I can't speak to the first and third examples, but the "painter's paradox" is silly, I think, not a paradox at all. It's easier to visualize in two dimensions: consider the function y = 1 / x ^ 2 (y equals one over x squared). We can start from 1 and head out to infinity, and the length of the line will be infinite, but the area under it will be finite. Extrude a fixed amount in and out of the page and we have an ugly three dimensional "horn" with the desired properties: finite volume and infinite surface area. But there's no paradox: you just have to keep painting in thinner and thinner coats the farther out you go. Of course, it soon becomes a tiny fraction of an atom's breadth in thickness, so no real paint would do, but there is no paradox.
I can't speak to the first and third examples, but the "painter's paradox" is silly, I think, not a paradox at all. It's easier to visualize in two dimensions: consider the function y = 1 / x ^ 2 (y equals one over x squared). We can start from 1 and head out to infinity, and the length of the line will be infinite, but the area under it will be finite. Extrude a fixed amount in and out of the page and we have an ugly three dimensional "horn" with the desired properties: finite volume and infinite surface area. But there's no paradox: you just have to keep painting in thinner and thinner coats the farther out you go. Of course, it soon becomes a tiny fraction of an atom's breadth in thickness, so no real paint would do, but there is no paradox.
I love that the popular response to nonsense is "mind blown!"
Excellent piece. Funny, I *just* read about "Hilberts Hotel" 2 days ago in a book by Barrow about "nothing." Like minds lol