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Apollonian sphere packing is the three-dimensional equivalent of the Apollonian gasket. 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 is approximately 2.473946 (±1 in the last digit).
Software for generating and visualization of the apollonian sphere packing: ApolFrac.
References
Apollonian sphere packing Wikipedia(Text) CC BY-SA