So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal abstract arguments are welcome as well! I want to invoke the unit quaternions here, but I can't think of a particularly simple way to do it.
If this question is too "soft" I sincerely apologize.