Floating metal spheres

9 Mar 2023

For some reason, while coming back from a trip abroad a couple of weeks ago I started to think about hollow metallic spheres.

Take a hollow sphere, made out of some metal, with thickness \(r > 0\), radius \(R > 0\) and filled with vacuum or a gas less dense than air. There exists a radius \(R^\star > 0\) such that the associated sphere is less dense than air. Would such a sphere float on air? It would appear so. This is a similar mechanism of how boats float on water. The gas inside may also be hot air. Balloons or globes operate similarly. But there is something bizarre about making the sphere out of metal.

Obviously, the radius \(R^\star\) would be huge. This may cause practical problems of how to build such a sphere, but it would also create other problems. The radius \(R^\star\) would be orders of magnitude bigger than the thickness r. Reading about the theoretical existence of these spheres, apparently this may cause structural issues where the sphere collapses onto itself due to atmospheric pressure.

Apparently, a writer called Buckminster Fuller thought of these spheres as possibly holding entire cities. He called them Cloud Nine, and postulated that a sphere a mile wide where the air inside is one degree hotter than the ambient air would be enough to make the sphere float.

The things we think about while travelling!