Floating Bodies of Equilibrium at Density 1/2 in Arbitrary Dimensions

TL;DR

This paper explores the mathematical conditions under which bodies of density 1/2 can float in all orientations across arbitrary dimensions, revealing a large family of solutions through expansions from spheres.

Contribution

It introduces a general framework for constructing bodies of density 1/2 that float in all orientations in any dimension, using power series expansions from spheres and water plane envelopes.

Findings

Existence of solutions in arbitrary dimensions

Multiple methods for constructing such bodies

Large solution manifold with deformable spheres

Abstract

Bodies of density one half (of the fluid in which they are immersed) that can float in all orientations are investigated. It is shown that expansions starting from and deforming the (hyper)sphere are possible in arbitrary dimensions and allow for a large manifold of solutions: One may either (i) expand r(n)+r(-n) in powers of a given difference r(u)-r(-u), (r(n) denoting the distance from the origin in direction n). Or (ii) the envelope of the water planes (for fixed body and varying direction of gravitation) may be given. Equivalently r(n) can be expanded in powers of the distance h(u) of the water planes from the origin perpendicular to u.