This reminds me of of one of my favorite math paradoxes: Gabriel's Horn
The video is a bit long but the headline is: using calculus you can solve that the shape has finite volume but infinite surface area. So we know how much it would take to fill it with paint but somehow that wouldn't cover the surface...
My favourite one is the napkin ring problem. For any two spheres of any size, if they were cored like an apple to the same height, the volume of both rings would be exactly the same. A 5cm high napkin ring made out of a billiard cueball has exactly the same volume as a 5cm high napkin ring made out of a neutron star.
Oh, yeah that's like fractals that have infinite perimeter, but finite area (i.e. Koch snowflake).
You CAN "cover the surface" with paint, if the thickness of the paint is very very small. If you fill the horn with paint, you are effectively painting it, the thickness of the paint being the radius.
It stops seeming so paradoxical when you realize it's not just the surface area of the horn that's infinite, the length of the horn is also infinite.
This reminds me of of one of my favorite math paradoxes: Gabriel's Horn
The video is a bit long but the headline is: using calculus you can solve that the shape has finite volume but infinite surface area. So we know how much it would take to fill it with paint but somehow that wouldn't cover the surface...
My favourite one is the napkin ring problem. For any two spheres of any size, if they were cored like an apple to the same height, the volume of both rings would be exactly the same. A 5cm high napkin ring made out of a billiard cueball has exactly the same volume as a 5cm high napkin ring made out of a neutron star.
this one makes sense to me, if you core a small sphere it's going to have a fat cross-section
Oh, yeah that's like fractals that have infinite perimeter, but finite area (i.e. Koch snowflake).
You CAN "cover the surface" with paint, if the thickness of the paint is very very small. If you fill the horn with paint, you are effectively painting it, the thickness of the paint being the radius.
It stops seeming so paradoxical when you realize it's not just the surface area of the horn that's infinite, the length of the horn is also infinite.