Point configurations that are asymmetric yet balanced

TL;DR

This paper demonstrates that in high-dimensional spaces, there exist balanced point configurations on spheres that lack symmetry, challenging the assumption that symmetry is necessary for balance.

Contribution

The authors construct explicit counterexamples of balanced configurations with trivial symmetry groups in high dimensions, disproving the converse of Leech's classification.

Findings

Counterexamples with trivial symmetry are balanced in high dimensions

Symmetry is not necessary for a configuration to be balanced

The result extends the understanding of equilibrium configurations beyond symmetric cases

Abstract

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.