5

How can I calculate the maximum number of points that can be placed on the surface of a unit sphere, if any two points wont be closer than 1 unit?

Taner
  • 247
  • 1
    Just to make sure everyone's clear, are we measuring distance between two points using a straight line, or the length of the shortest line drawn between the two on the surface? – Dennis Meng Oct 11 '13 at 00:44
  • We are measuring along the surface of the sphere. – Taner Oct 11 '13 at 00:47

1 Answers1

4

This is essentially the generalized Kissing Number Problem, where the center sphere and outer spheres can have two different radii. If you are measuring distance along a chord between the points, then it is exactly the "classical" version of the problem and the answer is known to be 12. For other values of the outer sphere radii, the question is to my knowledge open, and considered hard. More info here: https://mathoverflow.net/questions/41844/optimal-packing-of-spheres-tangent-to-a-central-sphere

Community
  • 1
user7530
  • 49,280
  • I checked the topic from many sources. As you said the answer is 12 if the center and outer spheres have the same radius. But there is just a little more room needed to fit the 13th one in. In this question, the outer spheres have a smaller radius than the center one. Does this mean the 13th one will probably fit in? – Taner Oct 11 '13 at 12:05
  • Maybe? I haven't checked the formulas but they should give you the "expected" number of spheres that fit – user7530 Oct 11 '13 at 16:46
  • The problem is, there is no formula! I couldn't find any. There is only the values for different number of dimensions, when the spheres have the same radius. – Taner Oct 11 '13 at 22:03