Name
Abstract | ||
|---|---|---|
. 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. |
Year | DOI | Venue |
|---|---|---|
1998 | 10.1007/PL00009397 | Discrete & Computational Geometry |
Keywords | Field | DocType |
symmetry group,convex hull | Topology,Combinatorics,Symmetry group,Lattice (order),Ball (mathematics),Convex hull,Isometry,Invariant (mathematics),Isolated point,Delone set,Mathematics | Journal |
Volume | Issue | ISSN |
20 | 4 | 1432-0444 |
Citations | PageRank | References |
6 | 2.03 | 2 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nikolai P. Dolbilin | 1 | 13 | 5.56 |
| J. C. Lagarias | 2 | 563 | 235.61 |
| Marjorie Senechal | 3 | 17 | 6.08 |