3D Geometry Problem

Question

If we have 4 equal sized spheres with radius $R$ arranged surrounding another smaller sphere such as to make a triangular pyramid from the centers of the $4$ spheres with radius $R$. The radius of smaller sphere is changed just to make a perfect fit when all 4 spheres touch each other and the smaller sphere touches all other spheres.Now call this smaller sphere be in the tetrahedral void of the $4$ spheres.(Actually it will look like a tetrahedron from inside of the void.)Now we fit 4 more smaller spheres similarly between the 4 newly created tetrahedral void. What is the limit of the volume occupies by these spheres when we keep doing this process?

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EDIT

Apparently I have solved this problem but anyways I'll post a solution later, till then consider it as a challenge.

Hint

$r/R\approx0.225$

Answer 1

As the tetrahedral angle is $109^\circ28'$, from this figure with two bigger circles and one smaller one we have:

In $\Delta aoc$: $$ao=R+r,ac=R,\sin(\theta/2)=\frac{R}{R+r},\theta=109^\circ28'$$ Calculating $r\approx0.225R$ or we can take similiar triangles with spheres surrounding that void, whose radius we want and keep on forming a series of volume $\frac43\pi r_i^3$ to get the limit, unfortunately I have not completed that.

Answer 2

See here for pictures and other material on this problem:

http://mathworld.wolfram.com/ApollonianGasket.html

The two-dimensional version of this problem (with a large bounding disk) is called Apollonian circle packing, or Apollonian casket. It turns out that the points not lying in one of the open disks form a set $B$ of measure zero. The set $B$ has a fractal dimension of about $1.3$ which can only be computed numerically; see the references to the papers of Boyd at the above link.

Taking from this it seems that your small balls fill completely (measurewise) a certain volume whose outer boundary remains to be investigated.