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
CRk 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.
Received September 13, 1996, and in revised form September 27, 1996, February 6, 1997, and May 7, 1997.