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Given a square and a circle, can you dissect the square into pieces such that, using only similarity transformations, these pieces can be reassembled to form the circle?

Clarifications:

  • YOU CANNOT SCALE PIECES DOWN TO SIZE ZERO!

  • The pieces are not required to be polygonal, but each must have a well-defined boundary, either clearly shown in a diagram or explicitly described in the explanation.

  • All pieces must be used

  • Finite number of pieces

  • Please do not give a solution that uses about 10200 Borel pieces.

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I can do it in

four pieces.

Take the square, and cut out a circle with the same center that doesn't touch the sides. Cut the remainder of the square along a diagonal, without cutting the circle. Take the resulting wedges-with-holes, scale one of them down by a factor of 2, and place it alongside the other like so:
enter image description here
If you had infinitely many of these pieces, you could build a spiral-like shape, by shrinking down a wedge by a factor of 4 and placing it alongside the second piece in a similar fashion, shrinking down another by a factor of 8 and placing it alongside the third, and so on.
Out of the circle, cut a shape similar to the one that would be formed as a result of this infinite process.

The transformation:

Scale down the spiral by a factor of 4, and set it aside. Scale down the wedges so that they fit as the first and second pieces in the spiral's construction, then use the scaled-down spiral to fill in the rest.

Added figure:

figure for the transformation described above

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    $\begingroup$ I don't see a way to do better, unless you allow reveal spoiler pieces to be disconnected, which admits a fairly trivial two-piece solution. $\endgroup$ Commented Jan 7 at 11:05
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    $\begingroup$ Would you consider a piece to be connected if it’s made of two pieces that meet at a single point? $\endgroup$ Commented Jan 7 at 11:25
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    $\begingroup$ I'll leave that consideration to the OP, but my personal inclination is to say "no", since these sorts of puzzles tend to be physical in nature. $\endgroup$ Commented Jan 7 at 17:07
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    $\begingroup$ Hope you like the figure! $\endgroup$ Commented Jan 7 at 17:31
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    $\begingroup$ @Pranay It's perfect, thank you! $\endgroup$ Commented Jan 7 at 23:08
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3 pieces

I made this animation a few years ago. (Probably for this site?! I'm not even sure if this was my idea.)

enter image description here enter image description here

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  • $\begingroup$ Wow. Doesn’t even involve rotations. Nice! $\endgroup$ Commented Jan 13 at 8:52

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