@srslypascal Stimme Dir grundsätzlich zu (aber nicht der Formulierung "menschlicher Bodensatz")

@jnfrd @Croco @geist

@sibrosan
Nobody criticizes your initial response from Saturday~7am (CET). divbyzero even acknowledged your point on Saturday~3:30pm when he realized that you were assuming planar cuts.

But the end of your Sunday~8am post sounds to me like finger-pointing at divbyzero, just because he didn't mention the conic cuts initially.

As a matter of taste, I prefer cooperative and productive discussions where people try to understand each other instead of finger-pointing at (repairable) inaccuracies.

@sibrosan
The post you were replying to was about the missing explicit requirement for conic cuts. With conic cuts one doesn't need any thickness limits or approximation as we have seen. (Or have you mixed up my remarks on "wrong" with my remarks on "negligible"?)

If you are not able or not willing to distinguish between
- plain wrong statements and
- oversimplified and thus technically incorrect, but easily repairable statements,
then we will not find a common ground.

Have a nice week!

@sibrosan @divbyzero

In more formal "Weierstraß style":

For *whatever* relative difference ε > 0 we might be willing to neglect, there is a thickness δ > 0 such that

|vol_planar - vol_conic| / vol_conic < ε

for all shells with thickness δ or thinner, that is, for any r with R-δ <= r < R.

@sibrosan @divbyzero

And when I called the cut slopes "negligible" for thin shells, this was not about a particular amount or percentage of difference I'm willing to neglect, but it was an informal/sloppy way to say that the difference between planar and conic cuts decreases faster than linearly if we make the shell thinner. So even the relative difference approaches zero by making the shell sufficiently thin.

@sibrosan
Good that we agree on the conic cuts now. But sorry, I can't resist to discuss the words "wrong" and "negligible" a bit.

I wouldn't call @divbyzero's initial statement "wrong". The conic cutting can be seen as a less interesting detail not worth mentioning in that context. (It might even have confused some readers.)

But yes, it can be misunderstood easily if one implicitly transfers the planes from Archimedes' statement on 2D surfaces to the 3D shells.

@sibrosan @divbyzero

The precise formula here is actually not important.
The important observation is that this volume depends only on the *area* of P. It depends neither on the shape of P nor on its position on the outer sphere.

So if you do steps 1 to 4 for two slices P and P' of the outer sphere with the same area, the resulting "rings" will have the same volume.

So, does this convince you?

@sibrosan @divbyzero
4. Notice that s is contained in S. If we remove s from S, the remaining solid is the part of the shell between P and the corresponding piece of the inner sphere. It has the volume

volume(S) - volume(s)
= volume(S) * (1 - (r/R)^3)
= area(P) * R / 3 * (1 - (r/R)^3)

@sibrosan @divbyzero

2. Construct a solid S consisting of all the points between C and some point of P. We get:

volume(S) := area(P) * R / 3.

3. Construct another solid s by contracting S towards C by the factor r/R. We get:

volume(s) = (r/R)^3 * volume(S)