Clark Wells, a mathematics professor at Grand Valley State University, presented his paper"Quadrature, the Geometric Mean, Hinged Dissections, and the Purpose of Proof"at the MAA's Mathfest in Hartford, CT on August 2, 2013. As he wrote in his description of the talk,"Mathematicians will generally agree that proofs are a good thing (why else would we be talking about our favorites?) and that rigor is important. But as educators, what is the purpose of proof? I would argue that in a perfect world a proof should not only verify the truth of a proposition, but should give insight into the proposition itself. A sad fact is that proofs often do not give insight, and worse, they can sometimes seem to students as if they were written to deliberately obscure insight. Sometimes, though, you can have both rigor and insight. Among my favorite geometry proofs are quadrature proofs, which I typically discuss in our senior capstone course, The Nature of Modern Mathematics. The idea of quadrature is to create (typically by compass and straight-edge construction) a square that is "the same size" as a given geometrical object. My very favorite geometry proof is the quadrature of the rectangle for several reasons. One is that the side length of the square obtained is the geometric mean of the side lengths of the rectangle, another is that it can be proven using hinged dissections and then animated using GeoGebra, as I will show in my talk, which leads to insight about what quadrature and the geometric mean really are. Furthermore, by taking a theorem due to the ancient Greeks and proving it using modern technology, I can emphasize the connectedness of mathematical ideas across centuries."The particular hinged dissection that Clark discussed was of a rectangle to a square. Such a dissection was illustrated in the Persian manuscript Interlocks of Similar or Complementary Figures, dating to around 1300 C.E. My book does not display this particular hinged dissection, because it is a specific example of the more general Q-slide technique discovered by Harry Lindgren and described in his 1964 book, Geometric Dissections. The dissection converts one quadrilateral to another quadrilateral of equal area and the same angles. (I renamed the Q-slide as the "Q-swing", because the four pieces can be hinged with swing hinges.) It's too bad that Clark wasn't aware of this material as discussed in my books, Dissections: Plane & Fancy, and Hinged Dissections: Swinging & Twisting.
Thank you, Clark, for pointing out the connection of hinged dissections with quadrature, the geometric mean, and the notion of proof.
Copyright 2014, Greg N. Frederickson.
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Last updated March 17, 2014.