Geometry and arithmetic of crystallographic sphere packings

Alex Kontorovich; Kei Nakamura

doi:10.1073/pnas.1721104116

Geometry and arithmetic of crystallographic sphere packings

Proc Natl Acad Sci U S A. 2019 Jan 8;116(2):436-441. doi: 10.1073/pnas.1721104116. Epub 2018 Dec 26.

Authors

Alex Kontorovich^{1

2}, Kei Nakamura³

Affiliations

¹ Department of Mathematics, Rutgers University, New Brunswick, NJ 08854; alex.kontorovich@rutgers.edu.
² School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540.
³ Department of Mathematics, Rutgers University, New Brunswick, NJ 08854.

Abstract

We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in, at most, 20 dimensions.

Keywords: Coxeter diagrams; arithmetic; crystallographic; polyhedra; sphere packings.

Publication types

Research Support, Non-U.S. Gov't
Research Support, U.S. Gov't, Non-P.H.S.