This paper is a joint paper of Nikolay Dolbilin, Alexey Garber, Undine Leopold and Egon Schulte.

In the paper we complete the proof of the upper bound of the 10R for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, we show that if all 10R-clusters of a Delone set X with parameters (r,R) are equivalent, then X is a regular system.

The paper will be soon available at arXiv.

The related source code is available under supporting files tab below.

Wolfram Mathematica files

• The Wolfram Mathematica source files used for the groups D_4d and S₈ in the Appendix: Wolfram Mathematica notebook (save as .txt and rename to .nb).