Gabriel’s Horn Paradox

Have you ever tried to paint something that has a finite volume but somehow lasts forever to paint? This paradox will leave you perplexed.

Welcome welcome math-followers. Today, I greet you with one of the best paradoxes I have encountered in my life. It is called the Gabriel’s Horn Paradox. The name refers to archangel Gabriel blowing the horn to announce Judgement Day. And the geometric shape that we are dealing with in this problem is basically a horn. The first studies on this paradox were made by Italian physicist and mathematician Evangelista Torricelli and, therefore, the paradox also goes by the name “Torricelli’s Trumpet”.

After some historical information on the problem. Let me take you through this paradox. If you will, I strongly recommend you to take a piece of paper and a pencil, and go along with my instructions as I continue to explain the problem. First, let’s draw the sketches of the functions 1/x.

Great! Now, to form this horn, I want us to only focus on the part of the sketch where x > 1 and rotate that part along the x-axis. We will then have Gabriel’s Horn staring at us. The point here, as you might have guessed, is that this horn is infinite. It does not end.

Say that I want to pour green paint into this horn and therefore paint the inside of the horn. Let’s calculate how much paint I need to do that. In other words, let’s calculate the volume of this horn by using infinite integrals. We have integral articles in Betamat, so if you are not familiar with integrals you can either rely on my calculations for this once or check out our previous articles and learn the topic and come back.

Excellent! I would need π units of paint to paint the inside of this horn. It’s a bit interesting to me that this horn does not have an end yet π units of paint is enough to fill this horn. But the real mind-blowing result is not this. We can say from what we have done until this moment, is that our horn has a finite volume. Great. Now, as a painter, I also want to paint the outside area of my horn. And as a mathematician, I want to first calculate how much paint I would need for that. In other words, I, now, calculate the surface area of our horn by, again, using infinite integrals.

Uh oh! Because the surface area is infinite, I can never buy enough paint to paint the surface area. But with π units of paint, I can fill the horn. And when I fill the inside of the horn with π units of paint, I also paint the inner surface area of the horn. But the surface area is infinite. Then how on earth am I able to paint the inside of the horn but never the outside of the horn?

That is the paradox. Amazing, right?

See you on other paradoxes!