Gabriel's horn

3D-model of Gabriels Horn

Gabriel's horn (also Torricelli's trumpet ) is a body described by Evangelista Torricelli that has an infinite surface area but a finite volume . The name derives on the one hand from the shape resembling a wind instrument, on the other hand from the tradition of viewing the Archangel Gabriel as the angel who blows the horn to announce the Last Judgment , whereby infinity is associated with divinity.

Mathematical definition

The left end of Gabriel's horn

Plot of the hyperbolic function y = 1 / x

Gabriels Horn results when the graph of the function with the domain of definition (to avoid the pole at x = 0) is rotated around the x-axis. The volume ( ) and surface area ( ) of this solid of revolution are calculated as follows: ${\displaystyle y={\tfrac {1}{x}}}$ ${\displaystyle x\geq 1}$ ${\displaystyle V}$ ${\displaystyle A}$

${\displaystyle V=\pi \int _{1}^{\infty }{1 \over x^{2}}\mathrm {d} x=\pi }$

${\displaystyle A=2\pi \int _{1}^{\infty }{\frac {1}{x}}{\sqrt {1+{\frac {1}{x^{4}}}}}\mathrm {d} x>2\pi \int _{1}^{\infty }{\frac {1}{x}}\ \mathrm {d} x=\infty }$

In the case of the volume - as with all functions with - the improper integral from 1 to exists , that is, it converges to a finite value because it is ${\displaystyle \,f(x)={x}^{-n}}$ ${\displaystyle n>1}$ ${\displaystyle \infty }$

${\displaystyle \int _{1}^{\infty }x^{-n}\mathrm {d} x=\left[{\frac {1}{1-n}}x^{1-n}\right]_{1}^{\infty }={\frac {1}{n-1}}.}$

In order to be able to estimate the size of the lateral surface, the function must be integrated. An antiderivative is the natural logarithm and the integral over the range from 1 to a fixed one is: ${\displaystyle f(x)={\tfrac {1}{x}}}$ ${\displaystyle a}$

${\displaystyle \int _{1}^{a}{\frac {1}{x}}\mathrm {d} x=\left[\ln x\right]_{1}^{a}=\ln a.}$

Since the natural logarithm is unlimited, there is no finite limit of this integral for , so that the surface area of ​​the body is infinitely large. ${\displaystyle a\to \infty }$

Amount of color when filling and covering with color

Since Gabriel's horn has a finite volume, it can be filled with a finite amount of color. However, to cover an infinitely large area you need an infinitely large amount of paint. If you look at the inside of the horn, on the one hand - because of the infinitely large area - an infinite amount of color seems to be required to cover it. On the other hand, when the horn is filled, the inside is completely covered, for which only a finite volume is required.

This apparent paradox does not take into account the fact that the paint layer has a certain thickness when it is actually covered with paint. When this finite thickness becomes greater than the radius of the horn, the color fills the entire cross-section of the horn. Then the required amount of paint is no longer determined by the surface, but by the volume. The required amount of paint cannot therefore be determined by multiplying the infinitely large area by a finite thickness of the paint layer. If, on the other hand, one assumes an infinitely thin layer of paint without volume properties, one cannot compare its non-existent volume with the volume of the body.

See also

• Pseudosphere
• Zeno's paradoxes of multiplicity

Web links

• Thomas Peters: Gabriels Horn (PDF; 104 kB)
• Torricelli's Trumpet on PlanetMath
• Eric W. Weisstein : Gabriel's Horn . In: MathWorld (English).

Individual evidence

1. ↑ Johanna Heitzer: Spiralen, a chapter of phenomenal mathematics. Ernst Klett Schulbuchverlag, Leipzig 1998, p. 48