The Pythagorean Theorem - Educational Outreach

In the century following Perigal, both Pythagorean Tilingand tiling phenomena in general were extensively studiedby mathematicians resulting in two fascinating discoveries:1] Pythagorean Tiling guaranteed that the existence ofcountless dissection proofs of the Pythagorean Theorem.2] Many previous dissection proofs were in actuality simplevariants of each, inescapably linked by Pythagorean Tiling.Gone forever was the keeping count of the number of proofsof the Pythagorean Theorem! For classical dissections, thecontinuing quest for new proofs became akin to writing thenumbers from 1 ,234, 567 to 1 ,334, 567 . People started toask, what is the point other than garnering a potential entryin the Guinness Book of World Records? As we continueour Pythagorean journey, keep in mind Henry Perigal, for itwas he (albeit unknowingly) that opened the door to thismore general way of thinking.Note: Elisha Loomis whom we shall meet in Section 2.10, publisheda book in 1927 entitled The Pythagorean Proposition, in which hedetails over 350 original proofs of the Pythagorean Theorem.We are now going to examine Perigal’s novel proofand quadrilateral filling using the modern methodsassociated with Pythagorean Tiling, an example of which isshown in Figure 2.29 on the next page. From Figure 2.29,we see that three items comprise a Pythagorean Tilingwhere each item is generated, either directly or indirectly,from the master right triangle.1) The Bride’s Chair, which serves as a basic tessellationunit when repeatedly drawn.2) The master right triangle itself, which serves as an‘anchor-point’ somewhere within the tessellation pattern.3) A square cutting grid, aligned as shown with thetriangular anchor point. The length of each line segmentwithin the grid equals the length of the hypotenuse for themaster triangle. Therefore, the area of each square holeequals the area of the square formed on the hypotenuse.63