Crystallographic Sphere Packings

Created by:

• Debra Chait
• Alisa Cui
• Zachary Stier

This work was funded by NSF grant DMS-1802119 and the Rutgers University Math Department at the DIMACS REU at Rutgers University-New Brunswick. Thank you to Professor Alex Kontorovich for his guidance and mentorship on this research.

A crystallographic sphere packing as defined by Kontorovich-Nakamura is a collection of spheres in \(\mathbb{R}^n\) that:

• have mutually disjoint interiors
• densely fill up space, so that any ball in \(\mathbb{R}^n\) intersects the interior of a sphere in the packing
• its limit set is the limit set of a geometrically finite reflection group

As a consequence of the Structure Theorem, all crystallographic packings are generated by the action of a cocluster of spheres on a cluster of spheres, wherein all spheres in the cluster are mutually tangent or disjoint to one another, and are mutually tangent, disjoint, or orthogonal to all spheres in the cocluster.

Kontorovich and Nakamura's Finiteness Theorem proves that there are finitely many commensurability classes of superintegral crystallographic packings, all of which exist in dimension \(n<21\). Kontorovich and Nakamura identified three sources that give rise to crystallographic sphere packings: polyhedra, the extended reflective Bianchi groups, and higher-dimensional quadratic forms. This website systematically catalogues the crystallographic packings that arise from polyhedra on up to seven vertices, all crystallographic packings that arise from the extended Bianchi groups, and all known crystallographic packings that arise from higher-dimensional quadratic forms.

Click here to download the "header" needed to run the code for generating the data in Tables 2-4.

Polyhedral Circle Packings

Polyhedron	Strip Supercluster	Strip Packing	Gramian Matrix	Inversive Coordinates	Bend Matrices	Mathematica File
Tetrahedron					Integral	Code
Square Pyramid					Integral	Code
Triangular Bipyramid					Integral	Code
6v6f_1					See code for bend matrices. Not integral	Code
Pentagonal Pyramid					Not integral	Code
6v7f_1					Not integral	Code
6v7f_2					Integral	Code
6v8f_1					Integral	Code
Octahedron					Integral	Code
7v7f_1					See code for bend matrices. Not integral	Code
7v7f_2					See code for bend matrices. Not integral	Code
7v7f_3					Not integral	Code
7v7f_4					Integral	Code
7v7f_5					See code for bend matrices. Not integral	Code
7v7f_6					See code for bend matrices. Not integral	Code
7v7f_7					See code for bend matrices. Not integral	Code
Hexagonal Pyramid					Integral	Code
7v8f_1					See code for bend matrices. Not integral	Code
7v8f_2					See code for bend matrices. Not integral	Code
7v8f_3					Not integral	Code
7v8f_4					Not integral	Code
7v8f_5					See code for bend matrices. Not integral	Code
7v8f_6					Integral	Code
7v8f_7					Integral	Code
7v8f_8					See code for bend matrices. Not integral	Code
7v8f_9					Not integral	Code
7v8f_10					See code for bend matrices. Not integral	Code
7v8f_11					See code for bend matrices. Not integral	Code
7v9f_1					Not integral	Code
7v9f_2					Not integral	Code
7v9f_3					Not integral	Code
7v9f_4					Not integral	Code
7v9f_5					See code for bend matrices. Not integral	Code
7v9f_6					Not integral	Code
7v9f_7					Not integral	Code
7v9f_8					Not integral	Code
7v10f_1					Integral	Code
7v10f_2					Integral	Code
7v10f_3					Integral	Code
7v10f_4					Not integral	Code
7v10f_5					Not integral	Code
Cuboctahedron					Integral	Code
Pentagonal Prism					Not Integral	Code
Hexagonal Prism					Integral	Code
Octagonal Prism					Not Integral	Code

