# Sphere Covering Problem

Is it possible that one can cover a sphere with 19 equal spherical caps of 30 degrees(i.e. angular radius is 30 degrees)? A table of Neil Sloane suggests it is impossible, but I want to know if anyone could give some theoretical evidence supporting this suggestion.

P.S. L.Fejes Toth gives a quite sharp bound of covering radius in his book "Lagerungen in der ebene auf der Kugel und im Raum".The covering radius $r_n$ is bounded by $\cos{r_n}\leq\frac{1}{\sqrt{3}}\cot{\frac{n\pi}{6(n-2)}}$,where n is the number of spherical caps. But when n=19,the inequality is not strong enough to get what I want.($r_{19}\geq 29.4^{\circ}$)

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That page says "We give putatively optimal coverings" -- so I suspect these are just the best they found? – joriki Jun 21 '12 at 11:36
@joriki:The word "putatively" is subtle,I must say. Because Neil Sloane says "For example, we have found conjecturally optimal packings of N spherical caps on a sphere in n dimensions...and coverings..." in his book "Sphere Packings,Lattices and Groups". – y zh Jun 21 '12 at 11:51
Just curious, what do you need this for? – user1708 Jun 21 '12 at 14:33
@Broflowski:I want to solve a sphere covering problem in $\mathbb{R^3}$,and this is the difficulty I am trying to overcome. – y zh Jun 21 '12 at 14:42
Just curious, do you have a link (in English?) to the proof of the $r_n$ bound you mention? Does it scale up to higher dimensions than 3? – Thomas Ahle Oct 19 '15 at 9:08