Equilibria in non-Euclidean geometries

The Balancing Act: A Guide to the "Wobble" of Shapes

Imagine you have a perfectly round marble. If you place it on a flat table, it stays put. If you try to roll it, it rolls smoothly in any direction. It is the ultimate "stable" object. Now, imagine a heavy, lopsided rock. If you set it down, it might tip over, roll, and eventually settle into a specific position where it feels "stuck."

This paper is about the mathematical science of equilibrium—the study of how shapes settle, tip, and balance in different kinds of "worlds."

1. The Core Concept: The "Sticking Points"

In geometry, every shape has a centroid (its center of gravity). An equilibrium point is a spot on the surface of a shape where, if you rested it on a flat floor, it wouldn't want to tip over.

Think of these points like the "feet" of a chair:

Stable points: These are like the feet of a sturdy chair. If you nudge the chair, it wobbles but snaps back to its original position.
Unstable points: These are like trying to balance a pencil on its tip. It’s technically "balanced," but the slightest breath will make it fall.
Saddle points: These are like a mountain pass. If you move one way, you go up; if you move another way, you go down.

2. The "Gomböc" Mystery

For a long time, mathematicians wondered: What is the simplest possible shape that can only sit in one way?

In our normal 3D world, most shapes have many "feet" (equilibrium points). If you have a cube, it can sit on any of its six faces. But researchers discovered a magical shape called the Gomböc. It is a special, egg-like shape that has only one stable point and one unstable point. It is the "loneliest" shape possible—it only has one way to sit, and it's always looking for that one perfect spot.

3. Changing the Rules of the Universe

The "magic" of this paper is that the authors didn't just look at our normal, flat Euclidean world. They asked: "What if the rules of space itself changed?"

They tested these shapes in three different "alternate universes":

The Spherical Universe: Imagine living on the surface of a giant balloon. Everything curves inward.
The Hyperbolic Universe: Imagine living on a giant, infinite saddle or a piece of curly kale. Everything curves outward.
The Normed Universe: Imagine a world where "distance" is measured strangely—perhaps walking diagonally is just as easy as walking straight, or perhaps the "straightest" path is actually a zig-zag.

4. What did they find? (The "Spoiler Alert")

The researchers wanted to see if the "lonely" Gomböc shape could exist in these weird universes.

In 2D (Flatland): They proved that in any of these worlds, if you have a flat, 2D shape, it must have at least four "feet" (equilibrium points). You can't have a 2D "lonely" shape; it will always have at least four ways to settle.
In 3D (The Weird Worlds): They proved that even in the curved, strange universes of spheres and saddles, you can still build a Gomböc! They showed that you can take a standard ball and "dent" it just a tiny, microscopic amount to create a shape that is almost a perfect sphere but possesses that magical, single-point balance.

Summary: Why does this matter?

While this sounds like pure math, it’s actually about the fundamental relationship between geometry (the shape of things) and physics (how things move). By proving that these "lonely" shapes exist even in curved or strange spaces, the authors are showing that the laws of balance are much more universal and resilient than we previously thought.

Whether you are in a flat room, on a giant balloon, or in a warped dimension, the math of the "wobble" still holds true.

Technical Summary: Equilibria in Non-Euclidean Geometries

Authors: Zsolt Lángi and Shanshan Wang

1. Problem Statement

The paper investigates the existence and minimum number of equilibrium points and centroids for convex bodies in non-Euclidean geometries—specifically spherical space, hyperbolic space, and normed spaces.

In Euclidean geometry, a point on the boundary of a convex body is an equilibrium point if the supporting hyperplane at that point is orthogonal to the segment connecting the point to the body's centroid. While classical results (e.g., by Domokos, Papadopoulos, and Ruina) established that every non-degenerate plane convex body in Euclidean space has at least four equilibria, and the discovery of the Gömböc proved that 3D Euclidean bodies can be "mono-monostatic" (having only one stable and one unstable equilibrium), the authors seek to determine if these properties hold when the underlying geometry is changed.

2. Methodology

The authors employ several advanced mathematical frameworks to generalize Euclidean concepts:

Centroid Definitions: Since the linear structure of $\mathbb{R}^d$ is absent in non-Euclidean spaces, the authors use definitions based on constant curvature models (the unit sphere $S^d$ for spherical space and the hyperboloid model for hyperbolic space) and axiomatic properties (isometry invariance and the "rule of the lever"). In normed spaces, they utilize the linear structure of the vector space combined with Haar measures.
Gnomonic and Projective Projections: To bridge the gap between non-Euclidean and Euclidean analysis, the authors use the gnomonic projection for spherical space (mapping the hemisphere to a Euclidean plane) and the projective ball model for hyperbolic space.
Morse Theory and Poincaré-Hopf Theorem: To classify and count equilibrium points (stable, unstable, and saddle-type), the authors treat the distance function from the centroid as a Morse function on the boundary of the convex body.
Perturbation Construction: To prove the existence of mono-monostatic bodies, the authors construct a two-parameter family of star-shaped bodies $K(c, d)$ . They use a specific radial function $\rho_{c,d}$ to perturb a standard ball, then apply the Intermediate Value Theorem and Lebesgue’s Dominated Convergence Theorem to ensure the centroid remains at the origin while maintaining minimal equilibrium points.

3. Key Contributions and Results

The paper provides two primary theorems that generalize Euclidean findings:

Theorem 1.1 (The Plane Case): Every plane convex body in a space of constant curvature (spherical/hyperbolic) or in a normed plane (with a smooth, strictly convex unit disk) has at least four equilibrium points. This result is a significant generalization as it removes the regularity assumptions required in previous Euclidean proofs.
Theorem 1.2 (The 3D Mono-monostatic Case): In 3-dimensional spherical space, hyperbolic space, or rotationally symmetric normed spaces, there exist mono-monostatic convex bodies (bodies with exactly one stable and one unstable equilibrium point) that can be arbitrarily close to any closed ball in the Hausdorff distance.
Conjecture 1: The authors propose that mono-monostatic bodies exist in any 3D normed space where the unit ball has a $C^2$ -class boundary and strictly positive Gaussian curvature, suggesting their current rotational symmetry requirement is merely a technicality.

4. Significance

This research is significant for several reasons:

Mathematical Generalization: It successfully extends the theory of static equilibrium from the flat Euclidean plane to curved manifolds and non-isotropic normed spaces, filling a gap in the literature regarding the interplay between geometry and mechanics.
Unification of Disciplines: The work connects pure geometry (convexity, Riemannian manifolds) with classical mechanics and physics (centroids, stability).
Foundational Extension: By proving that the "four-equilibrium" rule for planes and the "mono-monostatic" possibility for 3D volumes are not artifacts of Euclidean space but are intrinsic to the nature of convexity and curvature, the authors provide a more universal understanding of the stability of physical objects.