point system that has at most k orbits under the action of Sym(X) on Rn . In particular, ideal crystals have perfect local rules under isometries.