Paradoxes: Gabriels Horn

Gabriel’s horn is the function \frac{1}{x} on the interval [1,\infty] rotated around the x-axis. Intuitively, it should require less paint to fill the horn than to paint the surface (imagine painting a box or a ball). With Gabriel’s Horn this is not the case. In fact, There is an infinite surface area and only a finite volume. Then, this means you could pour paint inside the horn and fill it up (quite quickly) yet you could never finish painting the entire surface. What will follow is the mathematical explanation of this.

GabrielHorn

In calculus the volume of a surface of revolution is defined to be

\int_a^b A(x)dx

where A is the area of the disk with radius f(x).  So in order to find the volume of Gabriel’s Horn, we note that the disk has radius \pi (\frac{1}{x})^2. Since we are integrating on [1,\infty] we must evaluate the improper integral

\pi \int_1^\infty \frac{1}{x^2}dx

So,

\pi \int_1^\infty \frac{1}{x^2}dx = \lim_{b\to\infty}\pi\int_1^b\frac{1}{x^2}dx =\lim_{b\to\infty} \pi(1-\frac{1}{b}) = \pi.

Thus we see that Gabriel’s Horn does in fact have finite volume. Let us take a look at the surface area formula,

SA = 2\pi\int_a^b f(x) \sqrt{1-f'(x)^2} dx.

Putting it into our context we have

SA = 2\pi\int_1^\infty \frac{1}{x} \cdot \sqrt{1-\frac{1}{x^4}} dx

This might look like a mess but our ultimate goal is to prove this evaluates to infinity. To this, we see that the integrand is \frac{1}{x} multiplied by a number greater than 1. Hence we can bound our integral from below as follows

2\pi\int_1^\infty \frac{1}{x} \cdot \sqrt{1-\frac{1}{x^4}} dx \geq 2 \pi \int_1^\infty \frac{1}{x} dx.

 

Evaluating the right hand side we get

2\pi\int_1^\infty \frac{1}{x} dx = \lim_{b\to\infty}2\pi\int_1^b \frac{1}{x} dx =\lim_{b\to\infty}2\pi ln(b) = \infty

And there we have it an infinite surface area. Obviously if we brought the horn to the physical realm, the horn would get so small that not even a molecule could pass through. But how does this compare with Zeno’s paradoxes? Zeno’s paradox seems to point toward a continuity of the physical realm, but Gabriel’s Horn seems to indicate that this can’t actually happen either. Further advancements in quantum physics seems to share the same ideology as Zeno in that the physical realm is indeed made of discrete fibers and distances. Bringing the realm of the metaphysical to the physical is an impressive goal that science seems to be attempting to answer today. Who knows, maybe we will find the answer soon enough.

 

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