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 In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

to each other, it is then possible to construct two more spheres that are cotangent to four of them, resulting in an infinite

sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...

. The

fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...

is approximately 2.473946 (±1 in the last digit). ApolFrac is software for generating and visualization of Apollonian sphere packing.ApolFrac
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References

Fractals Hyperbolic geometry Packing problems {{hyperbolic-geometry-stub