A puzzle: dissect a square into congruent pieces, all touching the center

There's a math problem I've been working on for some time that looks like it's finally been solved. To explain it to people I usually start with a warm-up:

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

As I say, this is a warm up. It's fairly easy to do, and serves to introduce the ideas involved. So we do that, and we move on.

Over the weekend, though, I wondered how many solutions there are to the above problem. I rapidly came up with a small number, and was reasonably content.

Then a friend showed me another. Then another friend found another. Now I have rather more than I thought or expected. Today, a work colleague came up with yet another.

How could I miss so many? What are they all?

I invite the HN community to explore this puzzle/problem. Maybe I still haven't got all the answers.

Who will start me off?

Yes, but simply saying there are infinitely many solutions is not enough. There are infinitely many solutions, but there is more you can say.

It's a shame this will sink without making it to the front page - I'd really like to get the HN input before writing the blog post.

So you have two infinite families - the generality is what I originally missed.

There are more ...

That's one solution. There are more ...

Suppose the square can be dissected into n pieces. Then any symmetry of the square induces a permutation of those n pieces. Hence, the symmetry group of the square (D_8) must be embeddable as a subgroup of S_n. Hence there is no dissection into three pieces, since |D_8| = 8, which does not divide |S_3| = 6.

I suspect this line of argument can be taken further, but wanted to post it before someone else did :-)

Edit: no, that doesn't work, because we already have a dissection into two pieces, and |S_2| = 2. Dammit. Not all of the symmetries of the square must map pieces to pieces.