What does a shape with infinite surface area but finite volume look like?
An example of such a mind-bending shape is Gabriel’s Horn. It was first studied in the 17th century. It is also infinitely long.
The shaded region below illustrates a two-dimensional cross-section of the horn:
See this link from Wikipedia to visualize the horn in three dimensions.
In this post, I’ll describe why this shape has infinite surface area but finite volume.
The horn is obtained by taking the graph of \( y = 1/x \) (when \( x \geq 1 \)) and rotating it about the x-axis. Note that in the above picture, the top boundary is \( y = 1/x \) and the bottom boundary is \( y = -1/x \).
Let \( f(x) = 1/x \). It turns out that the volume of Gabriel’s Horn is:
$$ V = \lim_{a \to \infty} \pi \int_1^a f(x)^2 \, dx $$
and it turns out that the surface area of Gabriel’s Horn is:
$$ A = \lim_{a \to \infty} 2\pi \int_1^a f(x) \sqrt{1 + (f'(x))^2} \, dx $$
Let’s calculate the volume. Firstly,
$$ f(x)^2 = \bigg{(} \frac{1}{x} \bigg{)}^2 = \frac{1}{x^2} $$
Secondly, it turns out that the following is true:
$$ \int_1^a \frac{1}{x^2} \, dx = 1 \;\; – \, \frac{1}{a} $$
So,
$$ \pi \int_1^a f(x)^2 \, dx = \pi \;\; – \, \frac{\pi}{a} $$
So the volume is:
$$ V = \lim_{a \to \infty} \pi \;\; – \, \frac{\pi}{a} = \pi $$
In conclusion, the volume of Gabriel’s Horn is around 3.14.
Now, let’s calculate the area. It turns out that
$$ f'(x) = \frac{-1}{x^2} $$
So,
$$ f(x) \sqrt{1 + (f'(x))^2} = \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} $$
What is significant about this is that (when \( x \geq 1 \))
$$ \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} \; > \; \frac{1}{x} $$
We can use the above inequality. It turns out that
$$ A \geq \lim_{a \to \infty} 2\pi \int_1^a \frac{1}{x} \, dx $$
Moreover, it turns out that
$$ \int_1^a \frac{1}{x} \, dx = \ln(a) $$
So,
$$ \lim_{a \to \infty} 2\pi \int_1^a \frac{1}{x} \, dx = \lim_{a \to \infty} 2 \pi \ln(a) = \infty $$
In conclusion, Gabriel’s Horn has infinite surface area.
A peculiar feature of this shape is that almost all of its surface area is concentrated in the narrow part of this horn (the part that looks almost like a line). You can imagine endless amounts of space that are crammed into thinner and thinner regions.
Interesting shapes, such as Gabriel’s Horn, appear in all sorts of places. What this post hopefully does is give you a taste of what these shapes are like.
