The Pythagorean Theorem - Educational Outreach

One of my favorites was Boxel, a game where the playerhad to pack boxes into a variety of convoluted warehouseconfigurations. In a sense, I believe this is precisely whatLiu Hui did: he perceived the Pythagorean Proposition as apacking problem and succeeded to solve the problem by themasterful dismemberment and reassembly as shown above.In the spirit of Liu Hui, actual step-by-step confirmation of the‘packing of the pieces’ is left to the reader as a challengingvisual exercise.Note: One could say that Euclid succeeded in packing two squaresinto two rectangles, the sum of which equaled the square formed onthe hypotenuse.So what might have been the origin of Liu Hui’spacking idea? Why did Liu Hui use such odd-shaped pieces,especially the two obtuse, scalene triangles? Finally, whydid Liu Hui dissect the three squares into exactly fourteenpieces as opposed to twenty? Archimedes (287BCE-212BCE), a Greek and one of the three greatestmathematicians of all time—Isaac Newton and Karl Gaussbeing the other two—may provide some possible answers.Archimedes is commonly credited (rightly orwrongly) with a puzzle known by two names, theArchimedes’ Square or the Stomachion, Figure 2.15 on thenext page. In the Stomachion, a 12 by 12 square grid isexpertly dissected into 14 polygonal playing pieces whereeach piece has an integral area. Each of the fourteen piecesis labeled with two numbers. The first is the number of thepiece and the second is the associated area. Two views ofthe Stomachion are provided in Figure 2.15, an ‘artist’sconcept’ followed by an ‘engineering drawing’. I would liketo think that the Stomachion somehow played a key role inLiu Hui’s development of his magnificent packing solutionto the Pythagorean Proposition. Archimedes’ puzzle couldhave traveled the Silk Road to China and eventually foundits way into the hands of another ancient and great out-ofthe-boxthinker!47