A 2D shape that can tile itself with smaller, equally sized copies of itself is called a rep-tile and is said to be rep-.[more]
The order of a rep-tile is the smallest number such that the tile is rep-.
Any polysquare that tiles a square is obviously a rep-tile. These trivial cases are omitted here.[less]
A shape that tiles itself using different sizes is called an irregular rep-tile (or irreptile). Irreptiles are investigated in a separate Demonstration.
 M. Gardner, "Rep-Tiles," The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, New York: W. W. Norton & Company, Inc., 2001 pp. 46–58.