Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should start on Monday, and end on Sunday.
4. Unflag the two options "Notify me of new top level comments on this article" and "
You can come arbitrarily close by choosing any tiling shape and making it as small as necessary.
No, because the degree of failure of the tiling is judged against the area of the covering shape.