next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |
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 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, |
![]() ![]() |
See also the "Gabriel's
cake", the tower
with constant pressure, and the second
tractroid.
next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |
© Robert FERRÉOL 2022