Its not really a paradox. The horn is being conceptualized as being an infinitely thin surface (otherwise as x->infinity the inside of the horn would close up). The thing to ask yourself is how much of an infinitely thin surface fits in some given volume. Picture an extremely long piece of paper with basically no thickness (some infinitesimally small thickness). Now picture a normal 3d box. How much of that paper can fit in the box? Well, we can just keep folding it and folding it, and we can do this infinitely. Infinite surface area can fit in that box. Now, if you can picture this, we solved the paradox. If I dump 20 cans of paint in that box, assuming the paint can fit in those infinitesimally small crevices of the paper folds, they are covered. In fact, if we think about Gabriel's horn like this, we get a completely different answer than the answer of infinity everyone else assumes. If I just want paint to surround the surface, I just need to put in a volume of paint that will touch the surface. If my paint can be infinitely thinned, then I can just subtract the volume of the 1/x horn with one that has been shifted down some super small amount (say 1/x -.0000001). The function 1/x - .000000001 will create a horn with a slightly lower volume. My paint just needs to fit in between the volume of these two horns. It's going to be a very very small amount. I can actually do even better if I add more zeroes with something like 1/x - .00000000000000000000000001. In fact, I can create a function with an infinitesimally small difference... and then I'd need an infinitesimally small amount of paint. So, my answer is I'd need 1/ infinity volume of paint.
Evaluating Gabriel’s Horn Paradox . Area of horn= infinite square units. . Volume of horn= pi cubic units. . Paint required to cover the area= infinite × thickness approaches zero = zero cubic units. . The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi. . Note: no surprising for an object has area bigger than volume, for instance, a cube with one unit length, has one volume unit and six area units. .The misunderstanding of Gabriel's Horn has existed after We had compared volume unit with area unit. We should not compare different units.
Its not really a paradox. The horn is being conceptualized as being an infinitely thin surface (otherwise as x->infinity the inside of the horn would close up). The thing to ask yourself is how much of an infinitely thin surface fits in some given volume. Picture an extremely long piece of paper with basically no thickness (some infinitesimally small thickness). Now picture a normal 3d box. How much of that paper can fit in the box? Well, we can just keep folding it and folding it, and we can do this infinitely. Infinite surface area can fit in that box.
Now, if you can picture this, we solved the paradox. If I dump 20 cans of paint in that box, assuming the paint can fit in those infinitesimally small crevices of the paper folds, they are covered. In fact, if we think about Gabriel's horn like this, we get a completely different answer than the answer of infinity everyone else assumes. If I just want paint to surround the surface, I just need to put in a volume of paint that will touch the surface. If my paint can be infinitely thinned, then I can just subtract the volume of the 1/x horn with one that has been shifted down some super small amount (say 1/x -.0000001). The function 1/x - .000000001 will create a horn with a slightly lower volume. My paint just needs to fit in between the volume of these two horns. It's going to be a very very small amount. I can actually do even better if I add more zeroes with something like 1/x - .00000000000000000000000001. In fact, I can create a function with an infinitesimally small difference... and then I'd need an infinitesimally small amount of paint. So, my answer is I'd need 1/ infinity volume of paint.
Evaluating Gabriel’s Horn Paradox
. Area of horn= infinite square units.
. Volume of horn= pi cubic units.
. Paint required to cover the area= infinite × thickness approaches zero = zero cubic units.
. The paint of pi cubic units will cover the inner side with zero amount of paint ( infinite × thickness approaches zero ) and fill the horn with all amount of pi.
. Note: no surprising for an object has area bigger than volume, for instance, a cube with one unit length, has one volume unit and six area units.
.The misunderstanding of Gabriel's Horn has existed after We had compared volume unit with area unit.
We should not compare different units.
Great video! But check out the pronunciation of "Torricelli". The "c" sounds more like "ch" in English.