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The beautiful math behind the properties of the horn of infinity might just blow your mind!

Horn Of Infinity: The Beauty Of Mathematical Paradoxes!

Many mathematical problems and statements have confounded and confused people from time to time. The bulk of mathematics lies in its formulation of beautiful, yet abstract concepts and their applications in real life. One such concept is the concept of solid of revolution, a method to determine the volume of an object.

Basic integration to the rescue

The method of definite integration is a mathematical procedure used to calculate the area under a curve bounded by an axis and two limits.  This can be extended to finding volume as well. By definite integration, we can find areas of thin circular disks. Now, imagine stacking up all of these disks on top of each other; they contribute to a solid figure. Addition in this case gives the volume of the solid obtained thus. If the disks are infinitesimally thin, integration is used instead of addition.

Hence finding the volume of a solid is as simple as finding a curve such that the rotation of the chosen curve about an axis reconstructs the solid, which when subjected to an operation, viz. integration will give the volume. At the same time there is a caveat which demands the solid in consideration must be easily decomposable into spherical disks.

The established formulae used in mensuration to compute columes can be verified using this method.

As soon as the method was put to practice, mathematicians tried to experiment with various curves, found the volumes bounded by them, to check the validity of this method and further establish it as a standard method for the calculation of volumes. Thus the method has proved advantageous for the mathematicians, industrialists and many other people.

An equation that has been of great interest is as follows:

As we know, as x tends to infinity, y tends to 0 asymptotically, which means that it gets very close to zero and converges after infinite distance. Thus common sense says that the length of the solid obtained shall also tend to infinity.

The shape obtained from rotating the equation about the x-axis resembles a trumpet. Hence, the solid so obtained is called Torricelli’s trumpet or Gabriel’s horn or the horn of infinity. The scientist and mathematician, Evangelista Torricelli studied the properties of this solid in the 17th century.

The paradox of the horn of infinity, or the painter’s paradox :

As the length of this solid is infinity, the surface area of this solid should also be infinity, which means that it would require an infinite amount of paint to cover the outer or inner surface of this solid.

Using the method of solid of revolution, it was established that the volume of the horn of infinity is π cubic units.  π not  only a unique, interesting and exotic value, but also the finite volume of a solid with infinite surface area. Thus, the horn can hold finite amount of paint but at the same time the paint won’t be enough to paint the horn, which is referred to as painter’s paradox!

To explore more about solid of revolutions, do look at the resources mentioned.

Resources:

  1. http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx
  2. http://www.dummies.com/how-to/content/how-to-find-the-volume-and-surface-area-of-gabriel.html

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