Building a monostable tetrahedron (2506.19244v1)

Abstract: In this short note we describe what we believe to be the first working model of a monostable tetrahedron.

• The paper demonstrates that a tetrahedron can be monostable through a non-uniform mass distribution and the obtuse path criterion.
• The construction method uses a composite of carbon fiber, tungsten carbide, and epoxy to achieve a critical density ratio.
• Numerical analysis shows larger loading zones for Type I patterns, confirming feasibility for engineering applications such as lander design.

Building a Monostable Tetrahedron: Construction, Theory, and Implications

This paper presents the first documented construction of a physical monostable tetrahedron—a tetrahedron that, due to its mass distribution, is stable on only one face. The work addresses a longstanding question in geometric mechanics and convex polyhedral stability, originally posed by Conway and Guy in 1966, regarding the possibility of such an object with straight edges and a non-uniform mass distribution.

Theoretical Foundations

The existence of a monostable tetrahedron requires a non-homogeneous mass distribution. The key theoretical advance, building on prior work, is the characterization of "loadable" tetrahedra: those whose geometry admits a mass distribution yielding monostability. The necessary and sufficient condition is the presence of an "obtuse path"—a sequence of three obtuse edges—allowing the center of mass to be positioned such that only one face is stable. The set of possible centers of mass for a given falling pattern forms a convex region (the "loading zone") within the tetrahedron.

The paper distinguishes between two types of falling patterns (Type I and Type II), each corresponding to different sequences of face transitions as the object tumbles to its unique stable face. Numerical analysis reveals that Type I loading zones are significantly larger than Type II, with the former exceeding the latter by at least an order of magnitude in volume. This has direct implications for the feasibility of constructing physical models.

Construction Methodology

The authors detail the construction of a monostable tetrahedron using a composite approach:

• Frame: Pultruded carbon fiber tubes (outer diameter 1 mm, inner diameter 0.5 mm, density 1.36 g/cm³) form the lightweight skeleton.
• Core: Tungsten carbide (density 14.15 g/cm³) is used for the heavy component.
• Joints: Epoxy glue (density 1.3 g/cm³) secures the frame.

The design leverages a planar interface between the heavy and light materials, with the interface geometry determined by the desired loading zone. The density ratio between the heavy and light components is critical; the model functions as intended when the heavy part is at least three orders of magnitude denser than the light part. The final object is constructed at a scale 10% larger than the computed minimum to ensure robustness.

Empirical results confirm that the constructed tetrahedron is stable only on one face (face D). When placed on any other face, it tumbles deterministically to the stable face, following a directed sequence of transitions. The observed falling patterns are consistent with theoretical predictions and are documented in supplementary video material.

Numerical Results

The paper provides quantitative data on the volumes of the loading zones for different falling patterns:

Falling Pattern	Type	Volume (cm³)
B → A → D ← C	I	1.4318
C → D → A ← B	I	0.5716
B → A → D → C	II	0.0199
C → D → A → B	II	0.0067

Given the total tetrahedron volume of 668.624 cm³, the practical construction is only feasible for Type I patterns due to the much larger loading zones.

Implications and Applications

The paper has both theoretical and practical implications:

• Theoretical: The explicit construction validates the sufficiency of the obtuse path criterion for monostability in tetrahedra with straight edges and non-uniform mass distribution. The work also clarifies the limitations imposed by material densities, showing that certain falling patterns (Type II) are unattainable with terrestrial materials.
• Practical: The results are directly relevant to engineering problems involving the stability of non-convex, non-smooth objects. The convex hull of such objects, often a polyhedron with straight edges, determines their equilibrium behavior when supported on a plane. The paper highlights the relevance of this analysis to the design of lunar landers, referencing recent incidents where landers tipped over and rolled on their convex hulls. The findings suggest that, while universal passive self-righting may be unattainable, designing for monostability on a horizontal surface is feasible and may inform future lander designs.

Future Directions

Potential avenues for further research include:

• Extending the analysis to polyhedra with more faces and to non-convex geometries.
• Investigating the robustness of monostability under manufacturing tolerances and real-world perturbations.
• Exploring active or hybrid strategies for self-righting in robotic and aerospace applications, informed by the geometric and mass distribution principles elucidated here.

The construction of a monostable tetrahedron with straight edges and a non-uniform mass distribution represents a significant step in understanding the interplay between geometry, mass distribution, and stability in polyhedral objects. The practical methodology and quantitative analysis provided in this work offer a foundation for both theoretical exploration and engineering application in stability-critical domains.