# Delone Sets in ℝ^{3} with 2*R*-Regularity Conditions

- 13 Downloads

## Abstract

A regular system is the orbit of a point with respect to a crystallographic group. The central problem of the local theory of regular systems is to determine the value of the regularity radius, which is the least number such that every Delone set of type (*r,R*) with identical neighborhoods/clusters of this radius is regular. In this paper, conditions are described under which the regularity of a Delone set in three-dimensional Euclidean space follows from the pairwise congruence of small clusters of radius 2*R*. Combined with the analysis of one particular case, this result also implies the proof of the “10*R*-theorem,” which states that if the clusters of radius 10*R* in a Delone set are congruent, then this set is regular.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.I. A. Baburin, M. Bouniaev, N. Dolbilin, N. Yu. Erokhovets, A. Garber, S. V. Krivovichev, and E. Schulte, “On the origin of crystallinity: A lower bound for the regularity radius of Delone sets,” Acta Crystallogr. A
**74**(6), 616–629 (2018); arXiv: 1804.05035 [math.MG].CrossRefGoogle Scholar - 2.B. N. Delone, N. P. Dolbilin, M. I. Shtogrin, and R. V. Galiulin, “A local criterion for regularity of a system of points,” Sov. Math., Dokl.
**17**(2), 319–322 (1976) [transl. from Dokl. Akad. Nauk SSSR**227**(1), 19–21 (1976)].zbMATHGoogle Scholar - 3.N. P. Dolbilin, “Crystal criterion and locally antipodal Delaunay sets,” Vestn. Chelyab. Gos. Univ., No.
**3**, 6–17 (2015).MathSciNetGoogle Scholar - 4.N. Dolbilin, “Delone sets with congruent clusters,” Struct. Chem.
**27**(6), 1725–1732 (2016).CrossRefGoogle Scholar - 5.N. P. Dolbilin, “Delone sets in R3: Regularity conditions,” Fundam. Prikl. Mat.
**21**(6), 115–141 (2016).Google Scholar - 6.N. P. Dolbilin and A. N. Magazinov, “Locally antipodal Delaunay sets,” Russ. Math. Surv.
**70**(5), 958–960 (2015) [transl. from Usp. Mat. Nauk**70**(5), 179–180 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar - 7.N. P. Dolbilin and A. N. Magazinov, “Uniqueness theorem for locally antipodal Delaunay sets,” Proc. Steklov Inst. Math.
**294**, 215–221 (2016) [transl. from Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk**294**, 230–236 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar - 8.R. P. Feynman, R. B. Leighton, and M. Sands,
*The Feynman Lectures on Physics*(Addison Wesley, Reading, MA, 1964), Vol. II, Ch. 30.CrossRefzbMATHGoogle Scholar - 9.M. Shtogrin, “Bound on the order of spider’s axis in a locally regular Delone system,” in Geometry, Topology, Algebra and Number Theory, Applications: Int. Conf. Dedicated to the 120th Anniversary of Boris Delone, Moscow, 2010: Abstracts (Steklov Math. Inst., Moscow, 2010), pp. 168–169.Google Scholar