The D₄ Root System Is Not Universally Optimal

Abstract

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.

Keywords:

• Primary 52C17, 05B40
• Secondary 52A40
• 24-cell
• D ₄ root system
• potential energy minimization
• spherical code
• spherical design
• universally optimal code