# Does the 13th sphere fit in?

I've been searching for a formula, but couldn't find any. Here is the question:

What is the highest number of equal nonoverlapping spheres that touch a unit sphere? The distance between the points that the outer spheres touch to the inner sphere wont be any smaller than 1 unit, and this distance measuring will be along the surface of the inner sphere.

• That only explains when all the spheres have the same size. I don't think that's the case here. The outer ones have a smaller radius if I'm not mistaken. – Taner Oct 12 '13 at 17:58
• What's the role of the "outer spheres" if they're allowed to be smaller than the inner sphere? Are you just asking about how many points can be placed on the surface of a sphere, such that the distance (angle) between any two points is at least one radian? – mjqxxxx Oct 12 '13 at 18:04
• @Taner You have to state the radius of the outer ones then. Else, we could have many many extremely small spheres. Note that I don't think this is a solved question for any other values of $r$ that are not very close to 1. – Calvin Lin Oct 12 '13 at 18:05
• @mjgxxxx Yes that's what i'm trying to say, and I'm sorry for my bad english. – Taner Oct 12 '13 at 18:12
• en.wikipedia.org/wiki/Tammes_problem – Will Jagy Oct 12 '13 at 18:15

One radian is about $57.29577951^\circ.$ The maximum mutual distance of 13 points on the unit sphere is about $57.1367^\circ,$ which is very slightly smaller. As a result, it is not possible to place 13 points on the unit sphere, such that the angular distance between any two of the points is at least one radian.

See the first three pages in Musin and Tarasov

This is generally called the http://en.wikipedia.org/wiki/Tammes_problem

They refer to chapter 6, section 4 of L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und in Raum, Springer-Verlag,1953; Russian translation, Moscow, 1958

• Thank you very much. I really appreciate it. – Taner Oct 12 '13 at 18:46

The kissing number in three dimensions is $12$, but the upper bound is not trivial.

• When they all have the same size, the answer is 12. But, there is a lot of space left. I think in this question, the outer spheres are a bit smaller. But I can't figure out if they are small enough for a 13th one to fit in. – Taner Oct 12 '13 at 18:00
• @Taner, what question are you talking about? You give no indication at all of any radius for the 12 or 13 outer ones, other than the same radius as the central one. Read what you posted. – Will Jagy Oct 12 '13 at 18:04
• en.wikipedia.org/wiki/Tammes_problem – Will Jagy Oct 12 '13 at 18:16