Sometime back, I watched the Numberphile video (referenced below) and immediately thought, this is not a paradox at all.  Any volume has an infinite surface area, and it seemed quite puzzling to see mathematicians confusing area and volume in this way.

For a quick run-down, Gabriel's horn equation is Y = 1 / x starting at Y = 1.  This function is then revolved around and the surface area and volume are calculated.  The surface area is infinite, while the volume is equal to pi.

It became quite apparent to me that a function which started out with a bell larger than one and whose volume was '2 pi' would satisfy the thickness of paint required to paint the horn.  The obvious example being the square root of two over one (2 ^ (1/2) / 1). However, this function has a substantial decreasing paint thickness from bell to mouthpiece.

This is where I became stuck.  I want to find a function whose revolved volume would be '2 pi', but would have the same offset from the original Y = 1 / x.  That is, you wind up with a consistent layer of "paint".

Now being a practical person, who lives in a mostly realistic world, I figure that one could start with a rather large value for 'one' (say an astronomical unit) and use the carbon atom as the minimum limit where the mouthpiece of the horn would start where a single carbon atom could no longer pass through.  I would, also, assume that the minimum wall would be a carbon nano tube expanding to a graphene sheet.  The mouthpiece. then would have a single atom in the middle surrounded by six more.  That is, a five atom deficit at the very small end.  However, as you go to the larger end, there winds up being quite a surplus.  We can simplify things just a bit, and make the inside wall and thickness being our carbon paint.  That is, the mathematical infinitely small wall is on to which our carbon paint is applied.

It is, at this point, I find my math skills to be quite inadequate. I've been thinking about this off and on for quite a few months, without a reasonable answer.  One could start out with a better unit for 'one' like a kilometer and still wind up with something that trends to the ideal solution.

Your statement that "any volume has an infinite surface area" is incorrect, on its face.  I am not sure what you are considering to be the "surface area" of a volume, but in mathematics it is the interface between what is inside your volume and what is outside it. In the case of a cube of side length L, for instance, the surface area is 6 x L^2, in that there are 6 sides to the cube, each having an area of L^2.  The surface area of a sphere is also finite, being 4*Pi*R^2.

As for your search for a revolved function with a minimum thickness of paint, you can't have one fitting your criteria.  If the paint thickness never approaches zero, then the area bounded by the curve always stays above a minimum value, and with an infinite length the resulting volume goes to infinity.  In order to have a minimum "depth" and constant volume, your function has to be bounded on all sides, as in it must be a geometric shape of finite dimensions, such as the square or sphere above.

Mike

Consider a cylinder of diameter D and a length of X.  Now slice the cylinder into wafers of dx thick.  As dx approaches zero you are left with an infinite number of wafers which each have an area of 'pi x D/4 ^ 2"  Therefore a volume has an infinite area.  Now when you consider a volume of an ideal fluid, you can likewise spread out that fluid over an infinite area with an infinitesimal thickness.   Which is why the painter's "paradox" just isn't a paradox at all.

The surface area of your cylinder is

A = PI * D  * X + PI * D^2 / 2

When you take dx to zero and get infinitely many slices, you have to really be careful about what you are doing because it is easy to do something that feels intuitive with infinities but results in paradox. Formally write it out with limits and you will find that the surface area is not infinite.