Obvious Theorems (Which Are False)

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[ScarletEmber]

We learned about this in our recent class
An interesting thing that was said was
"You can fully fill this horn with paint, but you can never fully paint it"

[Anonymous-Indian..]

Yes....
For any closed

36πV² ≤ S³ (equality holds for sphere)

V=volume, S=surface area
S is finite implies V is also finite
V is infinite implies S is also infinite

But it is possible that if V is finite then S can be infinite
We can say that one drop of colour can paint the entire earth

But I'm not sure whether V=0 and S=∞ can

[Blaqjaqshellaq]

On the other hand, finite surface area DOES mean finite volume!

[brandonklein]

It is not specific to 2D surfaces either. Consider the graph of e^-x from 0 to infinity. We clearly have an infinite line but there is finite area underneath it.

[JaybeePenaflor]

My field of study is differential geometry and I love Gabriel's Horn. Apart from the surface area and the volume, we also studied the Gaussian and mean curvatures of this amazing mathematical object.

[PlanesAndGames]

Finally, a loop that isn't a crappy "now you know that (beginning of the video)"

[dennishou]

I love how this same technique is used to show that the harmonic series is divergent even though it's terms converge to 0

[harindertaliyan]

we can see by example of gold.
One ounce, which is about the size of a cube of sugar, can be beaten into sheet nearly 100 square feet in size

[OptimusPhillip]

Is there a surface analogy for fractals? I feel like that would also be a good argument if there was, since we know there exist fractals with infinite lengths that enclose finite areas

[leesweets]

The real paradox is when you consider painting Gabriels Horn. You can fill the entire volume up with paint but you can never paint its internal surface area.

[God-ldll]

was recently myself thinking about "nondualifying" dimentions by, for ex. To "integrate" {R}^2 into {R}^3 by (m^2)*delta(m), where delta(m) is the "added spacial dimension" or an accounting for the "thiness" of the horn. At least roughly speaking.

[BenjaminSenk]

That's like saying convergent integral means function on a finite interval

[sharpfang]

You don't even need infinite length. Take Koch's Snowflake: an equilateral triangle where each side grows an extra equilateral triangle 1/3 its length, in the middle. Repeat for each line produced, infinitely. Every division increases the circumference by 1/3, while the area is still enclosed by a circle circumscribed on the original triangle. So, infinite circumference, finite area. Extrude into a 3D shape, and you have finite volume, infinite area.

[anandsuralkar]

Finally a perfect loop for real this time

[EDoyl]

The intuitive clay example is itself something that feels right but isn't.

[kinshuksinghania]

Some things are so counter-intuitive . You can fill the entire volume with a finite amount of paint yet you can’t cover the entire surface with that amount of paint.

And even more counter-intuitive is the question - Isn’t the paint already covering the entire surface area when the entire volume of the Gabriel Horn is filled with paint??

[friedrichhayek]

I come here for a false theorem, no for a counterexample.

[illumino]

So... This is not a Theorem about Fuel?

[insouciantFox]

Fractals are also a good example. In 2D they clearly have finite comvergent area, but their arclength is infinite.

[BootyRealDreamMurMurs]

So basically, THEORETICALLY, You can stretch out something like a clay to the point of infinity with the same volume. its like the same principle of ultimately making the infinitely thinnest pasta ever.