Franz Denk published a cute dissection proof that a2 b2 = (a+b)(ab) in 1938. The dissection uses only two pieces (just like the ellipse-to-heart dissection), or three, if you have a piece for the b-square. It is twist-hingeable, though there is nothing to suggest that Denk was aware of that fact. (See the webpage update for Chapter 22.) Below is Denk's photo from the Oberwolfach Photo Collection, which Douglas Rogers brought to my attention.
Copyright 2008, Greg N. Frederickson.
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Last updated December 30, 2008.