Gabriel's Horn

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Gabriel's Horn (also called Torricelli's trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. The name refers to the tradition identifying the archangel Gabriel with the angel who blows the horn to announce Judgement Day, associating the infinite with the divine.

Illustration of left hand end of Gabriel's Horn (or Torricelli's trumpet)

1 Mathematical definition
2 Explanation
3 See also
4 External link

[edit] Mathematical definition

Gabriel's horn is formed by taking the graph of $y= \frac{1} {x}$ , with the domain $x \ge 1$ (thus avoiding the asymptote at x = 0) and rotating it in three dimensions about the x-axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, where a > 1. Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find the volume $V$ and the surface area $A$ :

$V = \pi \int_{1}^{a} {1 \over x^2}\mathrm{d}x = \pi \left( 1 - {1 \over a} \right)$

$A = 2\pi \int_1^a \frac{\sqrt{1 + \frac{1}{x^4}}}{x}\mathrm{d}x > 2\pi \int_1^a \frac{\sqrt{1}}{x}\ \mathrm{d}x = 2\pi \ln a$

$a$ can be as large as required, but it can be seen from the equation that the volume of the part of the horn between $x = 1$ and $x = a$ will never exceed $π$ ; however, it will get closer and closer to $π$ as $a$ becomes larger. Mathematicians say that the volume approaches $π$ as $a$ approaches infinity, which is another way of saying that the horn's volume equals $π$ . Expressed using the limit notation of calculus:

$\lim_{a \to \infty}\pi \left( 1 - {1 \over a} \right) = \pi$

As for the area, the above shows that the area is greater than $2π$ times the natural logarithm of $a$ . There is no upper bound for the natural logarithm of $a$ as it approaches infinity. That means, in this case, that the horn has an infinite surface area. That is to say;

$2 \pi \ln a \rightarrow \infty$ as $a \rightarrow \infty$

At the time this was discovered, it was considered paradoxical as, by rotating an infinite area about the x-axis, an area of finite volume is obtained.

[edit] Explanation

The explanation for this paradox is related to the dimensions of the quantities involved in the calculations. The dimension of length is 1, area 2 and volume 3 ( $m$ , $m 2$ and $m 3$ respectively).

When calculating the surface area of a graph which has been rotated, we suppose that the result is composed of small strips of a one-dimensional quantity - "rings" whose radii are equal to the graph's height at a given point. When these are integrated along (i.e. added up), the result is a two-dimensional quantity - the surface area. Similarly, measuring the volume of this rotated graph sums the total of many circles whose radii are the height of the graph; the result is a three-dimensional quantity (volume).

The paradox arises because the strips of length on the "rings" being added to give the surface area are of a lower dimension (1 vs. 2) than the disks of area being used to find the volume. As $x \to \infty$ :

$\pi\frac{1}{x^2} \ll 2\pi\frac{\sqrt{1 + \frac{1}{x^4}}}{x}$

Essentially, this means that as x becomes larger and larger, the numerical size of the two-dimensional disks that are added is so much smaller than the one-dimensional rings that they decrease far too quickly to ever bring up the volume of the entire horn to anywhere past a volume of $π$ . When integrated (as above), it should be apparent that the volume quickly converges on $π$ .