Gabriel’s Horn Paradox

September 20, 2022

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Some people just die too soon. One such person was Evangelista Torricelli, an Italian mathematician who died at the age of 39 in the year of 1647. Had Torricelli lived longer, he just might have discovered calculus, before Sir Isaac Newton and Gottfried Leibniz.

Although Torricelli’s contributions to science and mathematics are too numerous to list, it is worth stopping to think about his mystery of comparing areas and volumes, called Torricelli’s Paradox. This paradox was discovered in 1641. In fact, it is even more amazing when you consider that Torricelli did not have the modern tools of integral calculus, as Newton’s Principia for example was published in 1697. Torricelli used a technique called the Cavalieri method of indivisibles, a predecessor to the integral, to demonstrate this paradox.

The paradox involves a surprising result related to the area under the curve; see Figure 1. y = \frac{1}{x}, 1 < x < \infty. Any student of calculus knows that the are under the. Any student of calculus knows (or should know) that the are under the curve and above the x-axis can be found by computing the improper integral

    \[Z = \int_{1}^{\infty} \left(\frac{1}{x}\right) dx \equiv \lim_{K\to \infty} \int_{1}^{K} \left(\frac{1}{x}\right) dx.\]

Computing this integral gives:

    \[Z= \lim_{K\to \infty} \int_{1}^{K} \left(\frac{1}{x}\right) dx\]


    \[= \lim_{K\to \infty} (ln x) \Big|_1^K \]

    \[\lim_{K\to \infty} (ln K - ln 1)\]


    \[= \lim_{K\to \infty} (ln K)\]


    \[= \infty\]

In other words, the area under the curve is infinite. What does that mean? It means that the area increases without bound as we move the right edge of the shaded area more and more to the right, although it reaches infinity very slowly. Actually, when one moves all the way to x = 1,000,000, the area under the curve is only ln(1, 000, 000) = 13.8.

So, where is the paradox..? After finding out that the area under the curve is infinity, Torricelli decided to find the volume of the region obtained by rotating this region around the x-axis which one can observe in Figure 2.

Roughly, this volume can be found by adding up the areas of all the circles which is drawn in the depicted horn. Furthermore, since the area of a circle with radius r is Z \pi r^{2} and because the radius of the circle at location x is \frac{1}{x}, the area of a cricle at x is

    \[Z(x) = \pi r^{2} = \pi \left(\frac{1}{x}\right)^{2} = \frac{\pi}{x^{2}}\]

Subsequently,

    \[V = \int_{1}^{\infty} \left(\frac{\pi}{x^{2}} \right) dx = -\frac{\pi}{x} \Big|_1^\infty = - \pi \left(\frac{1}{\infty} - \frac{1}{1}\right) = \pi\]

where V denotes the volume of Gabriel’s Horn. So, the volume of Gabriel’s Horn is \pi cubic units!

To give you more practical insight. You fill Gabriel’s Horn with pi cubic units of paint, but there is not enough pain in the world in order to paint its outside. So, why has mathematics gone wrong? Well, it really is not wrong. It is simply invalid to compare numbers in different dimensions. In fact, area is a two-dimensional property, whereas volume is a three-dimensional property.

References:

Vengatesan, K. (2020). Gabriel’s horn: Perceiving the infinite in the Finite! From https://shasthrasnehi.com/gabriels-horn-perceiving-the-infinite-in-the-finite/

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