Surprising Uses of the Pythagorean Theorem(betterexplained.com)
betterexplained.com
Surprising Uses of the Pythagorean Theorem
http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/
11 comments
this is why i surf hacker news...
"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?
Hi, you don't have to split a triangle into integers. For example, you could have
50^2 = 25^2 + 43.3^2
Strictly speaking, if you have a circle of radius 50, you can take the sine and cosine of any angle and get a possible way to split the length.
50^2 = 25^2 + 43.3^2
Strictly speaking, if you have a circle of radius 50, you can take the sine and cosine of any angle and get a possible way to split the length.
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? =)
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.
You might want to try that math again; it doesn't work for spheres.
What math?
I really like the concept surrounding sorting.
It really helps you visualize why you need to separate your lists into smaller piles before sorting if you can, since it takes the same amount of time to sort a 50 element list as it does sorting a 40 and a 30 both.
It really helps you visualize why you need to separate your lists into smaller piles before sorting if you can, since it takes the same amount of time to sort a 50 element list as it does sorting a 40 and a 30 both.