Gabriel’s Horn and the Painter’s Paradox

Gabriel’s Horn is a three-dimensional horn shape with the counterintuitive property of having a finite volume but an infinite surface area.

This fact results in the Painter’s Paradox — A painter could fill the horn with a finite quantity of paint, “and yet that paint would not be sufficient to coat [the horn’s] inner surface” [1].

If the horn’s bell had, for example, a 6-inch radius, we’d only need about a half gallon of paint to fill the horn all the way up. Even though this half gallon is enough to entirely fill the horn, it’s not enough to even coat a fraction of the inner wall!

The mathematical explanation is a bit confusing if you haven’t taken a first course in calculus, but if you’re interested, you can check it out here.

Mathematica code:

x[u_, v_] := u
y[u_, v_] := Cos[v]/u
z[u_, v_] := Sin[v]/u
Manipulate[ParametricPlot3D[{{x[u, v], y[u, v], z[u, v]}},
    {u, 1, umax}, {v, 0, 2*Pi},
    PlotRange -> {{0, 20}, {-1, 1}, {-1, 1}},
    Mesh -> {Floor[umax], 20}, Axes -> False, Boxed -> False],
    {{umax, 20}, 1.1, 20}]

Additional source not linked above.