Consider Gabriel's Horn, the mathematical object formed by a surface of revolution of the curve x= 1/x from x=1 to infinity. It is known that one can fill the horn with a volume of Pi cubic units of paint but it would take an infinite amount to paint the surface. I think they usually mean the outer surface. But what of the inner surface which must be by definition already *covered in paint? And for a purely mathematical object how can the inner surface differ from the outer surface? Consider a sphere or box with a 1D plane defining the surface. Aren't the inner and outer areas the same? I'm concerned with only the mathematics of the situation and not practical aspects such as the size of atoms. Thanks. * Of course, this may be a mistaken assumption on my part! Contemplating the infinite is tricky.