Was Pythagoras wrong? - related heresies
- Judah L Schwartz
Please look carefully at the applet “Was Pythagoras Wrong?” before exploring this applet. In that applet we established that there is a sequence of paths that come closer and closer the to the hypotenuse of an isosceles right triangle. Each of the paths in the sequence has the same length; i.e., the sum of the lengths of the two perpendicular legs of the triangle. In an isosceles right triangle with hypotenuse of length a, the length of each of the other legs is a Thus the sum of the lengths of the two perpendicular legs is a and the perimeter of the isosceles right triangle is a The perimeter of the shape formed in this applet, i.e., the region bounded by the two perpendicular legs (green) and the “sawtooth hypotenuse” (blue) is a. Challenge: What is the area enclosed by the shape formed? What do you think it means for a curve to be smooth? If the distance between two points that are normally thought of as being a distance a apart can be as much as a apart, what are the implications for other polygons that have such “sawtooth” edges?