Cutting a rectangle into an odd number of congruent pieces

Golomb's book Polyominoes has a section on this. Call the smallest odd number of copies of a polyomino that can tile a rectangle its "odd-order". Then Golomb says there are polyominoes of odd order 1, 11, and 15+6t for all $t \ge 0$. The polyomino of odd order 11 is due to Klarner [1], and is illustrated here by Michael Reid.

Reid has lots of pictures of tilings of rectangles with polyominoes. In particular the 15+6t family can be seen: here are polyominoes with odd-order 15, odd-order 21, odd-order 27, and so on. Reid has shown [3] that other odd orders exist, including 35, 49, and 221, but I don't know if there's a general pattern.

Finally, Stewart and Wormstein [2] proved that polyominoes of order 3 do not exist. (Stewart's book Another Fine Math You've got Me Into suggests that Wormstein is a fictional character.)

[1] David A. Klarner, Packing a rectangle with congruent N-ominoes, J. Combin. Theory 7 (1969) 107-115, [2] Stewart, Ian N. and Wormstein, Albert. Polyominoes of order 3 do not exist. J. Combin. Theory Series A 61 (1992) 130-136.
[3] Michael Reid. Tiling Rectangles and Half Strips with Congruent Polyominoes. J. Combin. Theory Series A 80 (1997) 106-123.

I am posting the 11 pieces solution shown in the article cited by Michael (it is not freely available online).

     (source)

This is the smallest known number of pieces. Some remarks:

• The question is open for 5, 7, or 9 pieces. Get your pencils!
• Everything so far is with polyominoes. Any suggestion with more complicated shapes?
• Unlike the other solution I posted, this one cannot be resized along the $x$ or $y$ axis.

Ok. It's easy to see that rectangle cannot be cut into odd number of equal triangles: Monsky proved [here] that a square cannot be cut into an odd number of triangles with equal areas. But if we have rectangle divided into odd number of equal triangles, we could shrink it in one direction and get a square divided into an odd number of triangles with equal areas - contradiction.

Update: If the tile could be cut into odd number of triangles with the same area, then obvious that rectangle cannot be cut into odd number of such tiles. Example: