$\pi$ -ing the square

Find the minimum amount of pieces that the square with side $1$ needs to be cut into, if a rectangle with side $\pi$ is to be formed. All the pieces need to be used, with no overlaps. If no solution exists, prove so.

NOTE: I posed this as a problem a few hours ago, thinking I have had a solution. Once I started writing it, I realized that I have had a fault in my proof. I did not manage to find a correct proof on my own, so I offer it as a collaborative adventure into the plena.

#Geometry

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Check this out

Tarksi's Circle Squaring Problem

Michael Mendrin - 5 years, 3 months ago

Hi, I posed a different problem. I am quite sure about the solution of my problem - $5$ . On Dissections, there is nicely depicted the dissection. However, they fail to prove that it is necessary to have as many pieces. That is what I am missing.

Stanislava Sojáková - 5 years, 3 months ago

Yes, I realize there's a difference between turning an unit circle into an unit square, or into a rectangle with sides $(1,\pi)$ Nevertheless, the nature of the proof is suggested. Probably far beyond the scope of a Note in Brilliant.org.

As a matter of fact, once one has any rectangle, it's almost a trivial exercise to dissect it into a finite number of pieces, and reform the pieces to make a square. This would dispute Tarski's findings, unless one accepts the idea of dissecting the circle into pieces that "cannot be cut from the circle with scissors".

Rectangle Into Square

Michael Mendrin - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

π\piπ-ing the square

Comments

$\pi$ -ing the square