Computational Construction and Engineering Evaluation of Verified Mono-Monostatic Bodies
This work provides the first openly verified mono-monostatic geometry, solving a long-standing geometric design problem for orientation-dependent systems, though the method is incremental (extending existing parameterization).
The paper introduces an Equilibrium Count Score (ECS) oracle to verify mono-monostatic bodies, finding that Sloan's Gomboc parameterization fails to produce such bodies, and constructs three verified mono-monostatic geometries via Fourier extension and optimization. The primary instance is the first openly published, computationally verified mono-monostatic geometry, with applications showing 349x precision improvement in IMU calibration and elimination of 20-67% germination loss in seed pods.
Many engineering failures in orientation-dependent systems are geometric failure modes: changing the geometry can eliminate what changing the material merely delays. The mono-monostatic property (exactly one stable equilibrium under gravity) is mathematically proven to exist in convex homogeneous bodies, but no verified geometry has been openly published. We introduce an Equilibrium Count Score (ECS) oracle measuring stable equilibria via drainage basin analysis on the center-of-mass height landscape. Applying this oracle to Sloan's (2023) analytical Gomboc parameterization, we find that no tested parameter value produces a mono-monostatic body. The surface function has two critical points as proven, but the COM height landscape exhibits 4-11 local minima. Surface critical points are necessary but not sufficient for mono-monostatic behavior. We close this gap by extending the Sloan phase function with Fourier terms and optimizing via differential evolution, constructing three verified mono-monostatic bodies with ECS=1 confirmed across merge thresholds from 0.5% to 10%. The primary instance (beta=0.023, a1=0.234) is the first openly published, computationally verified mono-monostatic geometry. The central result: conventional geometries cannot achieve ECS=1 through ballast alone. Cylinders retain multiple equilibria even at 30% bottom-weighted mass. Applied to IMU calibration housing (349x precision improvement, zero prior art), aerial reforestation seed pods (eliminating 20-67% germination loss from orientation), and marine buoy self-righting. Cross-layer scoring confirms the Gomboc is 11.8x worse than the cylinder on contact distribution while optimal on equilibrium stability, demonstrating framework discrimination across three invariant classes.