Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere Consider the following problem:
How many regular tetrahedra of edge length 1 can be packed inside a unit sphere with each one has a vertex located at the origin?
The answer is at least 20, forming an icosahedron. On the other hand, it is at most 22, which can be shown by dividing the surface area of the unit sphere by that of a spherical triangle generated from the three vertices of a tetrahedron. Can further progress be made? Thanks in advance.
Edit: as suggested by Wlodek Kuperberg, an equivalent formulation is: 
What is the maximum number of equilateral spherical triangles of edge length $π/3$ that can be packed on the unit sphere?
 A: I'm not sure how to answer this question, but I'll suggest another approach to get an upper bound. 
Considering the problem of packing equilateral $\pi/3$ triangles on a unit sphere, one may convert this to a packing problem in the unit tangent bundle $T^1(S^2)$. Take a unit vector $v$ on the equilateral triangle (say at the center, pointing toward a vertex). Then the unit vector associated to a disjoint equilateral triangle will lie outside of some region $R\subset T^1(S^2)$. The group of rotations $SO(3)$ acts transitively on $T^1(S^2)$, homeomorphic to $\mathbb{RP}^3$. There is a biinvariant metric on $SO(3)$ which we may therefore transfer to $T^1(S^2)$. Then we get a region $R'\subset R$ of half size with respect to this metric surrounding the vector $v$ at the center of a triangle which must be disjoint from the $R'$ regions surrounding the vectors of the other equilateral triangles in some packing. If one could estimate the volume of $R'$, then one would be able to get a packing estimate. I don't know if this gives an improvement, but it's possible that it could given that it takes into account the orientation of triangles, not just their area. The region $R'$ should be described piecewise by algebraic equations, and hence one ought to be able to compute it to arbitrary approximation to get an estimate of the volume of $R'$.  
