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GABRIEL'S HORN

Surface studied by Roberval
and Torricelli in 1641.
Gabriel: archangel (and hornist in his spare time...). Other name: acute hyperbolic solid (name given by Torricelli), hyperbolic funnel. 
Cylindrical equation: .
Cartesian equation: . Quartic surface. Cartesian parametrization: . Volume of the portion obtained for : . Area of this portion: . 
Gabriel's horn is the surface of revolution obtained by rotating a rectangular hyperbola around one of its asymptotes.
This surface displays a really surprising paradox: if we want to fill the tube formed by this surface (which has an infinite length), we will only need a finite amount of liquid, but if we want to paint it, we will need an infinite amount of paint!


This paradox can be solved in the following way: we are assuming that the layer of paint has a constant thickness. But we could very well paint the trumpet with a finite amount a paint and with the thickness of the layer going to 0 at infinity. This is precisely what would happen if we filled the inside with paint!
The surface, quite close to the previous one, with cylindrical equation , can be encountered in daytoday life, especially in kitchens; it is indeed the shape assumed by a trickle of water coming out from a faucet with constant output.
Indeed, it can be proved that the radius r of the trickle of water is given as a function of z (the origin of the axis being at the faucet, the axis pointing downward, being the output speed of the water) by the formula . 
See also the tower with constant pressure, and "Gabriel's cake".
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© Robert FERRÉOL 2017