This paper gives several conditions in geometric crystallography which force a structure X in R n to be an ideal crystal. An ideal crystal in R n is a finite union of translates of a full-dimensional lattice. An (r, R)-set is a discrete set X in R n 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 R n 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 {y i : 1 ≤ i ≤ 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 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, ideal crystals have perfect local rules under isometries. 478 N. P. Dolbilin, J. C. Lagarias, and M. Senechal

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