An Interactive Visualization of the Fascinating Book by David Mumford, Caroline Series and David Wright. [ Video | Article ]

The book studies tessellations of the plane made by a pair of generators made of mobius transformations, creating an action consisting of a succession of scaling, roation, translation and circle inversions.

By applying this action to infinity, we complete a tessellation of the plane.

When the circles are tangent, the limit set of the group (intersecton of all circles generated) creates a curve. In this case the limit set is a circle.

Other arrangements of circles + generators yield more complex curves as limit sets, the book calls this one Indra's necklace.

The Apollonian Gasket can be defined as the limit set of a special group of generators.

The Apollonian Gasket is related to other known sets like the Ford circles via a Mobius transformation.

One way to think about the Mobius transformation is by taking a look at rotations in the Riemman Sphere that get conformally mapped into the plane.

The study of limit sets can take us to sets where circles are no longer the seeding shapes used by generators. Here is a "bending" of the Apollonian Gasket.

The Maskit slice (chart at the right) is the boundary where these limit sets become non-discrete. In these cases the sets are doubly cusped.

Use the slider on the right to explore the limit sets at the boundary of the Maskit slice.