Mathematicians Found A Shape You Can Fill But Never Paint

Added: 2026-05-19

[00:00:01]I have a shape to show you. You can hold it in  your hands. You can fill it with paint, but you can never paint its surface. Not with all the  paint in the universe. There is no mathematical trick here and it's a precise mathematical  calculation. This shape has a finite volume but an infinite surface area. It seems to contradict  common sense and it feels like mathematics itself must be wrong. It has even a name worthy of  its strangeness. It's called Gabriel's horn, named after the angel whose trumpet call announces  the end of the world. And today we are going to calculate exactly why it works. In 1641, the  Italian mathematician Evangelista Torricelli was studying a simple curve, the hyperbola, which is  y = 1 /x. He took the portion from x= 1 stretching to infinity to the right. He rotated the shape,  the curve around the x-axis. The result is a shape like an infinitely long trumpet. And he calculated  two things: the volume and its surface area. And the results.. they were shocking. Check the radius  of the disc at x= 1. The radius is 1. At x= 2 the radius is 1/2 and so on. So to find the volume of  this shape, we use the disc method, slice the horn into infinitely thin circular discs perpendicular  to the x-axis. Each disc at the position x has a radius of r= 1 /x. We pick a portion and around  the center we see the volume of one infinitesimal thin disc and the total volume is calculated by  this given formula which is nothing but almost pi r² along the dx and the volume calculated from  here after a simple integration is nothing but pi cubic units. So the volume of this shape is  finite. The surface area of a solid of revolution uses a different formula. Instead of discs, we use  thin bands. Each band is a slice of cone wrapped around the surface. Since y = 1 /x, we know what  is the dy/dx and we have - 1 /x². And this is exactly what we needed to calculate the ds. So  let's look at the final form of the integral.

[00:02:38]We have this square root which is larger than 1  for all the x values. So we can use a lower bound and what we obtain after this integration is you  have 2 pi ln of x and you have to integrate from these boundaries. So you have the surface area of  this shape larger than infinity which means it is infinite. The critical difference between the  volume and the surface area is r versus r². And what we mean by this is the following. As we see  from the integral the volume integrates over pi r². So the volume integrates over pi r² which is  pi /x² dx. And this converges because the integral of 1 /x² converges. And the surface integrates  over 2 pi r ds. Once you do the integration, what you see is there is the integral of 1 /x dx  which is divergent. The horn narrows but not fast enough for the surface to be finite. Here if you  calculate what you get for the volume is pi and what you get for the surface area is infinite.  You could fill Gabriel's horn with exactly pi cubic units of paint. But if you try to paint the  surface, all the paint in the universe, however thin we make the layer of the paint, would not  be enough to coat the horn. So Gabrielle's horn is a lesson about where the intuition fails. And  our intuition says that an infinitely long object must have an infinite volume, but it's wrong. It's  finite. Then our intuition says well if the volume is finite the surface area must be finite too  but it's wrong again because it's infinite. So mathematics never stops surprising us regardless  of if we are talking about angel's horn or what's inside of black holes. What a great compass  mathematics is against our common sense. And if this ignited your curiosity, subscribe to Breaking  Spacetime and I'll see you in the next one :)