PAINTING INFINITY

PARADOX
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).

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