3 ms·

Considering a Sphere

21 points by ColinWright 5y ago

not2b 5y ago
"We know that for a circle, for some reason, the circumference is the derivative of the area, and similarly for the sphere, the surface area is the derivative of the volume ..." To understand why, imagine the circle or sphere starting at a point and growing. For the circle, the circumference is 2 pi r, so the area of a strip dr wide is (2 pi r) dr. Integrate r from 0 to R and you sum it all up. You get pi r ^ 2. Likewise for the sphere.
- not2b 5y ago
  ... and that's roughly what the article does.
  - ffibdinfk 5y ago
    You're missing the basic mystery, which is "How does this generalize? Why does it work for a square of side 2r, but not a square of side r?"
    - not2b 5y ago
      No, I am not missing it: you need an r that corresponds in some way to the radius: it should be half of the side length, so if you start from the center and expand, you integrate over r. That's why you need the side to be 2r. Unfortunately hacker news doesn't give me a way to draw you a picture to illustrate it. Picture a growing square that goes from a point to the square that you want the area of.
      - ffibdinfk 5y ago
        Really, the business of Euclidean constructions involving apothems and polygons does not touch what's going on here. It is trivial and doesn't generalize. And there's no reason why a square with apothem r is more natural than a square with side r. See Tong's paper.
ffibdinfk 5y ago
The author wants feedback. Well, the article by Tong which is mentioned here https://math.stackexchange.com/a/2092943/124777 https://math.stackexchange.com/a/2092943/124777 gives a deep and surprising reason for the relationship between surface area and volume. The referenced paper is currently available for download at https://www.maa.org/sites/default/files/0746834257453.di020775.02p0287y.pdf https://www.maa.org/sites/default/files/0746834257453.di0207... The same explanation is given by a video by Mu Prime Math on YouTube: https://youtu.be/cCwD38lw4ow https://youtu.be/cCwD38lw4ow IMO arguments involving the apothem of a polygon are obsoleted by Tong's insight. It becomes clear that it's just a coincidence that the "natural" parameterizations of these regular shapes give a neat answer. The basic observation that area scales as the square of a representative length, for any shape, comes into this discussion as well: The Pythagorean theorem: what is it about? Alexander Givental http://math.berkeley.edu/~giventh/papers/eu.pdf http://math.berkeley.edu/~giventh/papers/eu.pdf

21 points by ColinWright 5y ago

"We know that for a circle, for some reason, the circumference is the derivative of the area, and similarly for the sphere, the surface area is the derivative of the volume ..." To understand why, imagine the circle or sphere starting at a point and growing. For the circle, the circumference is 2 pi r, so the area of a strip dr wide is (2 pi r) dr. Integrate r from 0 to R and you sum it all up. You get pi r ^ 2. Likewise for the sphere.

not2b 5y ago
... and that's roughly what the article does.
- ffibdinfk 5y ago
  You're missing the basic mystery, which is "How does this generalize? Why does it work for a square of side 2r, but not a square of side r?"
  - not2b 5y ago
    No, I am not missing it: you need an r that corresponds in some way to the radius: it should be half of the side length, so if you start from the center and expand, you integrate over r. That's why you need the side to be 2r. Unfortunately hacker news doesn't give me a way to draw you a picture to illustrate it. Picture a growing square that goes from a point to the square that you want the area of.
    - ffibdinfk 5y ago
      Really, the business of Euclidean constructions involving apothems and polygons does not touch what's going on here. It is trivial and doesn't generalize. And there's no reason why a square with apothem r is more natural than a square with side r. See Tong's paper.

ffibdinfk 5y ago

The author wants feedback. Well, the article by Tong which is mentioned here https://math.stackexchange.com/a/2092943/124777 https://math.stackexchange.com/a/2092943/124777 gives a deep and surprising reason for the relationship between surface area and volume. The referenced paper is currently available for download at https://www.maa.org/sites/default/files/0746834257453.di020775.02p0287y.pdf https://www.maa.org/sites/default/files/0746834257453.di0207... The same explanation is given by a video by Mu Prime Math on YouTube: https://youtu.be/cCwD38lw4ow https://youtu.be/cCwD38lw4ow IMO arguments involving the apothem of a polygon are obsoleted by Tong's insight. It becomes clear that it's just a coincidence that the "natural" parameterizations of these regular shapes give a neat answer. The basic observation that area scales as the square of a representative length, for any shape, comes into this discussion as well: The Pythagorean theorem: what is it about? Alexander Givental http://math.berkeley.edu/~giventh/papers/eu.pdf http://math.berkeley.edu/~giventh/papers/eu.pdf