The problem of tiling a region, or the whole plane, using congruent
copies of a single polyomino has been well studied.
Karl Scherer has
introduced the problem of using similar copies of a single polyomino.
(He also considers other shapes besides polyominoes in his privately
published book,
A Puzzling Journey To The Reptiles And Related Animals.)
We consider two tiling problems,
using similar copies of a single polyomino,
tiling a rectangle, and tiling a larger copy of the original shape.
We also suggest several directions for further investigation.