I am considering a unit sphere with center at $(0, 0, 0)$ and spherical triangles congruent to the one with the vertices in $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$. If you allow for triangles to have overlapping sides, then it's obvious that you can put at most $8$ of them. What happens if I want them to be completely disjoint? What is the maximal number of triangles one can fit on a sphere?
Guided by intuition, I thought you could put $7$ of them. But, the best I could do with a bit of imagination is $4$.
If the question appears to be too trivial, I am interested in generalisations in higher dimensions as well. Any ideas or references are appreciated.