[pozorvlak]

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 further subdividing the corners (into 1/3s, say) would mean some pieces don't touch the centre (another lacuna).

[ColinWright]

OK, that's now the set of solutions I've got. You've also gone some way to showing them to be complete.

More to do, though.

And now do it for an equilateral triangle.

[ColinWright]

OK - I've now seen a "solution" with 16 "pieces."

My head hurts.