Curry Triangle Paradox: Rearranging Pieces to Figure of Less Area?

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SUMMARY

The Curry Triangle Paradox illustrates how rearranging pieces can lead to a figure that appears to have less area than the original. The discussion highlights that the green and red triangles in the paradox are not similar due to differing angles and side ratios, specifically 2/5 and 3/8. This discrepancy indicates that the construction is impossible if absolute care is taken. Additionally, the Banach-Tarski Paradox is mentioned as a mathematically sound example that further complicates the understanding of area and volume in geometry.

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Because there are liberties being taken with the pieces - you are presming that everything fits exactly, yet the green triangle and the red triangle are not similar triangles as they ought to be. The ratios of the non-hyptoneuse sides 'appear' to be 2/5 and 3/8 so although it looks right, it isn't. if we were to take absolute care in constructing it we would see it is an impossible construction.

if you want to really mix your head up there is a genuine paradox called Banach-Tarskl, that is mathematically 'sound'
 
Right. Solving the triangles shows that the green triangle and the Red triangle have different angles and are not similar. Also, put a straight edge on the hypotenuse of each large triangle; one is concave, the other convex. This is also the same method that the government has used for over 200 years to calculate the budget.
 
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