Crystallographic Sphere Packings

Created by:

The paper describing this work is available here:

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:

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
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
Square Pyramid
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
Triangular Bipyramid
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
6v6f_1
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
Pentagonal Pyramid
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
6v7f_1
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
6v7f_2
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
6v8f_1
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
Octahedron
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v7f_1
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v7f_2
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v7f_3
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v7f_4
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v7f_5
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v7f_6
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v7f_7
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
Hexagonal Pyramid
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v8f_1
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v8f_2
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v8f_3
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v8f_4
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v8f_5
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v8f_6
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v8f_7
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v8f_8
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v8f_9
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v8f_10
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v8f_11
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v9f_1
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_2
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_3
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_4
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_5
planar graph
supercluster packing Gram matrix inversive coordinates See code for bend matrices.
Not integral
Code
7v9f_6
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_7
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v9f_8
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v10f_1
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v10f_2
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v10f_3
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
7v10f_4
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
7v10f_5
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not integral
Code
Cuboctahedron
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
Pentagonal Prism
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not Integral
Code
Hexagonal Prism
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Integral
Code
Octagonal Prism
planar graph
supercluster packing Gram matrix inversive coordinates one of the bend matrices
Not Integral
Code

Bianchi Group Packings

Group Visualization Coxeter Diagram Strip Packings Gramian Matrix Inversive Coordinates Bend Matrices Mathematica File
vizualization Bi(1)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(2)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
Bi(3) vizualization Bi(3)'s Coxeter diagram None as is due to missed symmetry: requires doubling gram inversive coordinates None Code
vizualization Bi(5)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(6)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(7)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(10)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(11)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(13)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(14)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(15)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(17)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(19)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(21)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(30)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(33)'s Coxeter diagram strip gram inversive coordinates bend matrix Code
vizualization Bi(39)'s Coxeter diagram strip gram inversive coordinates bend matrix Code

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

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

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

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

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

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