Earlier quoted context omitted.
You've found two infinite families and one sporadic solution. There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic. And yes, the infinities are uncountable.
puzzles Oh, right - you can vary the shape of the diagonal lines, provided each "arm" has rotational symmetry about its mid-point. OK, so we've got three infinite families and the trivial solution (only one piece - is that your sporadic solution?). I think that might be all: each piece can contain 4, 2, 1 or 1/2 of the original square's corners, since (lacuna) all pieces must contain the same number of corners and fu…
OK - I've now seen a "solution" with 16 "pieces."
My head hurts.