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 Judgment Day, associating the infinite with the divine.

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

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[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

This is so because as a approaches infinity, 1 / a approaches zero. This means the volume is π(1 - 0) which equals π. 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

or

\lim_{a \to \infty}2 \pi \ln a = \infty

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

[edit] The "painter's paradox"

This is sometimes called the "painter's paradox" since it takes an infinite amount of paint to paint an infinite area. But if you fill the horn with paint you will need a finite amount. 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, m2 and m3 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 to π.

A non-symbolic way of saying the same thing is the following: to "paint" the surface of the horn does indeed require an infinite surface area of paint, so that the sense in which it is infinite is as a two-dimensional substance. But to "paint" the surface by filling the horn with paint is to obscure it by a three-dimensional object, so the sense in which the amount of paint is finite is as a three-dimensional substance. The paradox arises because real paint is not two-dimensional, and in fact has a discrete thickness, so that painting the surface actually requires an infinite three-dimensional quantity. However, when the horn is filled with paint it is not the outside but the inside surface that is painted. To paint the inside surface of the horn with a layer of paint having a discrete thickness is impossible; once the horn becomes too narrow the paint will not fit. it is also impossible to fill the horn with such paint, so that in both cases, only a finite extent of the horn is covered and the paradox vanishes.

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