A floating body with no preferred orientation: an experimental realization

This paper presents an experimental realization of a heart-shaped floating object based on Zindler curves that maintains neutral equilibrium in any orientation when its density is half that of the surrounding liquid, demonstrating the interplay between geometry and buoyancy while highlighting the effects of density variations and capillarity.

The Gist

Imagine you have a toy boat. Usually, if you push it sideways or tilt it, it wobbles and then snaps back to a specific "comfortable" position. Maybe it wants to float with its nose pointing north, or maybe it likes to lie flat. It has a favorite way to sit in the water.

But what if you had a magical boat that didn't care? What if you could spin it, tilt it, or turn it upside down, and it would just stay exactly where you put it, perfectly happy in any position? It wouldn't try to right itself, nor would it flip over. It would be perfectly balanced in every single direction.

That is exactly what the scientists in this paper managed to build. They created a floating heart-shaped object that can sit in water in any orientation without trying to move.

Here is the story of how they did it, broken down into simple concepts:

1. The "Magic Shape" (The Zindler Curve)

The secret isn't just the material; it's the shape. The scientists used a special geometric shape called a Zindler curve.

Think of a pizza. If you cut a slice right through the middle, you get two equal halves. Now, imagine a magic pizza where no matter how you cut it through the center, as long as you split the area in half, the length of the cut is always exactly the same.

That is the weird, magical property of this heart shape. Because every "cut" that splits the heart into two equal halves is the same length, the water doesn't "feel" any difference when the object rotates. The physics of the water pushing up (buoyancy) stays perfectly balanced no matter how the heart is turned.

2. The "Goldilocks" Density

Having the right shape isn't enough; you also need the right weight.

• If the object is too heavy, it sinks.
• If it's too light, it floats too high.
• The scientists needed the object to be exactly half as dense as the water.

Think of it like a seesaw. If the object is half the weight of the water it displaces, the "up" force and the "down" force are perfectly matched. In this specific "Goldilocks" zone, the object becomes neutrally stable. It's like a ball sitting on a perfectly flat, infinite table. If you push it, it doesn't roll back, and it doesn't roll away; it just stays where you put it.

3. The "Sandwich" Trick

You might ask, "Why didn't they just 3D print a plastic heart?"
The problem is that 3D printers aren't perfect. Tiny air bubbles or slight variations in the plastic make the object slightly uneven. Even a tiny bit of unevenness makes the heart want to tip over to one side.

To solve this, the scientists built a sandwich:

• They 3D printed a thin, dark outline of the heart.
• They sandwiched this outline between two clear, flat plastic sheets.
• This created a "2D" object that was perfectly flat and uniform.

By adjusting how thick the plastic sheets were versus the printed outline, they could fine-tune the weight until it was exactly half the density of the water.

4. What Happens When You Mess Up the Balance?

The scientists also played a game: "What happens if we get the weight slightly wrong?"

• If it's a tiny bit too heavy: The heart suddenly develops "favorite" positions. It will wobble and then snap into one of three specific angles to rest.
• If it's a tiny bit too light: It snaps into a different set of three favorite angles.

This showed them that the "perfect balance" is a very delicate state. It's like balancing a pencil on its tip. If you are perfect, it stays. If you are off by a hair, it falls to one side.

5. The Invisible "Sticky" Water

In the real world, water isn't just a smooth force; it has "stickiness" (surface tension) at the edges where the water meets the plastic. This creates tiny forces that usually mess up these perfect experiments.

However, the scientists found that because their shape was so mathematically perfect, these sticky forces canceled each other out almost entirely. The only reason the heart didn't float perfectly at exactly 50% density was that the water's surface tension pulled it down just a tiny bit, so they had to make the object slightly lighter (about 49% density) to compensate.

The Big Takeaway

This experiment is a beautiful marriage of math and physics.

• Math gave them the shape (the heart/Zindler curve).
• Physics told them the weight (half-density).
• Engineering helped them build it (the sandwich method).

It proves that a simple geometric idea—something that looks like a fun puzzle on a piece of paper—can actually exist in the real world. It's a floating heart that doesn't care which way it faces, a perfect example of how nature loves symmetry and balance.

Technical Summary

1. Problem Statement

The paper addresses a classical problem in fluid mechanics and geometry known as Ulam's floating body problem. Specifically, it investigates whether non-spherical, homogeneous solids can float in hydrostatic equilibrium in any orientation.

• Theoretical Context: While a sphere is the obvious solution, the mathematical question is whether other shapes exist. In 2D (cross-sections of cylinders), it was proven by Auerbach (1938) that non-circular shapes exist if the relative density ($\alpha = \rho_{obj}/\rho_{liq}$) is exactly $1/2$.
• Geometric Condition: These shapes are bounded by Zindler curves, defined by the property that all chords dividing the area into two equal parts have the same length.
• The Gap: Despite extensive mathematical literature, there were no prior experimental realizations of such objects. The authors aimed to bridge the gap between abstract geometric theory and physical reality, testing the robustness of the theory against physical imperfections like density inhomogeneities and capillary forces.

