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?