Computational Construction and Engineering Evaluation of Verified Mono-Monostatic Bodies

TL;DR

This paper introduces a computational method to design and verify mono-monostatic bodies, achieving the first openly published geometry with exactly one stable equilibrium, with broad applications in engineering and design.

Contribution

It extends the Sloan phase function with Fourier terms and uses optimization to construct verified mono-monostatic bodies, filling a gap in geometric design verification.

Findings

First verified mono-monostatic geometry published.

Conventional geometries cannot achieve ECS=1 through ballast alone.

Achieved 349x precision improvement in IMU calibration housing.

Abstract

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.