2. Methodology

Theoretical Framework

The authors derived the conditions for orientation-independent floating using an energy-based approach:

• Potential Energy ($E_p$): The system's potential energy depends on the vertical distance between the center of mass of the emerged part ($C_1$) and the immersed part ($C_2$).
• Equilibrium Condition: For the object to float in equilibrium at any angle $\theta$, $E_p$ must be independent of $\theta$. This requires that as the object rotates, the center of mass of the immersed region ($C_2$) moves along a circle centered on the object's total center of mass ($G$).
• Geometric Implication: This circular motion of $C_2$ implies that the length of the waterline chord ($L$) must be constant for all orientations that bisect the area. This confirms the definition of a Zindler curve.

Experimental Fabrication

Direct 3D printing of homogeneous cylinders failed to achieve the necessary density homogeneity ($\alpha \approx 0.5$). The authors adopted a "sandwich" fabrication approach:

• Structure: A thin, black 3D-printed profile (defining the heart-shaped Zindler curve) was sandwiched between two transparent PMMA plates.
• Density Control: The effective density was tuned by adjusting the thickness of the low-density printed core relative to the high-density PMMA plates.
• Liquid Tuning: To achieve the critical relative density $\alpha = 0.5$ precisely, the liquid medium was a mixture of water and ethanol. The ratio was adjusted until the floater exhibited neutral equilibrium.

Experimental Setup

• Imaging: A glass tank with a dark background was used. A lateral camera captured the floating object from a distance to minimize parallax.
• Analysis: An image-processing routine extracted the object's contour, the waterline, and the centers of mass ($G$, $C_1$, $C_2$) for various orientations.
• Perturbation: The object was manually rotated using thin rods and released to observe its relaxation dynamics.

3. Key Results

Neutral Equilibrium at $\alpha \approx 0.5$

• Observation: When the relative density was tuned to $\alpha \approx 0.49$ (slightly below the theoretical 0.5), the heart-shaped object remained in equilibrium in any orientation. It did not return to a preferred angle upon release.
• Geometric Verification:
  • The waterline length remained constant ($\approx 40.4$ mm) across all orientations, with variations $< \pm 0.9$ mm.
  • The center of buoyancy ($C_2$) traced a circular path centered on the object's center of mass ($G$), confirming the theoretical prediction.
  • The envelope of the waterlines formed a triangular caustic, matching the geometric construction of the specific Zindler curve used.

Deviation from Critical Density ($\alpha \neq 0.5$)

• Energy Landscape: When $\alpha$ was set to $0.45$ (lighter) or $0.55$ (denser), the potential energy became orientation-dependent.
• Stable Orientations: The energy profile revealed three stable equilibrium positions separated by $\approx 60^\circ$.
• Dynamics:
  • Objects released from non-equilibrium positions exhibited damped oscillations toward the nearest energy minimum.
  • Measured oscillation frequencies were $\approx 1.15$ Hz (lighter) and $1.50$ Hz (denser), consistent with a damped harmonic oscillator model derived from the potential energy curvature.
  • A specific case of a denser floater released near $165^\circ$ showed a higher frequency ($\approx 4$ Hz), attributed to a stiffer vertical potential well coupled with rotational motion.

Role of Physical Effects

• Capillarity: Surface tension creates a vertical force and a torque. The authors estimated this torque to be $\sim 10^{-5}$ N·m.
• Contact Line Pinning: Near $\alpha = 0.5$, the energy landscape is nearly flat. The authors found that contact line pinning (where the meniscus gets stuck on surface imperfections) creates a threshold torque. If the residual capillary torque is below this threshold, the object remains in an arbitrary orientation rather than rotating to a specific angle. This explains the observed "neutral" behavior despite the presence of capillary forces.
• Density Discrepancy: The experimental neutral equilibrium occurred at $\alpha \approx 0.49$ rather than $0.5$, likely due to the downward pull of the meniscus (capillary force) effectively increasing the object's apparent weight.

4. Significance and Contributions

1. First Experimental Realization: This work provides the first physical demonstration of a Zindler curve floating in neutral equilibrium, validating a century-old mathematical concept.
2. Robustness of Geometry: The experiment demonstrates that the geometric condition for orientation-independent floating is robust. Even with minor density inhomogeneities and the presence of capillary forces, the system maintains its unique behavior.
3. Methodological Innovation: The "sandwich" fabrication technique offers a simple, reproducible platform for creating 2D floating bodies with precisely tunable effective densities, overcoming the limitations of 3D printing homogeneity.
4. Pedagogical Value: The study serves as a tangible illustration of the interplay between geometry, buoyancy, and stability, making abstract concepts like potential energy landscapes and Zindler curves accessible through tabletop experiments.
5. Insight into Non-Ideal Systems: By analyzing deviations from the ideal case, the paper elucidates the subtle roles of surface tension and contact line pinning in determining the stability of floating bodies, providing a more complete picture of real-world hydrostatics.