There is an apocryphal story that someone from the Manhattan Project asked a mathematician how to uniformly distribute 100 points on a sphere. The mathematician replied that it couldn’t be done, and the project leader thought the mathematician was being uncooperative.
If this story is true, the mathematician’s response was correct but unhelpful. He took no initiative but strictly answering the question that was posed to him. Maybe he wasn’t trying to be uncooperative, but he was showing a lack of imagination, a lack of empathy.
Applied mathematicians are continually asked to do things that are impossible, strictly speaking. But rather than state what can and cannot be done exactly and with certainty, we reply by saying what can be done approximately or with high probability.
Here’s a more helpful response:
You can only evenly distribute points on a sphere if the points are the vertices of a regular solid. So you can exactly evenly space 4, 6, 8, 12, or 20 points on a sphere. Oh, also there’s the degenerate case of 2 antipodal points. If you want to put more than 20 points on a sphere, they can’t be exactly evenly spaced—some points will necessarily be closer to their neighbors than others—but there are ways to approximately position points on a sphere.
There are several ways to quantify how evenly points are distributed on a sphere. If you can tell me more about your problem, I can determine which metric best maps to your need. I can try a few ideas and let you know how well they do according to that metric, then you can let me know whether any of the solutions are good enough for your purposes.
To read more about how you might actually distribute points on a sphere as evenly as you can, see a recent article that describes the Fibonacci lattice method, and a variation called the offset Fibonacci lattice.