An L-shape is a polyomino with 6 vertices (5 convex, 1 concave).
I am investigating polyominoes that can tile some L-shape.
Two non-square or three square rectangles can be put together to make an L-shape. So any polyomino that can tile a rectangle can tile an L-shape.
Are there any other polyominoes that can tile L-shapes?
Some L-shapes do not tile rectangles (eg. V-pentomino), so these are trivial examples. Are there others?
To summarize, I am looking for polyominoes with more than 6 vertices that can tile L-shapes but not rectangles. (Note that this implies the L-shape region must also not tile a rectangle.)
This is part of a bigger investigation in how the vertices (number and arrangement of convex and concave) affect tileability. For example, it seems to me difficult to find polyominoes with more than $2m$ vertices that tile regions with $2m$ vertices, but not tile regions with less than $2m$ vertices. The problem above is the simplest instance. It also seems almost to good to be true that this is impossible. It is a little bit related to another (difficult, open) problem: whether a reptile can always tile a rectangle.
The bounty expires soon, and since there are no answers I thought I would give some more thoughts on this problem. As I mentioned, this problem is related to the problem of whether all reptiles can tile rectangles. (A reptile is a polyomino that can tile a scaled copy of itself.) All the (polyomino) reptiles we know tile rectangles. Polyominoes that tile rectangles are automatically reptiles too (rectangles can be stacked to create squares, and squares can be stacked to create any scaled polyomino). It seems strange that the converse should be true, being a reptile does not seem to have to do with anything about rectangles.
But I have been wondering if it is not really something like this that is true: "If a polyomino can tile polyomino with all sides larger than some $k$, it can tile a rectangle." The motivation behind this is for a polyomino to tile a scaled version of a polyomino, it needs to be able to tile a corner, and indeed an edge of minimum length, and indeed it needs to tile a straight edge that goes around to the beginning. This is a fairly difficult thing for polyominoes to do, and it may be just as difficult as actually tiling some rectangle. (This $k$ must be suitably large to avoid trivial cases.)
Now if the above is true, it also implies that any polyomino that tiles an L-shape (big enough) tiles a rectangle. So in this way the current question is a specific instance of the more general thing.
