The question of 06/98 is the problem 19 in the Scottish Book, which contains a large number of mathematical problems posed by mathematicians in Lviv (Lemberg) in the nineteen thirties.

Recently I considered the question (a): Are there long logs of non-circular cross-section with half the density of water which can float in any position without tendency to rotate?

It turns out, that there are non-circular solutions to this problem.

However, after finding the solution Yacov Kantor informed me, that this part of the problem was already solved by Auerbach in 1938.

Next I considered the problem (b) with densities ρ different from one half. There are one-parameter families of cross-sections for ρ not equal 1/2 which have a p-fold rotation axis. For given p they exist for p-2 densities ρ. There are strong indications, that for all p-2 densities one has the same family of cross-sections.

All this can be found in physics/0203061

Meanwhile I have found the differential equation which describes the boundary curve. The conjecture that one has the same family of cross-sections for all p-2 densities proved true. A short account can be found here , and more in physics/0205059

The main results of physics/0203061 and physics/0205059 are published in Studies in Applied Mathematics 111 (2003) 167-183

Deborah Oliveros-Braniff and Luis Montejano Peimbert looked for special solutions of this problem. Revista Ciencias 55-56 (1999) 46-53 They restricted themselves to the case where the edges of an equilateral pentagon constitute five waterlines. They call these 'volantines'. They imagine five cyclists each in the middle of an edge moving in the direction of the edge, and the five corners moving along the boundary (figure 1 and 2). Thus they restrict themselves to the ratio of circumference over length of the boundary line below water equal to five. They found a solution with p=7 which, however, was not sufficiently convex (figure 9 of their paper).

Recently Sergei Tabachnikov informed me that there is an equivalent problem which goes by the name of tire track problem. The problem consists in determining the direction of bicycle motion from the tire tracks of the bicycle wheels. If this is impossible then the curves are called bicycle curves. The track of the front wheel corresponds to the boundary of the floating log, and the track of the rear wheel to the envelope of the waterlines.

David L Finn has considered this problem and gives a construction if a certain piece of the track of the rear wheel is given. However, he does not give a solution for a closed path which would be necessary for the floating-body problem.

On the other hand the differential equation derived in my paper yields solutions for bicycle curves which are not closed, too.

In physics/0603160 the explicit solution in terms of the differential equation derived earlier is given. It is shown explicitly, that this yields solutions to the floating body and the tire track problem.

A more elegant version is given in physics/0701241 which contains figures of a large number of curves. Meanwhile I have prepared an introduction under the title Three Problems - One solution nearly without mathematics, but with many animations.

Es gibt auch eine deutsche Version dieser Einführung unter dem Titel Drei Probleme - Eine Lösung fast ohne Mathematik, aber mit vielen Animationen.

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