Neutrally Floating Objects of Density ½ in Three Dimensions


P. L. Várkonyi, Budapest University of Technology and Economics, Műegyetem rkp. 1–3, 1111 Budapest, Hungary; e-mail:


This paper is concerned with the Floating Body Problem of S. Ulam: the existence of objects other than the sphere, which can float in liquid in any orientation. Despite recent results of F. Wegner pointing towards an affirmative answer, a full proof of their existence is still unavailable. For objects with cylindrical symmetry and density 1/2, the conditions of neutral floating are formulated as an initial value problem, for which a unique solution is predicted in certain cases by a suitable generalization of the Picard–Lindelöf theorem. Numerical integration of the initial value problem provides a rich variety of neutrally floating shapes.