Multiregular Point Systems (original) (raw)

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 \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(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.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, 42 Vavilov St., 117466 Moscow, Russia nikolai@dolbilin.mian.su , , , , , , RU
    N. P. Dolbilin
  2. Room 2C-373, AT&T Labs—Research, 180 Park Avenue, Florham Park, NJ 07932-0971, USA jcl@research.att.com , , , , , , US
    J. C. Lagarias
  3. Department of Mathematics, Smith College, Northampton, MA 01056, USA senechal@minkowski.smith.edu, , , , , , US
    M. Senechal

Authors

  1. N. P. Dolbilin
  2. J. C. Lagarias
  3. M. Senechal

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

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Dolbilin, N., Lagarias, J. & Senechal, M. Multiregular Point Systems.Discrete Comput Geom 20, 477–498 (1998). https://doi.org/10.1007/PL00009397

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