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.

Image:GabrielsHorn.png

Gabriel's horn is formed by taking the graph of <math>y= \frac{1} {x}</math>, with the domain <math>x \ge 1</math> (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 obviously a > 1. Using integration (see Solid of revolution and Surface of revolution for details), it is possible to find that the volume <math>V = \pi(1 - 1/a)</math>, and that for the surface area <math>A</math>:

<math>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</math>

<math>a</math> can be as large as required, but the volume of the part of the horn between <math>x = 1</math> and <math>x = a</math> will never exceed <math>\pi</math>, and the volume will get closer to <math>\pi</math> as <math>a</math> becomes larger. Mathematicians say that the volume approaches <math>\pi</math> as <math>a</math> approaches infinity, which is another way of saying that the horn's volume equals <math>\pi</math>. As for the area, the above shows that the area is greater than <math>2\pi</math> times the natural logarithm of <math>a</math>. There is no upper bound for the natural logarithm of <math>a</math> as it approaches infinity. That means, in this case, that the horn has an infinite surface area.

At the time this was discovered, it was considered a paradox. The apparent paradox has been described informally by noting that it seems it would take an infinite amount of paint to coat the interior surface, but it also seems that it would be possible to simply fill the interior volume with a finite amount of paint and so coat the interior surface. The resolution of the paradox is that the implication, that an infinite surface area requires an infinite amount of paint, presupposes that a layer of paint is of constant thickness; this is not true in theory in the interior of the horn, and in practice much of the length of the horn is inaccessible to paint, especially where the diameter of the horn is less than that of a paint molecule. - If the paint is considered without thickness, it would further take infinitely long time for the paint to run all the way down to the "end" of the horn.

Another way this "paradox" is often proposed is by the statement that one could "fill the horn with paint, but not have enough to paint the outside".

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