Gabriel’s horn (the surface formed by revolving the curve 1/x around the x-axis from x=1 to ∞) has an infinite surface area but a finite volume π. This seems strange enough, but if one really wants a mental stretch, think about the following problem (posed, I can only assume, by a bored but brilliant house-painter of old).
To paint the infinitely large surface of Gabriel’s horn, an infinite volume of paint must be required. Because the horn’s volume is finite, this infinite amount of paint cannot be contained inside the horn. Yet if the horn is filled with paint, the entire horn should be covered, with some additional paint not directly in contact with the horn to spare. One can thus coat an infinite surface area with a finite amount of paint (in which case, the aforementioned painter need only ever buy one bucket of paint to last the remainder of his career).
To understand the flaws of the above paradox, one must first understand that Gabriel’s horn is a surface of infinite two-dimensional area but no depth. The size of a segment of the horn’s surface can be represented in units squared, and the area of part of the horn’s “inside” is equal area of that part of the horn’s “outside,” the two areas really being different measures of the same surface. The paradox exploits this fact, assuming that paint on the surface inside the horn is equivalent to paint on the surface outside the horn. A layer of paint is equated to a two-dimensional surface.
One must further keep in mind that the volume within the horn is three-dimensional and can be represented in units cubed. Paint, as dealt with in this problem, must also be a three-dimensional quantity, for it is said to be contained in a horn of like space. These three dimensions result from the two-dimensional surface of the horn and an added dimension of depth—the thickness of the paint coating the horn.
If the horn is painted from its outside, depth can be set arbitrarily and need not be constant. The only value depth cannot approach as the horn lengthens to infinity is zero, for such a lack of depth would then make the paint a two- dimensional area, as opposed to a volume that can be fitted inside a container.
If, however, the horn is painted from its inside, depth at any given point is limited to the ordinate of the curve revolved to generate horn. As the horn lengthens to infinity, the curve’s ordinate approaches zero (lim as x→∞ of 1/x), stripping the paint of its needed depth-dimension. The horn thus cannot be coated three-dimensionally from its inside.
Herein, then, lies the crux of the paradox. Whereas surface is identical whether looked from the inside or outside of the horn, paint is not. Two-dimensional areas cannot be discussed interchangeably with three-dimensional volumes. Given a finite section of the horn, one could paint with depth its inside surface. Taken to infinity, the horn’s form approaches that of a line and thus prevents a coat of paint from having depth. Without depth there can be no volume, and without volume there can be no paint contained within Gabriel’s horn. Shame on that painter who made these mistakes: he clearly ought to have paid more attention in his calculus class. (Although if he had, perhaps he wouldn’t have had to go into home-improvement in the first place.) In any case, math rules the day!
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