Apollonian sphere packing is the three dimensional equivalent of the Apollonian gasket

Apollonian gasket

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...

. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them.

The fractal dimension

Fractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

is 2.473946.

Software for generating and visualization of the apollonian sphere packing: ApolFrac.