Assuming I have a cube of size a which already filled with 8 spheres with radius of a/4. I would like to find the radius of the sphere tangent to all this 8 and fill the hole at the center of cube. Next, I would like to find a smaller sphere which fill the holes again (similar to Apollonian packing). The question is what is the radius of spheres and how much the void space can be reduced in this packing.

  • $\begingroup$ Hint: The distance from the center of the cube to the centers of each of the $8$ spheres is $a\frac{\sqrt3}4$. $\endgroup$ – robjohn Sep 12 '18 at 15:00
  • $\begingroup$ Thanks, but can I assume the radius of internal sphere is $$a\frac{(\sqrt3-1)}4$$? In that case, how I can find the spheres that fill new holes? $\endgroup$ – Bita Sep 12 '18 at 15:07
  • $\begingroup$ Look at the 2D case at Descartes theorem and its generalisation. In an Apolonian packing of spheres holes are filled by spheres that touch to four already existing pair-wise touching spheres and there is a simple relation between their radii. Your initial condition of eight spheres on a cubic lattice do not form that type of cavities yet, and some care needs to be taken. $\endgroup$ – Ronald Blaak Sep 13 '18 at 15:39

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