Neutrally floating objects of density 1/2 in three dimensions

TL;DR

This paper investigates the existence and shapes of three-dimensional objects with density 1/2 that can float neutrally in any orientation, using mathematical formulation and numerical methods to explore their properties.

Contribution

It formulates the floating problem as an initial value problem for cylindrically symmetric objects and demonstrates the existence of diverse neutrally floating shapes through numerical integration.

Findings

Numerical solutions reveal a variety of neutrally floating shapes.

The problem is formulated as an initial value problem with a unique solution in certain cases.

Supports the hypothesis of the existence of non-spherical neutrally floating objects.

Abstract

This paper is concerned with the Floating Body Problem of S. Ulam: the existence of objects other than the sphere, which can float in a 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-Lindel\"of theorem. Numerical integration of the initial value problem provides a rich variety of neutrally floating shapes.