Another 3D-printing suggestion from Colm Mulcahy:
Archimedes proved that if you slice a sphere of radius r with two planes a distance h apart, the surface area is 2πrh—the same as a cylinder of the same radius and height. So, it doesn't depend on where the slicing occurs.
1/2
Idea: Produce 3D-printed slices of a (hollow) sphere, all of the same height. Print with 100% fill. They should all weigh the same.
Bam!
I propose you try it with a hollow sphere of 30.0 cm diameter and a shell thickness of 14.9 mm, printing slices of 1mm thickness, one taken near the middle, and one just at the top of the sphere.
If you did it with the dimensions I suggested, you would find that the slice from the middle weighed much more than the slice from the top.
In your case (with a much thinner shell) the difference might be small enough to dismiss as measurement error, but if accurately printed and weighed it will be there.
With a thin shell the slopes of the slicing surfaces are negligible.
With a thicker shell (outer diameter D and inner diameter d) such a surface should have a slope such that the outer edges have vertical distance D/n and the inner edges have vertical distance d/n. Thus a slicing surface is a part of a cone with its apex at the center of the spheres. This way the slices should have the same weight again.
Edit: Ok, I was too slow. You already wrote essentially the same.
With a thick shell it's also negligible, it just depends on how much you are willing to neglect...
Anyway, my point is that the claim that the weight/volume of any slice is the same, is simply not true, no matter how thin the shell.
I think I did get your point. There are differences in weight/volume with planar cuts. (And the diffs are small for thin shells and get bigger for thick shells.)
But these differences go away *completely* with the conic cuts.
I don´t doubt that you can adjust the inner shape of the slices to get them the exact same volume. Whether the cone cut, as you describe, does that, i'm not so sure. You're a mathematician, right? Can you prove it?
@sibrosan @divbyzero
Well, I'm not a mathematician, but I can prove it. I don't know into how much detail a proof must go to convince you, but here is an attempt:
We have a shell delimited by an outer sphere with radius R and an inner sphere with radius r. The spheres have the same center C.
(In your example R = 150 mm and r = 1 mm, but we need not restrict ourselves to these values.)
1. Select some piece P of the outer sphere.
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)
@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)
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?
Yes, I can see it.
I can now also see that @divbyzero understood this, yet triumphantly presented the 3D print as a model to illustrate something he already knew was wrong
🙄
@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.
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.
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.
The statement plainly claimed the slices would have the same volume, no limits on shell thickness, no "approximately". That is demonstrably not true.
@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!
When I first responded it was about planar sliced cuts claimed to have the same weight, no mention of conical cuts and no approximation caveats.
Otherwise I would not have taken exception. The claim, as it was, was faulty, as could be demonstrated.
That's just a simple fact. Don't shoot the messenger.
Anyway, thanks for explaining how it *is* correct with the conical cuts.
@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.
Well, somehow I thought of mathematics of an exact science whose practitioners spend much attention to correctly formulating theorems and constructing proofs for those.
Imagine my desillusion when someone who profiles themself as a mathematician claims something in their field of expertise that is demonstrably wrong *while already knowing that it is incorrect*
So it turns out I was mistaken. Sorry for that 🙄
That I didn't.
I learned that you cannot just trust what a mathematician claims. It pays off to double check.
Dave Richeson
Sibrosan
Heribert Schütz