The Painter’s Paradox is based on the fact that Gabriel’s horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel’s horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel’s horn has a convergent series associated with volume and a divergent series associated with surface area. The dimensions of this object, which are in the heart of the above mentioned paradox, were first studied by Torricelli. To situate the paradox historically, we provide a brief overview of the development of the notion of infinity in mathematics and the debate around Torricelli’s discovery.
Gabriel and His Horn
Gabriel was an archangel, as the Bible tells us, who “used a horn to announce news that was sometimes heartening (e.g., the birth of Christ in Luke l) and sometimes fatalistic (e.g., Armageddon in Revelation 8-11)”.
Torricelli’s Long Horn
In 1641 Evangelista Torricelli showed that a certain solid of infinite length, now known as the Gabriel’s horn, which he called the acute hyperbolic solid, has a finite volume. In De solido hyperbolico acuto he defined an acute hyperbolic solid as the solid generated when a hyperbola is rotated around an asymptote and stated the following theorem:
THEOREM: An acute hyperbolic solid, infinitely long [infinite longum], cut by a plane [perpendicular] to the axis, together with a cylinder of the same base, is equal to that right cylinder of which the base is the latus transversum of the hyperbola (that is, the diameter of the hyperbola), and of which the altitude is equal to the radius of the base of this acute body.
The Queer Volume
Construct the surface of revolution given by rotating the function f(x) = 1/x on [1,∞) around the x-axis. The volume of a surface of revolution given by rotating the function f(x), defined on the interval [a; b], around the x-axis is
where A(x) is the cross-sectional area at x ∈ [a, b]. Due to the construction, this is ALWAYS a circle, of radius f(x), and hence A(x) = π (f(x))2, so that
In our case, the interval of integration is infinite, and hence the integral we define is improper. Nevertheless, we find that
Hence, even though the Horn extends outward along the x-axis to ∞, the improper integral does converge, and hence there is finite volume “inside” the Horn. One can say that one can fill the Horn with π-units of a liquid. (This is oddly satisfying)
The Infinite Surface Area
The surface area of a surface of revolution is the subject. For a surface formed by revolving f(x) on [a, b] around the x-axis, the surface area is found by evaluating
This formula basically says that one can nd surface area by multiplying the circumference of the surface of evolution at x, which is a circle again, with circumference 2πf(x), by the arc-length along the original function f(x) (this is the radical part of the integrand). In our case, we get an improper integral again (call the surface area SA):
This integral is not such an easy calculation. However, we really do not need to actually calculate this quantity using an antiderivative. Instead, we make the following observation: Notice that on the interval [1;1), we have that
Thus we can say that, if
on the interval [1;1), then
by the properties of integrals. And by the Comparison Theorem for improper integrals, we can conclude that, if the integral of the smaller one (with h(x) as the integrand) diverges, then so does the integral of the larger function g(x). Indeed, we find by comparison that
But we have already evaluated this last integral in class. We get
Hence this last integral diverges, and hence by comparison so does the former integral. But
this implies that the surface area of Gabriel’s Horn is infinite!
So we have a surface with infinite surface area enclosing a finite volume.