Torricelli’s Trumpet & the Painter’s Paradox

 

Torricelli-03

Torricelli’s Trumpet

 

Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve {y=1/x} for {x\ge1} is rotated in 3-space about the x-axis.

Torricelli-01A

Functions y=1/x and y=1/[x] for x>1. Torricelli’s Trumpet is generated by rotating the smooth curve about the x-axis.

The volume contained within the surface is obtained from an elementary integral:

\displaystyle V = \pi\int_1^\infty \left(\frac{1}{x}\right)^2 \, d x = \pi \left[\frac{-1}{x}\right]_1^\infty = \pi 

On the other hand, the integral giving the surface area is awkward. However, it can easily be seen to diverge. Clearly, the surface must be greater than twice its projection onto the {xy}-plane.

Torricelli-02

The surface area of Torricelli’s Trumpet is greater than twice its projection on the xy-plane (shaded).

But the area between the curves in the above figure is

\displaystyle A = 2\int_1^\infty \frac{1}{x} \, d x = 2\lim_{b\rightarrow\infty}\left[\log x\right]_1^b = \infty

so the surface area of the trumpet is infinite.

The fact that a geometric form extending to infinity could have a finite volume came as a great surprise to Torricelli. It gave rise to discussion and to some controversy over the nature of infinity, which involved many of the key scholars of the day, with Thomas Hobbes, John Wallis and Galileo Galilei amongst them.

Gabriel’s Wedding Cake

The surface generated when the curve {y=1/x} is replaced by the step-function {y=1/[x]} is shown in the figure below. The volume is comprised of a series of cylinders of unit length and radii decreasing as {1/n}. It is easy to calculate the volume by adding up the volumes of all the cylinders. This is

\displaystyle V = \pi \sum_{n=1}^{\infty} \frac{1}{n^2}\times 1

The sum here is that of the famous Basel Problem, solved by Euler in 1734. He showed that the sum of the square reciprocals of the natural numbers is {\pi^2/6}. Thus, we find that the total volume is  {V=\pi^3/6}.

Torricelli-04

Gabriel’s Wedding Cake.

On the other hand, the surface area is infinite. This is easily seen by adding the areas of the curved surfaces of the cylinders:

\displaystyle S = 2\pi \sum_{n=1}^{\infty} \frac{1}{n}\times 1

The sum is the harmonic series, which is well-known to be divergent. By the way, inclusion of the flat annular surfaces between the cylinders makes no difference: if the volume is collapsed down the {x}-axis, the annuli collapse into a unit circle with area {\pi}.

Torricelli-05

Gabriel’s Wedding Cake as it might appear on the breakfast table.

An alternative name for Torricelli’s Trumpet is Gabriel’s Horn, the horn to be blown by the Archangel Gabriel to announce Judgement Day. The step-function surface, based on {1/[x]}, has been called Gabriel’s Wedding Cake (Fleron, 1999) as it resembles a wedding cake with an infinite number of tiers . While this cake has a finite volume, and so can be consumed by the wedding guests, the icing of the cake would pose difficulties as the surface area is unbounded.

The Painter’s Paradox

Now let us imagine filling the finite volume within Torricelli’s Trumpet with paint. Assuming SI units, a unit radius at the edge of the bell implies a horn of diameter 2 metres, enough to wake the dead. The volume is {\pi}, so about three cubic metres or three thousand litres of paint are required to fill it. However, once the vessel is filled, the inner surface will be completely covered. This is strange: the surface has infinite area, so how can a finite volume of paint cover it? This is the Painter’s Paradox.

Tile-Painting

In fact, it is quite feasible to cover an infinite area with a finite volume of paint (note: we are doing mathematics here, not physics!). Take the example of an area made up of an infinite row of square tiles, each of unit area (figure above). Paint the row so that the thickness of paint covering the first tile is 1, that of second is {1/2^2} and in general that of the {n}-th tile is {1/n^2}. Then the total volume of paint is

\displaystyle V = \sum_{n=1}^{\infty} \frac{1}{n^2}

This is Euler’s sum once again, so the total volume of paint is {V=\pi^2/6}.

Sources

{\bullet} Fleron, Julian F., 1999: Gabriel’s Wedding Cake. College Math. Journal, 30, 1, 35-38.

{\bullet} Wikipedia: Gabriel’s Horn.


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