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.
We prove that the (D_4) root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in (S^3), based on numerical computations suggesting that every 5-design consisting of 24 points in (S^3) is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the (D_4) root system.