Bianchi Group Packings

Group	Visualization	Coxeter Diagram	Strip Packings	Gramian Matrix	Inversive Coordinates	Bend Matrices	Mathematica File
Bi(1) {1}{3}							Code
Bi(2) {1}{3}							Code
Bi(3)			None as is due to missed symmetry: requires doubling			None	Code
Bi(5) {3}{4}{3_4}							Code
Bi(6) {1}{3}{4}{3_4}							Code
Bi(7) {3}{4}{3_4}							Code
Bi(10) {1}{3}{4}{7}{8}{9}{1_7}{1_8}{1_9}{3_4}{3_7}{3_8}{3_9}{4_7}{4_8}{4_9}{8_9}{1_8_9}{3_4_7}{3_4_8}{3_4_9}{3_8_9}{4_8_9}{3_4_8_9}							Code
Bi(11) {3}{4}{3_4}							Code
Bi(13) {3}{4}{9}{10}{3_4}{3_9}{3_10}{4_9}{4_10}{9_10}{3_4_9}{3_4_10}{3_9_10}{4_9_10}{3_4_9_10}							Code
Bi(14) {1}{3}{4}{7}{8}{9}{1_8}{1_9}{3_4}{3_7}{3_8}{3_9}{4_7}{4_8}{4_9}{7_9}{3_4_7}{3_4_8}{3_4_9}{3_7_9}{4_7_9}{3_4_7_9}							Code
Bi(15) {3}{4}{7}{8}{3_4}{3_7}{3_8}{4_7}{4_8}{7_8}{3_4_7}{3_4_8}{3_7_8}{4_7_8}{3_4_7_8}							Code
Bi(17) {3}{4}{8}{11}{12}{13}{3_4}{3_8}{3_11}{3_12}{3_13}{4_8}{4_11}{4_12}{4_13}{8_11}{8_12}{8_13}{11_12}{11_13}{12_13}{3_4_8}{3_4_11}{3_4_12}{3_4_13}{3_8_11}{3_8_12}{3_8_13}{3_11_12}{3_11_13}{3_12_13}{4_8_11}{4_8_12}{4_8_13}{4_11_12}{4_11_13}{4_12_13}{8_11_12}{8_11_13}{8_12_13}{11_12_13}{3_4_8_11}{3_4_8_12}{3_4_8_13}{3_4_11_12}{3_4_11_13}{3_4_12_13}{3_8_11_12}{3_8_11_13}{3_8_12_13}{3_11_12_13}{4_8_11_12}{4_8_11_13}{4_8_12_13}{4_11_12_13}{8_11_12_13}{3_4_8_11_12}{3_4_8_11_13}{3_4_8_12_13}{3_4_11_12_13}{3_8_11_12_13}{4_8_11_12_13}{3_4_8_11_12_13}							Code
Bi(19) {3}{4}{3_4}							Code
Bi(21) {3}{4}{9}{11}{3_4}{3_9}{3_11}{4_9}{4_11}{9_11}{3_4_9}{3_4_11}{3_9_11}{4_9_11}{3_4_9_11}							Code
Bi(30) {1}{3}{4}{8}{9}{10}{11}{1_9}{1_10}{1_11}{3_4}{3_8}{3_9}{3_10}{3_11}{4_8}{4_9}{4_10}{4_11}{8_11}{9_11}{10_11}{1_9_11}{1_10_11}{3_4_8}{3_4_9}{3_4_10}{3_4_11}{3_8_11}{3_9_11}{3_10_11}{4_8_11}{4_9_11}{4_10_11}{3_4_8_11}{3_4_9_11}{3_4_10_11}							Code
Bi(33) {3}{4}{7}{10}{11}{12}{13}{15}{3_4}{3_7}{3_10}{3_11}{3_12}{3_13}{3_15}{4_7}{4_10}{4_11}{4_12}{4_13}{4_15}{7_11}{7_12}{7_13}{7_15}{10_11}{10_12}{10_13}{10_15}{11_12}{11_13}{11_15}{12_15}{13_15}{3_4_7}{3_4_10}{3_4_11}{3_4_12}{3_4_13}{3_4_15}{3_7_11}{3_7_12}{3_7_13}{3_7_15}{3_10_11}{3_10_12}{3_10_13}{3_10_15}{3_11_12}{3_11_13}{3_11_15}{3_12_15}{3_13_15}{4_7_11}{4_7_12}{4_7_13}{4_7_15}{4_10_11}{4_10_12}{4_10_13}{4_10_15}{4_11_12}{4_11_13}{4_11_15}{4_12_15}{4_13_15}{7_11_12}{7_11_13}{7_11_15}{7_12_15}{7_13_15}{10_11_12}{10_11_13}{10_11_15}{10_12_15}{10_13_15}{11_12_15}{11_13_15}{3_4_7_11}{3_4_7_12}{3_4_7_13}{3_4_7_15}{3_4_10_11}{3_4_10_12}{3_4_10_13}{3_4_10_15}{3_4_11_12}{3_4_11_13}{3_4_11_15}{3_4_12_15}{3_4_13_15}{3_7_11_12}{3_7_11_13}{3_7_11_15}{3_7_12_15}{3_7_13_15}{3_10_11_12}{3_10_11_13}{3_10_11_15}{3_10_12_15}{3_10_13_15}{3_11_12_15}{3_11_13_15}{4_7_11_12}{4_7_11_13}{4_7_11_15}{4_7_12_15}{4_7_13_15}{4_10_11_12}{4_10_11_13}{4_10_11_15}{4_10_12_15}{4_10_13_15}{4_11_12_15}{4_11_13_15}{7_11_12_15}{7_11_13_15}{10_11_12_15}{10_11_13_15}{3_4_7_11_12}{3_4_7_11_13}{3_4_7_11_15}{3_4_7_12_15}{3_4_7_13_15}{3_4_10_11_12}{3_4_10_11_13}{3_4_10_11_15}{3_4_10_12_15}{3_4_10_13_15}{3_4_11_12_15}{3_4_11_13_15}{3_7_11_12_15}{3_7_11_13_15}{3_10_11_12_15}{3_10_11_13_15}{4_7_11_12_15}{4_7_11_13_15}{4_10_11_12_15}{4_10_11_13_15}{3_4_7_11_12_15}{3_4_7_11_13_15}{3_4_10_11_12_15}{3_4_10_11_13_15}							Code
Bi(39) {3}{4}{9}{10}{3_4}{3_9}{3_10}{4_9}{4_10}{9_10}{3_4_9}{3_4_10}{3_9_10}{4_9_10}{3_4_9_10}							Code

\( -x_0^2+\sum\limits_{i=1}^nx_i^2\)

n	Inversive Coordinates	Coxeter diagram	Gram matrix	Mathematica File
2				Code
3				Code
4				Code
5				Code
6				Code
7				Code
8				Code
9				Code
10				Code
11				Code
12				Code
13				Code
14				Code
15				Code
16				Code
17				Code

\( -2x_0^2+\sum\limits_{i=1}^nx_i^2\)

n	Inversive Coordinates	Coxeter diagram	Gram matrix	Mathematica File
2				Code
3				Code
4				Code
5				Code
6				Code
7				Code
8				Code
9				Code
10				Code
11				Code
12				Code
13				Code
14				Code

\(-3x_0^2+\sum\limits_{i=1}^nx_i^2\)

n	Inversive Coordinates	Coxeter diagram	Gram matrix	Mathematica File
2				Code
3				Code
4				Code
5				Code
6				Code
7				Code
8				Code
9				Code
10				Code
11				Code
12				Code
13				Code