The Pythagorean Theorem - Educational Outreach

The middle square (minus the donut hole) is still plainlyvisible and nothing has changed with respect to size ororientation. Moreover, in doing so, we have freed up fourplaying pieces, which can be used for further explorations.If we use the four lighter pieces to experiment withdifferent ways of filling the outline generated by the fourdarker pieces, an amazing discover will eventually manifestitself—again, perhaps after a few hours of fiddling andtwiddling or, perhaps after several years—Figure 1.9.Note: To reiterate, Thomas Edison tried 4000 different light-bulbfilaments before discovering the right material for such anapplication.Figure 1.9: A Discovery Comes into ViewThat the ancient discovery is undeniable is plain fromFigure 1.10 on the next page, which includes yet anotherpattern and, for comparison, the original square shown inFigure 1.7 comprised of all eight playing pieces. The 12 thcentury Indian mathematician Bhaskara was alleged tohave simply said, “Behold!” when showing these diagramsto students. Decoding Bhaskara’s terseness, one can createfour different, equivalent-area square patterns using eightcongruent playing pieces. Three of the patterns use half ofthe playing pieces and one uses the full set. Of the threepatterns using half the pieces, the sum of the areas for thetwo smaller squares equals the area of the rotated square inthe middle as shown in the final pattern with the threeoutlined squares.23