Floating Bodies of Equilibrium I

TL;DR

This paper investigates the geometric conditions under which variously shaped floating bodies can maintain equilibrium in all orientations, extending previous findings to non-circular cross-sections and different densities.

Contribution

It introduces new families of cross-sectional shapes capable of floating in all orientations, generalizing earlier results and exploring their dependence on density and symmetry.

Findings

Existence of non-circular cross-sections that float in all orientations.

Identification of p-fold symmetric shapes for specific densities.

Indications of a universal family of shapes across certain densities.

Abstract

A long cylindrical body of circular cross-section and homogeneous density may float in all orientations around the cylinder axis. It is shown that there are also bodies of non-circular cross-sections which may float in any direction. Apart from those found by Auerbach for rho=1/2 there are one-parameter families of cross-sections for rho not= 1/2 which have a p-fold rotation axis. For given p they exist for p-2 densities rho. There are strong indications, that for all p-2 densities one has the same family of cross-sections. (Proven in physics/0205059)