A taxonomy of crystallographic sphere packings

JOURNAL OF NUMBER THEORY

卷:207

Article

Chait-Roth, Devora¹  Cui, Alisa²  Stier, Zachary³ 

[1] Queens Coll, New York, NY 11367 USA

[2] Yale Univ, New Haven, CT 06520 USA

[3] Princeton Univ, Princeton, NJ 08544 USA

关键词: Crystallographic sphere packing;    Hyperbolic reflection groups;    Arithmetic groups;    Coxeter diagram;    Vinberg's algorithm;   

DOI  :  10.1016/j.jnt.2019.07.007

来源: Elsevier

【 摘 要 】

This paper seeks to catalogue and examine crystallographic sphere packings as defined by Kontorovich and Nakamura. There exist at least three sources which give rise to crystallographic packings, namely polyhedra, reflective extended Bianchi groups, and various higher dimensional quadratic forms. When applied in conjunction with the Koebe-Andreev-Thurston Theorem, Kontorovich and Nakamura's Structure Theorem guarantees crystallographic packings to be generated from polyhedra in n = 2. The Structure Theorem similarly allows us to generate packings from the reflective extended Bianchi groups in n = 2 by applying Vinberg's algorithm to obtain the appropriate Coxeter diagrams. In n > 2, the Structure Theorem when used with Vinberg's algorithm allows us to explore whether certain Coxeter diagrams in Hn+1 for a given quadratic form admit a packing at all. Kontorovich and Nakamura's Finiteness Theorem shows that there exist only finitely many classes of superintegral such packings, all of which exist in dimensions n <= 20. In this work, we systematically determine all known examples of crystallographic sphere packings. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free