Cohn, Henry; Conway, John H.; Elkies, Noam D.; Kumar, Abhinav The \(D_4\) root system is not universally optimal. (English) Zbl 1137.05020 Exp. Math. 16, No. 3, 313-320 (2007). Summary: 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. Cited in 15 Documents MSC: 05B40 Combinatorial aspects of packing and covering 94B35 Decoding Keywords:24-cell; \(D_4\) root system; potential energy minimization; spherical code; spherical design; universally optimal code PDF BibTeX XML Cite \textit{H. Cohn} et al., Exp. Math. 16, No. 3, 313--320 (2007; Zbl 1137.05020) Full Text: DOI arXiv Euclid