torricelli's trumpet

Torricelli’s Trumpet (Gabriel’s Horn)

Torricelli’s trumpet is a shape that has an infinite surface area while maintaining a finite volume.

Taking the graph of f(x) = 1/x for all x >= 1 and revolving it about the x axis creates the shape shown above. Computing the volume of the shape from 1 to any point greater than 1, a, the following can be shown:

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This shows that as a increases, the volume gets closer and closer to the value pi. Taking the limit of this solution as it approaches infinity, the value converges to pi–because a gets infinitely big, 1/a becomes 0. 

Calculating the surface area using the equation, similar to above, from 1 to any point a, gives the following:

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Unlike the solution to the volume equation, taking the limit of a as it approaches infinity for this solution diverges, because as aapproaches infinity, ln(a) approaches infinity. Therefore, with an infinite surface area, there is an finite volume.

Calculus, bitch.

*images of computations taken from wikipedia*

Torricelli’s trumpet encloses a finite volume but has an infinite area. It is formed by taking the graph of y=1/x, with the domain x≥1 (thus avoiding the asymptote at x = 0) and rotating it in three dimensions around the x-axis. It is also called Gabriel’s Horn; the name refers to Archangel Gabriel as he blows the horn to announce Judgement Day, thereby associating the infinite with the divine.

Evangelista Torricelli ( 1608 - 1647 ) was a student of Galileo.  As a young man he studied in Galileo’s home at Arcetri near Florence.  Upon Galileo’s death, Torricelli succeeded his teacher as mathematician and philosopher for their good friend and patron, the Grand Duke of Tuscany.

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Torricelli’s own words fully describe his amazement at discovering an infinitely long solid with a surface that calculates to have an infinite area, but a finite volume.  “It may seem incredible that although this solid has an infinite length, nevertheless none of the cylindrical surfaces we considered has an infinite length but all of them are finite."  This "incredible” paradox prompted Torricelli to try several alternate proofs.

Torricelli was born a bit too soon.  The study of infinitesimal was too new.  Recall that Newton was born the year Galileo had died (1642) and Leibniz was yet four years younger.  Unfortunately, Torricelli did not live to see the methods of calculus fully emerge to confirm his painstaking calculations largely based on his friend Cavalieri’s “summation of plane slices” method.