• A Taxonomy of Crystallographic Sphere Packings

• Project Year: 2018
• REU Student (s):   Debra Chait | Macaulay Honors College at Queens College NY   |   Alisa Cui | Yale University CT  
• Student 1 Institution: Macaulay Honors College at Queens College
• Student 2 Institution: Yale University
• Project Mentor: Alex Kontorovich
• Project Mentor Area: Mathematics
• Project Abstract: The Apollonian circle packing, generated from three mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. Recently, Kontorovich and Nakamura introduced the notion of crystallographic sphere packings, n-dimensional packings of spheres with symmetry groups that are isometries of Hⁿ⁺¹, which can be related to configurations of planes in Hⁿ⁺¹ for various quadratic forms in n+2 variables. 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<21. In this work, we systematically determine and enumerate crystallographic sphere packings arising from polyhedra on up to seven vertices, all crystallographic packings arising from Bianchi groups, and all known examples of crystallographic packings arising from higher dimensional quadratic forms.