# zbMATH — the first resource for mathematics

Multiregular point systems. (English) Zbl 0915.51017
Summary: This paper gives several conditions in geometric crystallography which force a structure $$X$$ in $$\mathbb{R}^n$$ to be an ideal crystal. An ideal crystal in $$\mathbb{R}^n$$ is a finite union of translates of a full-dimensional lattice. An $$(r,R)$$-set is a discrete set $$X$$ in $$\mathbb{R}^n$$ 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 $$\mathbb{R}^n$$ 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 $$\rho$$ of each $${\mathbf x}\in X$$ are isometric to one of $$k$$ given point configurations, and $$\rho$$ 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 $$\text{Sym}(X)$$ on $$\mathbb{R}^n$$. In particular, ideal crystals have perfect local rules under isometries.

##### MSC:
 51P05 Classical or axiomatic geometry and physics (should also be assigned at least one other classification number from Sections 70-XX–86-XX) 82D25 Statistical mechanical studies of crystals 51F15 Reflection groups, reflection geometries 51M05 Euclidean geometries (general) and generalizations
Full Text: