Gabriel's Horn
Gabriel's horn, also called Torricelli's trumpet, is the surface of revolution of the function 
about the x-axis for
. It is therefore given by parametric
equations
The surprising thing about this surface is that it (taking 
for convenience here) has finite volume
but infinite surface area, since
This leads to the paradoxical consequence that while Gabriel's horn can be filled up with 
cubic units of paint, an infinite number of square units
of paint are needed to cover its surface!
The coefficients of the first fundamental form are,
and of the second fundamental form are
The Gaussian and mean curvatures are
The Gaussian curvature can be expressed implicitly as
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(21) |
See also
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Cite this as:
Weisstein, Eric W. "Gabriel's Horn." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GabrielsHorn.html