Simply mind blowing
A paradox is something that seems completely absurd, but can be proved to be true. Paradoxes never fail to astound me and I came across one such mathematical paradox quite recently. Let’s have a look at it.
Consider the following function:
Such a simple function cannot possibly do any harm to us can it? Well this function gave me quite some trouble as I broke it down. First, let us graph this function:
As you can clearly see, from the equation or the graph, as x tends to 0, y tends to infinity. At the other end, as x tends to infinity, y tends to 0. For now, let us look at this function when its domain is x≥1.
If we were to calculate the area under this graph, we would compute the integral of this function with the lower limit as 1 and the upper limit as ∞:
Simplifying this equation, we would get:
Well that’s quite clear. The area under the graph of 1/x is undefined. But this is where the magic happens. Suppose we rotate the graph around the x-axis like this:
This structure looks sort of like the tube part of a very long trumpet like the one below.
We will now try to calculate the volume of this structure. By looking at the structure from a different angle, we notice that the graph is basically many many circles placed one after the other and decreasing in size.
So, we can use integration to calculate its volume by summing up the areas of all the circles using the formula:
We replace the r² in the formula with (1/x)² because the radius of all the circles is basically the y value of the graph at that x value as all the circles are centered at the origin. Solving this, we get:
Here, we see that the figure has a finite volume! But this can’t make sense can it? Intuitively, the volume of the figure can be said to be area under the original graph added many times. This would mean that the volume of the figure should also be undefined. But we just proved that the volume is finite. Quite crazy, right? Let’s now have a look at the surface area of the horn.
In the above image, you can see the Gabriel’s horn with a brown colored ring around it. You can see that the horn is made of many of these rings, so the surface area of the horn is just the sum of the surface areas of all those rings. Let us break down this image to help calculate the area of one ring:
As we can see in these two images, you can calculate the area of the ring by multiplying the circumference of the circle by dl. Assuming the ring to be as small as possible, dl will be almost a straight line. So, we can calculate dl using the Pythagorean theorem and you get:
Now that we have dl, we can calculate the surface area of the ring and then use an integral to find the total surface area:
To calculate this, we need to use this important observation:
This observation gives us this amazing discovery:
This shows us that the Gabriel’s horn not only has a finite volume, but it also has an infinite surface area! This shows how paradoxical this one structure truly is: a slice of it has an infinite area, the entirety of it has a finite volume, and it overall has an infinite surface area as well! This connects back to its original name: the Painter’s paradox. Imagine this horn as a paint bucket; you can completely fill it with paint as it has finite volume, but you cannot completely paint its surface!
Mathematics is filled with many such paradoxes which may seem unbelievable at first, but can be proved systematically using the tools we have. These paradoxes have always been there — they’re just waiting for us to discover them.