Point configurations that are asymmetric yet balanced

Understand

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.