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Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.

A classical example is perfect squaring of the square (https://en.wikipedia.org/wiki/Squaring_the_square) that works for some values of n, the number of pieces. By scaling a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

Obviously, all non-isosceles right triangles admit a self-similar perfect partition for any value of n.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces where n is such that a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.

A classical example is perfect squaring of the square (https://en.wikipedia.org/wiki/Squaring_the_square) that works for some values of n, the number of pieces. By scaling a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

Obviously, all non-isosceles right triangles admit a self-similar perfect partition for any value of n.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces where n is such that a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?

Note: there is some overlap between this post and Cutting polygons into mutually similar and non-congruent pieces but questions 2 and especially 3 seem new.

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.

A classical example is perfect squaring of the square (https://en.wikipedia.org/wiki/Squaring_the_square) that works for some values of n, the number of pieces. By scaling a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

Obviously, all non-isosceles right triangles admit a self-similar perfect partition for any value of n.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces when a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.

A classical example is perfect squaring of the square (https://en.wikipedia.org/wiki/Squaring_the_square) that works for some values of n, the number of pieces. By scaling a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

Obviously, all non-isosceles right triangles admit a self-similar perfect partition for any value of n.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces where n is such that a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually incongruent may be called a self-similar perfect partition.

Obviously, all right triangles that are not isosceles admit a self-similar perfect partition into any positive number n of pieces.

A more difficult and classical example is squaring the square (https://en.wikipedia.org/wiki/Squaring_the_square) where n cannot be any arbitrary integer. By a simple scaling of a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces when a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.

A classical example is perfect squaring of the square (https://en.wikipedia.org/wiki/Squaring_the_square) that works for some values of n, the number of pieces. By scaling a perfectly squared square, all rectangles can be seen to have such partitions at least for values of n for which the square allows such a partition.

Obviously, all non-isosceles right triangles admit a self-similar perfect partition for any value of n.

1. Which other convex planar regions admit self-similar perfect partitions (I don't know about isosceles right triangles)?

2. Are there rectangles that allow self-similar perfect partition into n pieces when a square does not allow perfect squaring into n pieces?

3. Given any n, can one construct planar regions (not necessarily convex) that allow self-similar perfect partitions into n pieces but not into less than n pieces?