Pappus' theorem intuition

  1. http://en.wikipedia.org/wiki/Pappus's_centroid_theorem (the second one)

    When we do this with a torus, wouldn't we gaps tiny gaps between the discs on the outer edge? The slices are closer together on the inner edge and this prevents them from getting any closer on the outer edge so shouldn't we have some gaps?
     
  2. jcsd
  3. Note that Pappus theorem originally comes from an integral proof. With integration, each "slice" has an infinitesimally small width, a width at which you could completely disregard the physical laws of "having gaps because the slice has a uniform width". This is the difference between a summation and an integral.

    To further illustrate things, imagine you have a curve and you want the area under it. If you take rectangular slices, you'll never be able to cover the entire area, and you're going to have gaps. However, if these rectangular slices are infinitesimally small, the gaps are going to grow smaller, until you reach a point where the width of the slice is 0, and the unaccounted gaps are therefore also 0.

    Here's an image for further comprehension of the idea. Notice how gaps get smaller. It's the same way with applying Pappus theorem to a torus.
    Riemann Sums GIF
     
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