Convex shapes with only one stable and one unstable balance point made explicitly
An explicit mono-monostatic polyhedron
Computational Geometry
Summary
Some special solid shapes can balance in only one stable way and one unstable way, which is rare and tricky to create. The authors built two such shapes made of flat faces, with one having over 56,000 faces, and proved exactly that they have this unique balancing property. They also explain why simply approximating smooth shapes doesn’t work and describe how they carefully constructed and checked their shapes. Interestingly, the entire process, from experiments to confirmation, was done with AI help guided by humans, showing a new method for making sure math results are exact.
convex bodymono-monostaticequilibrium positionstable equilibriumunstable equilibriumpolyhedroncentroidrational arithmeticcertificationAI-assisted mathematics
Authors
Tancredi Schettini Gherardini
Abstract
A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the Gömböc of Domokos and Várkonyi), and Lángi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.