Dissecting a square into congruent pieces that all touch the centre Edited for clarity:
I thought I had a complete set of solutions to this:

Cut a square into identical pieces so that they all touch the center point.

It became clear, after some discussions, that I was very, very wrong.
There are infinite families of solutions, and a sporadic. So I have two questions:


*

*What do you think is a complete set of solutions?

*What techniques and approaches can I use to prove that the ones I have are all there are?


I hope that's clearer. Thanks.
 A: What kind of cutting is allowed?  There is also a set of solutions starting from dividing the square into eighths with 45-45-90 triangles (assuming reflection is allowed).  You can then change the straight lines into curves as described by Phonon.  Maybe not in the spirit of the question is to consider all the rays from the center to the square.  Partition them into four sets making sure that each set of four rays at 90 degree angles go into distinct sets. $24^{\mathbf{c}}$ solutions!
A: Infinite set 1.  One line through the center at any angle.
Infinite set 2.  Two perpendicular lines through the center at any angle.
Sporadic set 1.  8 45-45-90 triangles.  
More interesting is what identical shapes can make a square.  Non-right triangles with angles $(\pi/8,\pi/4,5\pi/8)$, $(\pi/4,\pi/3,5\pi/12)$, and $(\pi/12,\pi/4,2\pi/3)$ will do it, as shown by Laczkovich. There are also rectifiable polyominoes, with pages by Reid, Clarke, and Friedman.
A: There's an infinite number of solutions. The general form involves a big + with its two lines crossing at the center of the square and perpendicular to each other. When you rotate it at any angle, it will cut the square into four equal parts. The shapes of the four pieces will also be the same provided we can rotate them. Because any angle works, there are infinitely many solutions.
Update: For that matter, because the four lines have the same distance from the center to a side of the square, they could be any identical curves going from the center outward at right angles to each other (this is probably better defined in terms of reflectional and rotational transformations than angles, but I assume the naive understanding of a right angle). As long as these curves always reach a side of a square and don't cross, you've got equal division. This also describes just one family, but it's broader I think.
