A floating body with no preferred orientation: an experimental realization

Abstract: We present a simple experimental realization of a two-dimensional floating body that can remain in equilibrium in any orientation. This system is based on a class of shapes known as Zindler curves, which possess the remarkable geometric property that all chords dividing their area into equal parts have the same length. Using a multilayer fabrication approach, we construct a heart-shaped floating object with an effective density close to one half of that of the surrounding liquid. We show experimentally that, under these conditions, the object exhibits neutral equilibrium with respect to rotation. When the density is slightly varied, preferred orientations emerge, consistent with a simple energy-based description. Our experiments highlight both the accessibility of this classical problem and the subtle role of physical effects such as density inhomogeneities and capillarity. They provide a simple platform to explore the interplay between geometry and buoyancy, and to test geometric results in a tangible setting.

• The paper demonstrates that a heart-shaped Zindler curve attains orientation-independent floating at a critical density ratio of 0.5 through rigorous experimental validation.
• The experimental methodology employs a 3D-printed multilayer design and a finely tuned water/ethanol mixture to control the object's density with an uncertainty of approximately 0.01.
• Findings reveal that slight deviations from the critical density introduce capillary effects and preferred orientations, with relaxation dynamics aligning with a harmonic oscillator model.

Experimental Realization of a Two-Dimensional Floating Body with No Preferred Orientation

Introduction and Background

The paper addresses a classical problem in hydrostatics and geometry, originated from Ulam’s floating body problem—a question asking whether only spheres (in three dimensions) or circles (in two dimensions) can float in equilibrium in all orientations, assuming homogeneity and a specific density. In the two-dimensional case, Auerbach and Zindler identified entire classes of non-circular solutions, known as Zindler curves, at the critical density ratio $a = \rho_{\mathrm{obj}}/\rho_{\mathrm{liq}} = 0.5$, where these shapes can float in neutral equilibrium for any orientation. Despite decades of mathematical progress, practical realization and direct experimental validation of these geometric bodies remained absent.

Theoretical Analysis of Orientation-Independent Floating

The paper develops the energy-based formalism of floating equilibrium. The analysis considers the division of the two-dimensional object by the waterline into emerged and submerged regions, with corresponding centers of mass. Hydrostatic equilibrium conditions demand that, for all orientations, the buoyancy force acts vertically below the center of mass, which is realized if and only if the locus of buoyancy traces a circle centered at the object's centroid. The critical geometric constraint is that all area-bisecting chords of the body must have equal length—precisely the defining property of Zindler curves.

A particularly symmetric situation arises at $a=1/2$, where this class admits an infinite family of solutions beyond the circle. The symmetry simplifies the energy potential, making it orientation-independent and providing neutral equilibrium. The heart-shaped Zindler curve constructed in the study demonstrates these theoretical properties.

Experimental Realization

Recognizing the sensitivity of the system to density inhomogeneities, the authors implement a multilayer fabrication: a 3D-printed thin contour of the target curve (heart-shape) is sandwiched between PMMA plates, enabling fine control over the effective density and mass distribution. The liquid medium is tuned using a water/ethanol mixture, setting the liquid density to match the targeted $a = 0.5$ condition. The effective densities are measured with approximately 0.01 absolute uncertainty.

The floating body is observed in a controlled tank, and image analysis yields quantitative reconstruction of the object’s contour, waterlines, centers of buoyancy, and mass in various orientations.

Empirical Results: Neutral and Preferred Orientations

For densities very close to the critical value ($a \approx 0.5$), the body exhibits orientation-independent floating; the object, when disturbed, retains the imposed orientation instead of relaxing to a unique equilibrium. Quantitative analysis shows constancy of the waterline length ($40.4\ \text{mm} \pm 0.9\ \text{mm}$, theoretical: $40.0\ \text{mm}$) and the center of mass almost always on the waterline, in excellent agreement with theory.

For densities slightly away from the critical point ($a=0.45$ and $a=0.55$), energy landscape computations display clear modulations and the emergence of three preferred orientations, differing by about $60^\circ$. The experimentally observed relaxation dynamics of the object after perturbations are consistent with a harmonic oscillator model defined by the curvature of the potential energy landscape, with extracted oscillation frequencies roughly $1.15$--$a=1/2$0 depending on the density deviation.

Physical Effects: Capillarity and Density Inhomogeneities

The study identifies critical sources of deviation from ideal behavior. Minor capillary forces and associated torques, though small (torque $a=1/2$1, vertical force $a=1/2$2 of the floater weight), shift the effective critical density from $a=1/2$3 to $a=1/2$4. At the flat energy landscape (near $a=1/2$5), contact line pinning is hypothesized to explain the lack of spontaneous reorientation despite the presence of small capillary torques; angular deviations in final orientations are found to be on the order of $a=1/2$6--$a=1/2$7.

Implications and Future Research

This work bridges a gap between geometric theory and physical implementation by constructing and validating a two-dimensional shape with orientation-independent floating, confirming the non-uniqueness of the circle at $a=1/2$8. The experimental platform demonstrates fine control over parameters and direct visual access to equilibrium phenomena, offering a didactic and research tool for studying geometry-buoyancy interactions, capillarity, and stability.

Potential directions include further exploration of other Zindler curves, systematic analysis of configurations for densities deviating from $a=1/2$9, and extension to three-dimensional analogues, which remain a significant theoretical challenge. The study also invites development in the fabrication of bulk homogeneous bodies and real-time analysis of capillary effects at the fluid-solid interface.

Conclusion

The paper delivers a rigorous experimental confirmation of the existence of two-dimensional floating bodies (specifically, Zindler curves at $a = 0.5$0) that exhibit no preferred orientation. The research provides both theoretical and practical insight into the geometric and physical elements of the floating body problem. This approach yields valuable avenues for pedagogical demonstration and advanced investigations at the intersection of geometry, fluid mechanics, and materials science, while highlighting the nuanced influence of non-idealities such as capillarity and density inhomogeneity.