Packability of Five Spheres on a Sphere Implies Packability of Six

* . . . f Whlle trylng to determlne how many nonoverlapplng spheres of radlus r2 can be tangent to (and outside of) a sphere of radius r1, we discovered a startling fact: If r1 and r2 are such that it is not possible to pack (i.e., place) six spheres as described, then it is not possible to pack five spheres either; the maximum number for which one can hope drops from six to four, skipping five, as rJr1 increases from 1 + 4. Stated alternately, whenever r1 and r2 are such that five spheres can be packed, then it is always possible to rearrange the packing to make room for a sixth sphere. We believe that this result is of interest because it can be understood by people with little mathematical background, and at the same time it illustrates how counterintuitive mathematics can be at times. The proof also nicely illustrates the usefulness of using different coordinate systems, among other things. The idea of packing spheres around another sphere has its beginnings from an apparent conversation between Isaac Newton and t)avid Gregory in 1694. The question that arose between them was: Can a rigid material sphere be brought into contact with 13 other spheres of the same size? Gregoxy thought the answer was yes, while Newton thought no. 180 years later in 1874, Newton's answer was shown to be the case [1]. Our results follow from the following two-part theorem.