What Scientific Term or Concept Ought to be More Widely Known?

Re:What Scientific Term or Concept Ought to be More Widely K... by systoll on Monday January 02, 2017 @01:06AM (#13299391)

I recall doing a proof of the form:

1. Prove the triangle inequality, & extend via induction to show that [A] a straight line is the shortest polygonal path, and that [B] adding a point to a polygonal path can only make it longer.

2. Arc length is defined as the limit of the length of a polygonal approximations to a curve. From (B), we know our approximations would approach the arc length from below.

So if an curve A->B has an arc length, there's a polygon A->B which is at least as short. And from [A] we know there's no polygon shorter than the straight line.

This is weaker than proofs you can do with calculus -- it doesn't prove uniqueness, and it completely ignores non-rectifiable curves, for instance. But I suspect that's what ams6110 is remembering.

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Re:What Scientific Term or Concept Ought to be More Widely K... by wfunction on Monday January 02, 2017 @01:58AM (#13299537)

2. Arc length is defined as the limit of the length of a polygonal approximations to a curve. From (B), we know our approximations would approach the arc length from below.

You call this "geometry"? Since when does geometry involve a function limiting and bounding process? We didn't even know what a limit was until calculus...

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