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[Math.MG] 7 Oct 2008 Cat(F)Udrgatsh 1503/4-1 EXPERIMENTAL STUDY OF ENERGY-MINIMIZING POINT CONFIGURATIONS ON SPHERES BRANDON BALLINGER, GRIGORIY BLEKHERMAN, HENRY COHN, NOAH GIANSIRACUSA, ELIZABETH KELLY, AND ACHILL SCHURMANN¨ Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima. [T]he problem of finding the configurations of stable equilibrium for a number of equal particles acting on each other according to some law of force...is of great interest in connexion with the rela- tion between the properties of an element and its atomic weight. Unfortunately the equations which determine the stability of such a collection of particles increase so rapidly in complexity with the number of particles that a general mathematical investigation is scarcely possible. J. J. Thomson, 1897 Contents 1. Introduction 2 1.1. Experimental results 4 1.2. New universal optima 8 2. Methodology 9 2.1. Techniques 9 2.2. Example 14 3. Experimental phenomena 15 arXiv:math/0611451v3 [math.MG] 7 Oct 2008 3.1. Analysis of Gram matrices 15 3.2. Other small examples 18 3.3. 2n + 1 points in Rn 20 3.4. 2n + 2 points in Rn 21 3.5. 48 points in R4 22 3.6. Hopf structure 23 3.7. Facet structure of universal optima 23 3.8. 96 points in R9 27 Date: September 8, 2008. Ballinger and Giansiracusa were supported by the University of Washington Mathematics Department’s NSF VIGRE grant. Sch¨urmann was supported by the Deutsche Forschungsgemein- schaft (DFG) under grant SCHU 1503/4-1. 1 2 BALLINGER, BLEKHERMAN, COHN, GIANSIRACUSA, KELLY, AND SCHURMANN¨ 3.9. Distribution of energy levels 28 4. Conjectured universal optima 30 4.1. 40 points in R10 30 4.2. 64 points in R14 32 5. Balanced, irreducible harmonic optima 33 6. Challenges 36 Acknowledgements 36 Appendix A. Local non-optimality of diplo-simplices 36 References 38 1. Introduction What is the best way to distribute N points over the unit sphere Sn−1 in Rn? Of course the answer depends on the notion of “best.” One particularly interesting case is energy minimization. Given a continuous, decreasing function f : (0, 4] R, define the f-potential energy of a finite subset Sn−1 to be → C ⊂ 1 2 Ef ( )= f x y . C 2 X | − | x,y∈C x6=y (We only need f to be defined on (0, 4] because x y 2 4 when x 2 = y 2 = 1. The factor of 1/2 is chosen for compatibility with| − the| ≤ physics literature,| | | | while the use of squared distance is incompatible but more convenient.) How can one n−1 choose S with = N so as to minimize Ef ( )? In this paper we report onC ⊂ lengthy computer|C| searches for configurations withC low energy. What distinguishes our approach from most earlier work on this topic (see for example [AP-G1, AP-G2, AP-G3, AWRTSDW, AW, BBCDHNNTW, BCNT1, BCNT2, C, DM, DLT, E, ELSBB, EH, F, GE, HSa1, HSa2, H, KaS, KuS, Ko, LL, M-FMRS, MKS, MDH, P-GDMOD-S, P-GDM, P-GM, RSZ1, RSZ2, SK, Sl, T, Wh, Wil]) is that we attempt to treat many different potential functions on as even a footing as possible. Much of the mathematical structure of this problem becomes appar- ent only when one varies the potential function f. Specifically, we find that many optimal configurations vary in surprisingly simple, low-dimensional families as f varies. The most striking case is when such a family is a single point: in other words, when the optimum is independent of f. Cohn and Kumar [CK] defined a configu- ration to be universally optimal if it minimizes Ef for all completely monotonic f (i.e., f is infinitely differentiable and ( 1)kf (k)(x) 0 for all k 0 and x (0, 4), as is the case for inverse power laws). They− were able≥ to prove universal≥ optimality∈ only for certain very special arrangements. One of our primary goals in this paper is to investigate how common universal optimality is. Was the limited list of exam- ples in [CK] an artifact of the proof techniques or a sign that these configurations are genuinely rare? Every universally optimal configuration is an optimal spherical code, in the sense that it maximizes the minimal distance between the points. (Consider an inverse power law f(r)=1/rs. If there were a configuration with a larger minimal distance, then its f-potential energy would be lower when s is sufficiently large.) However, universal optimality is a far stronger condition than optimality as a spherical code. ENERGY-MINIMIZING POINT CONFIGURATIONS 3 n N t Description 2 N cos(2π/N) N-gon n N n +1 