A floating (and open) problem

There are so many interesting books out there in the world, but among the most interesting are those contributed to by dozens, hundreds or even thousands of people bound by a common interest. A particular example of the species is a book of mathematical problems, a usually hefty tome left in a place liable to be frequented by mathematicians who are wise enough to know that what they don’t know dwarfs what they do know, and are inclined to inscribe in it for posterity some of the open problems that are vexing them — and offering for their solution the occasional alcoholic or even more exotic reward.

I’ve personally leafed through the book of this type maintained by the Mathematisches Forschungsinstitut Oberwolfach, but even this venerable institution is following in mightier footsteps: the Scottish Café of Lwów — frequented in the 1930s and 1940s by titans such as Stefan Banach, Stanisław Mazur, Hugo Steinhaus, Stanisław Ulam and many others — and its Scottish Book. Problem 19, posed by Ulam, is a perfect example of the kind of simply-posed yet very thorny question that takes one quite by surprise:

Is every solid of uniform density that will float in water in every position a sphere?

The first hint that the answer to Problem 19 might be “no” comes in dimension two, although it might be easier to visualize it in dimension three with an extra symmetry like this: is it true that every solid prism (body of constant cross-section) that will float stably in water with its central axis horizontal, but regardless of how you rotate it about that axis, must be a solid tube with circular cross-section like this ⬤?

Certainly not every prism has this property. For example, consider a prism with square cross-section and positive density much less than that of water. Then, by Archimedes’ Principle, when this square prism floats in water, it displaces very little water; it’s rather like an upside-down iceberg, with only a small bit below the waterline and most of its bulk above the waterline. Now try getting this prism to float not as a square like this ■ but with an edge uppermost as a diamond like this ◆; if you’re even slightly off plumb in the second situation, then there will be enough torque around the centre of mass (which will be above the waterline) to rotate the unstable ◆ about its central axis into the ■ configuration. In particular, this square prism does not float stably in every position.

However, in 1938, Herman Auerbach showed that there do exist non-circular prisms that will float in every horizontal position. He showed that the desired floating property will hold for any prism that

  1. is of density exact half that of water, and
  2. has a cross section of shape S such that every straight line that cuts S in such a way as to halve the perimeter of S must also halve the area of S — a simple example of a non-circular shape with this property is the British fifty pence piece, and Auerbach provided an infinite family of such shapes.

Amazingly, Problem 19 remains unsolved in full generality, even today! What we do know in arbitrary dimension n ≥ 2 is that if the solid floats as required, is centrally symmetric, and has density ½, then it must be a sphere. Beyond that… Well, if you feel adventurous and want to try to dispose of symmetry or the density ½ assumption, go ahead, and claim your prize from Ulam’s ghost!

  I would prefer to write “ball” rather than “sphere” since the problem obviously means the solid body (the ball) and not the hollow shell (the sphere), but my English translation of The Scottish Book uses “sphere”, so I’ll try to follow that convention for the rest of this post.

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