Cutting Polygons into Mutually Similar and non-Congruent Pieces

It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) – for example, see https://en.wikipedia.org/wiki/Squaring_the_square

Questions:

  1. Is there a non-rectangular polygon P that can be cut into a finite number of scaled down copies of P such that each copy is of a unique size (unique scale factor)?

  2. Is there a polygon P that can be cut into a finite number of pieces all of which are mutually similar and of different sizes but are not similar to P?