"Throughout our school life we think the Pythagorean Theorem is about triangles and geometry." Some of us still do. What he is talking about is Pythagorean triples. Well, maybe. Energy, for example, does not have to be an integer, so it's not really about Pythagorean triples. What is the point?
Fair enough. Still, what is the point of the article? Some additive quantities are squares of other quantities, you add them, and it looks like the Pythagorean Theorem, without triangles... Is that it?
Two main points, mostly to look at this old theorem in new ways.
1) The sides of a triangle (3-4-5) can represent portions of any shape. For example, circle area (radius 3) + circle area (radius 4) = circle area (radius 5).
Instead of radius, you could pick diameter, circumference, or any line segment and the relationship would hold.
2) You can use the Pythagorean Theorem to split any squared quantity into two smaller ones. This can yield surprising insights; is it immediately clear that a list of 50 takes as long to bubble sort as a list of 30 and 40? =)
There may be a point here, but I am still uncomfortable with the use of the Pythagorean theorem here. Even if applied to areas only, it seems like a stretch. In fact with this approach one can suggest other identities. For example, if you replace squares with cubes (or circles with spheres), you will have a^3+b^3=c^3. And this clearly has nothing to do with the Pythagorean theorem.
