Lagarias recognized the pattern as an example of an Apollonian packing a type of arrangement named after a Greek mathematician who studied tangent circles more than 2,000 years ago. Such an arrangement is created by repeatedly filling the spaces between touching circles with smaller touching circles. In Wilks' case, the Apollonian packing was made up of circles with integer curvatures. Such circle packings can be described by the curvatures of the first four circles that go into the pattern. Circles can be packed together in a variety of ways.
Colored circles highlight an interesting number pattern found among the curvatures.
Although some patterns are highly symmetric, others have no symmetry at all. To imagine these two realms, consider a triangle. If you draw any triangle on a flat sheet of paper and add up its three angles, the result is always 180 degrees. When you draw a triangle on the surface of a sphere, however, the angles invariably add up to more than 180 degrees a characteristic of spherical geometry. In hyperbolic geometry, represented in two dimensions by a saddle-shaped surface, the angles of a triangle add up to less than 180 degrees.