There are curious objects that possess properties strange enough to attract mathematicians. One of the best known is Gabriel's Horn, sometimes called Torricelli's Trumpet, a geometric figure that, despite having an infinite surface, has a finite volume. Conceived by Evangelista Torricelli, this object has given rise to an interesting paradox: an infinite amount of paint would be needed to paint its interior, yet it would be possible to fill that finite space with a few liters of pigment, thus covering its surface.
Can an infinitely large surface enclose a finite area? At first glance, it sounds impossible. But just as fractals showed that an infinitely long "coastline" could surround a finite area, it is also possible to enclose a finite volume with an infinitely large "shell". Despite this, there aren't many objects that meet these criteria, and one of the most famous is Gabriel's Horn, devised by Evangelista Torricelli more than 350 years ago.
The Paradox
This geometric figure, which some also call Torricelli's Trumpet, was initially considered a paradox. To paint an infinite surface, however thin the layer of pigment, an infinite amount of paint is required. If a painter with infinite time to tackle the task tried to paint Gabriel's Horn, they would find that there is not enough space in the universe to store the amount of paint they would have to use. But on the other hand, since its volume is finite, any mortal could fill it with a finite amount of paint. Where is the paradox? In filling Torricelli's Trumpet with paint, we are obviously painting its interior, the same one we just discovered was impossible to paint.
Flaws in the Reasoning
However, there are some flaws in this reasoning. First, it is impossible to build such an artifact, since there isn't enough material, and if there were, the time it would take would be infinite. But even if it existed, the layer of paint used would have to be of constant and finite thickness. This is impossible to achieve inside the horn, as most of the length of the figure (the end on the right in the previous image) is inaccessible to paint, because its diameter quickly becomes smaller than that of a paint molecule. And if someone managed to invent a "magic paint" made of atoms and molecules without thickness (overthrowing the structure of all known physics), we would need an infinite amount of time for it to reach the end of the horn.
The Theoretical Resolution
Suppose we can set aside the practical problems associated with building this artifact and focus on its theoretical properties. The inner diameter of the horn tends to zero, and current mathematicians know that when this happens, the thickness of the paint layer must necessarily be equal to or less than a certain value. We won't describe the calculation here, but it can be shown that the amount of "magic paint" needed is obtained through an improper integral, whose result is finite, and that from a certain point onward, a single drop of paint would suffice to cover the rest of the trumpet's surface, no matter how infinite its length and surface are.
Conclusion
In short, even without the appropriate mathematical tools it might seem otherwise (damn common sense!), the fact that the trumpet's surface is infinite does not imply that the amount of paint needed to cover it must also be infinite, destroying the apparent paradox. What is the practical use of all this? Probably none. But sometimes it's fun to come across these old problems that stimulate our imagination. Don't you think?