1/(N 1) simplex n ≤2n − 0− crosspolytope 3 12 1/√5 icosahedron 4 120 (1+ √5)/4 regular600-cell 5 16 1/5 hemicube/Clebsch graph 6 27 1/4 Schl¨afli graph/isotropic subspaces 7 56 1/3 equiangular lines 8 240 1/2 E8 root system 21 112 1/9 isotropic subspaces 21 162 1/7 (162, 56, 10, 24) strongly regular graph 22 100 1/11 Higman-Sims graph 22 275 1/6 McLaughlin graph 22 891 1/4 isotropic subspaces 23 552 1/5 equiangular lines 23 4600 1/3 kissing configuration of the following 24 196560 1/2 Leech lattice minimal vectors 3 q +1 3 2 q q+1 (q + 1)(q +1) 1/q isotropic subspaces (q is a prime power) Table 1. The known universal optima. There are optimal spherical codes of each size in each dimension, but they are rarely universally optimal. In three dimensions, the only examples are a single point, two antipodal points, an equilateral triangle on the equator, or the vertices of a regular tetrahedron, octahedron, or icosahedron. Universal optimality was proved in [CK], building on previous work by Yudin, Kolushov, and Andreev [Y, KY1, KY2, A1, A2], and the completeness of this list follows from a classification theorem due to Leech [L]. See [CK] for more details. In higher dimensions much less is known. Cohn and Kumar’s main theorem provides a general criterion from which they deduced the universal optimality of a number of previously studied configurations. Specifically, they proved that every spherical (2m 1)-design in which only m distances occur between distinct points is universally optimal.− Recall that a spherical d-design in Sn−1 is a finite subset of Sn−1 such that every polynomial on Rn of total degree d has the same average overC as over the entire sphere. This criterion holds for every known universal optimum C 4 except one case, namely the regular 600-cell in R (i.e., the H4 root system), for which Cohn and Kumar proved universal optimality by a special argument. A list of all known universal optima is given in Table 1. Here n is the dimension of the Euclidean space, N is the number of points, and t is the greatest inner product between distinct points in the configuration (i.e., the cosine of the minimal angle). For detailed descriptions of these configurations, see Section 1 of [CK]. Each is uniquely determined by the parameters listed in Table 1, except for the configurations listed on the last line. For that case, when q = pℓ with p an odd prime, there are at least (ℓ 1)/2 distinct universal optima (see [CGS] and [Ka]). Classifying these optima⌊ is− equivalent⌋ to classifying generalized quadrangles with 4 BALLINGER, BLEKHERMAN, COHN, GIANSIRACUSA, KELLY, AND SCHURMANN¨ nN t References 10 40 1/6 Conway, Sloane, and Smith [Sl], Hovinga [H] 14 64 1/7 Nordstrom and Robinson [NR], de Caen and van Dam [dCvD], Ericson and Zinoviev [EZ] Table 2. New conjectured universal optima. parameters (q, q2), which is a difficult problem in combinatorics. In the other cases from Table 1, when uniqueness holds, we use the notation UN, n for the unique N-point universal optimum in Rn. Each of the configurations in Table 1 had been studied before it appeared in [CK], and was already known to be an optimal spherical code. In fact, when N 2n +1 and n> 4, the codes on this list are exactly those that have been proved≥ optimal. Cohn and Kumar were unable to resolve the question of whether Table 1 is the complete list of universally optimal codes, except when n 3. All that is known in general is that any new universal optimum must have N ≤2n +1 (Proposition 1.4 in [CK]). It does not seem plausible that the current list≥ is complete, but it is far from obvious where to find any others. Each known universal optimum is a beautiful mathematical object, connected to various important exceptional structures (such as special lattices or groups). Our long-term hope is to develop automated tools that will help uncover more such objects. In this paper we do not discover any configurations as fundamental as those in Table 1, but perhaps our work is a first step in that direction. Table 1 shows several noteworthy features. When n 4, the codes listed are the vertices of regular polytopes, specifically those with≤ simplicial facets. When 5 n 8, the table also includes certain semiregular polytopes (their facets are simplices≤ ≤ and cross polytopes, with two cross polytopes and one simplex arranged around each (n 3)-dimensional face).
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