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

(1) (2) (3)

The surprising thing about this surface is that it (taking for convenience here) has finite volume

(4) (5) (6)

but infinite surface area, since

(7) (8) (9) (10) (11) (12)

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,

(13) (14) (15)

and of the second fundamental form are

(16) (17) (18)

The Gaussian and mean curvatures are

(19) (20)

The Gaussian curvature can be expressed implicitly as