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.


Rotational Stability

Time for an experiment! Find a book and secure it shut using tape or a rubber band. Now experiment with spinning the book while tossing it into the air. You’ll notice that when the book is spun about its longest or shortest axis it rotates stably, but when spun about its intermediate-length axis it quickly wobbles out of control.

Every rigid body has three special, or principal axes about which it can rotate. For a rectangular prism — like the book in our experiment — the principal axes run parallel to the shortest, intermediate-length, and longest edges, each going through the prism’s center of mass. These axes have the highest, intermediate, and lowest moments of inertia, respectively.

When the book is tossed into the air and spun, either about its shortest or longest principal axis, it continues to rotate about that axis forever (or until it hits the floor). For these axes, this indefinite, stable rotation occurs even when the axis of rotation is slightly perturbed.

When spun about its intermediate principal axis, though, the book also continues to rotate about that axis indefinitely, but only if the axis of rotation is exactly in the same direction as the intermediate principal axis. In this case, even the slightest perturbation causes the book to wobble out of control.

The first simulation above shows a rotation about the unstable intermediate axis, where a slight perturbation causes the book to wobble out of control. The second and third simulations show rotations about the two stable axes.

Unfortunately, as far as my understanding goes, there’s no intuitive, non-mathematical explanation as to why rotations about the intermediate principal axis are unstable. If you’re interested, you can find the stability analysis here.

Mathematica code posted here.

Additional sources not linked above: [1] [2] [3] [4]