Gabriel’s horn is formed by taking the graph of y=1/x with the domain x>= 1 and rotating it in three dimensions about the x-axis.
The converse of Torricelli’s acute hyperbolic solid is a surface of revolution that has a finite surface area but an infinite volume.
The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.
Torricelli’s original non-calculus proof used an object, slightly different to the aforegiven, that was constructed by truncating the acute hyperbolic solid with a plane perpendicular to the x axis and extending it from the opposite side of that plane with a cylinder of the same base.
Whereas the calculus method proceeds by setting the plane of truncation at x=1 and integrating along the x axis, Torricelli proceeded by to calculate the volume of this compound solid (with the added cylinder) by summing the surface areas of a series of concentric right cylinders within it along the y axis, and showing that this was equivalent to summing areas within another solid whose (finite) volume was known.
Although credited with primacy by his contemporaries, Torricelli had not been the first to describe an infinitely long shape with a finite volume/area. The work of Nicole Oresme in the 14th century had either been forgotten by, or was unknown to them.
Oresme had posited such things as an infinitely long shape constructed by subdividing two squares of finite total area 2 using a geometric series and rearranging the parts into a figure, infinitely long in one dimension, comprising a series of rectangles.
When the properties of Gabriel’s horn were discovered, the fact that the rotation of an infinitely large section of the xy-plane about the x-axis generates an object of finite volume was considered a paradox. While the section lying in the xy-plane has an infinite area, any other section parallel to it has a finite area. Thus the volume, being calculated from the “weighted sum” of sections, is finite.
Since the horn has finite volume but infinite surface area, there is an apparent paradox that the horn could be filled with a finite quantity of paint and yet that paint would not be sufficient to coat its surface.
However, this paradox is again only an apparent paradox caused by an incomplete definition of “paint”, or by using contradictory definitions of paint for the actions of filling and painting.
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