Circle packings are formed by starting with a small set of circles and a rule for filling the space that lies between these circles. By repeatedly applying a rule we get beautiful artwork that also has great mathematics involved. In this we present a collection of some of the art that comes from packing circles with various rules.

Apollonian circle packings are based off of filling a hole between three mutually tangent circles with a unique circle tangent to all three. This has the property that if the original three circles have integer curvatures a, b and c satisfying ab+ac+bc=t^2, for some integer t then all the circles in the packing have integer curvature. Note that this packing has at its root the tangency relationship of the tetrahedron (i.e., 4-sided die).

Octahedral circle packings are based off of filling a hole between three mutually tangent circles with a new set of three mutually tangent circles where each new circle is adjacent to precisely two of the original. This has the property that if the original three circles have integer curvatures a, b and c satisfying ab+ac+bc=2t^2, for some integer t then all the circles in the packing have integer curvature. Note that this packing has at its root the tangency relationship of the octahedron (i.e., 8-sided die).

Icosahedrol circle packings are based off of filling a hole between three mutually tangent circles with a new set of nine circles so that the resulting tangency graph is the skeleton of the icosahedron. This has the property that if the original three circles have curvatures a, b and c satisfying ab+ac+bc=t^2, where a,b,c and t all are in the ring of integers adjoin the golden ratio, then all curvatures are in this same ring. Note that this packing has at its root the tangency relationship of the icosahedron (i.e., 20-sided die).

We can use packing rules with infinitely many circles filling the space between the circles. A particularly lovely rule is based off of the hexagonal grid in the plane (which is a natural next step after the previous three). This has the property that if the original three circles have integer curvatures a, b and c satisfying ab+ac+bc=3t^2, for some integer t then all the circles in the packing have integer curvature.

We can start with any triangulation and form new packings. There are infinitely many possibilities, though some are more interesting than others (i.e., better symmetry). We also can find integer packings which do not have any odd cycles in the tangency graph. These can be used to form circle packings with gears that will actually turn.

We can also consider sphere packings. There are two beautiful packings based off of the tangency relationships of the 4-dimensional simplex and of the 4-dimensional cross polytope (also known as Apollonian and orthoplicial sphere packings, respectively). These packings can again have all curvatures integer if the initial seeds satisfy simple conditions. While it is hard to see the packing of the spheres, we can easily look at cross sections, getting us back to circle packings! Many of the above packing come as different cross sections, so that these different circle packings are not so different after all, but rather a different perspective on the same spherical packings.

The simplex packing as we rotate around z

The simplex packing as we move perpendicularly to z

The cross polytope packing as we rotate around z

The cross polytope packing as we move perpendicularly to z

In 2016 Steve Butler designed a series of three coasters for Department of Mathematics at Iowa State University based on some of these circle packings which also incorporated additional information (namely number theoretic properties; Hausdorff dimension; and matrices which generate the group to which the circle packings correspond).

Some coasters are still available and can be obtained by visiting the Department of Mathematics at Iowa State University, or by going to the Graduate Fair at the Joint Math Meetings.