Experimental study of energy-minimizing point configurations on spheres

Experimental Mathematics

Volumn 18, Issue 3, 2009, Pages 257-283

  Ballinger, Brandon ^a   Blekherman, Grigoriy ^b   Cohn, Henry ^b   Giansiracusa, Noah ^c   Kelly, Elizabeth ^d   Schürmann, Achill ^e  

^a GOOGLE INC   (United States)

^b MICROSOFT RESEARCH   (United States)

^c BROWN UNIVERSITY   (United States)

^d UNIVERSITY OF WASHINGTON   (United States)

^e DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Energy minimization; Polytopes; Universal optimality

Indexed keywords

EID: 70350031301     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2009.10129052     Document Type: Article

References (72)

1. E. L. Altschuler and A. Pérez-Garrido. “Global Minimum for Thomson’s Problem of Charges on a Sphere.” Phys. Rev. E 71 (2005), 047703: 1–4.
2. E. L. Altschuler and A. Pérez-Garrido. “Defect Free Global Minima in Thomson’s Problem of Charges on a Sphere.” Phys. Rev. E 73 (2006), 036108:1–6.
3. E. L. Altschuler and A. Pérez-Garrido, “Symmetric Four-Dimensional Polytope and Visualization Method in Four, Eight, and Sixteen Dimensions Using Hopf Maps.” Phys. Rev. E 76 (2007), 016705:1–6.
4. E. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tipton, R. Stong, F. Dowla, and F. Wooten. “Possible Global Minimum Lattice Configurations for Thomson’s Problem of Charges on a Sphere.” Phys. Rev. Lett. 78 (1997), 2681–2685.
5. N. N. Andreev. “An Extremal Property of the Icosahedron.” East J. Approx. 2 (1996), 459–462.
6. N. N. Andreev. “Location of Points on a Sphere with Minimal Energy” (Russian). Tr. Mat. Inst. Steklova 219 (1997), 27–31
7. translation in Proc. Steklov Inst. Math. 219 (1997), 20–24.
8. T. Aste and D. Weaire. The Pursuit of Perfect Packing, Second edition. New York: Taylor & Francis, 2008.
9. C. Bachoc and F. Vallentin. “Optimality and Uniqueness of the (4, 10, 1/6) Spherical Code.” J. Combin. Theory, Ser. A 116 (2009), 195–204.
10. E. Bannai, E. Bannai, and H. Bannai, “Uniqueness of Certain Association Schemes.” European J. Combin. 29 (2008), 1379–1395.
11. A. R. Bausch, M. J. Bowick, A. Cacciuto, A.D. Dinsmore, M. F. Hsu, D. R. Nelson, M.G. Nikolaides, A. Travesset, and D. A. Weitz. “Grain Boundary Scars and Spherical Crystallography.” Science 299 (2003), 1716–1718.
12. K. Böröczky Jr. Finite Packing and Covering, Cambridge Tracts in Mathematics 154. Cambridge: Cambridge University Press, 2004.
13. M. Bowick, A. Cacciuto, D. R. Nelson, and A. Travesset. “Crystalline Order on a Sphere and the Generalized Thomson Problem.” Phys. Rev. Lett. 89 (2002), 185502:1–4.
14. M. Bowick, A. Cacciuto, D. R. Nelson, and A. Travesset. “Crystalline Particle Packings on a Sphere with Long Range Power Law Potentials.” Phys. Rev. B 73 (2006), 024115:1–16.
15. A. E. Brouwer and W. H. Haemers. “Structure and Uniqueness of the (81, 20, 1, 6) Strongly Regular Graph.” Discrete Math. 106/107 (1992), 77–82.
16. A. E. Brouwer and W. H. Haemers. “The Gewirtz Graph: An Exercise in the Theory of Graph Spectra.” European J. Combin. 14 (1993), 397–407.
17. D. de Caen and E. R. van Dam. “Association Schemes Related to Kasami Codes and Kerdock Sets.” Des. Codes Cryptogr. 18 (1999), 89–102.
18. P. J. Cameron, J. M. Goethals, and J. J. Seidel. “Strongly Regular Graphs Having Strongly Regular Subconstituents.” J. Algebra 55 (1978), 257–280.
19. H. Cohn. “Stability Configurations of Electrons on a Sphere.” Math. Tables Aids Comput. 10 (1956), 117–120.
20. H. Cohn and A. Kumar. “Universally Optimal Distribution of Points on Spheres.” J. Amer. Math. Soc. 20 (2007), 99–148.
21. H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar. “The D4 Root System Is Not Universally Optimal.” Experimental Mathematics 16 (2007), 313–320.
22. J. Conway and N. J. A. Sloane. “The Cell Structures of Certain Lattices.” In Miscellanea Mathematica, pp. 71–107. Berlin: Springer, 1991.
23. J. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, third edition, Grundlehren der Mathematischen Wissenschaften 290. New York: Springer-Verlag, 1999.
24. E. R. van Dam. “Three-Class Association Schemes.” Journal of Algebraic Combinatorics 10 (1999), 69–107.
25. S. B. Damelin and V. Maymeskul. “On Point Energies, Separation Radius and Mesh Norm for s-Extremal Configurations on Compact Sets in Rn.” J. Complexity 21 (2005), 845–863.
26. 16.” Preprint, 1995, arXiv:hep-th/9501007.
27. P. D. Dragnev, D. A. Legg, and D. W. Townsend. “Discrete Logarithmic Energy on the Sphere.” Pacific J. Math. 207 (2002), 345–358.
28. M. Dutour Sikirić. “Polyhedral,” a package for GAP. Available online (http://www.liga.ens.fr/∼dutour/Polyhedral/index.html), 2006.
29. M. Dutour Sikirić, A. Schürmann, and F. Vallentin. “The Contact Polytope of the Leech Lattice.” Preprint, 2009, arXiv:0906.1427.
30. J. R. Edmundson. “The Distribution of Point Charges on the Surface of a Sphere.” Acta Cryst. A48 (1992), 60–69.
31. T. Einert, P. Lipowsky, J. Schilling, M. J. Bowick, and A. Bausch. “Grain Boundary Scars on Spherical Crystals.” Langmuir 21 (2005), 12076–12079.
32. T. Erber and G. M. Hockney. “Equilibrium Configurations of N Equal Charges on a Sphere.” J. Phys. A: Math. Gen. 24 (1991), L1369–L1377.
33. T. Ericson and V. Zinoviev. Codes on Euclidean Spheres, North-Holland Mathematical Library 63. Amsterdam: North-Holland Publishing Co., 2001.
34. L. Föppl. “Stabile Anordnungen von Elektronen im Atom.” J. Reine Angew. Math. 141 (1912), 251–302.
35. The GAP Group. GAP: Groups, Algorithms, and Programming, Version 4.4. Available online (http://www.gap-system.org/), 2006.
36. E. Gawrilow and M. Joswig. “Polymake: A Framework for Analyzing Convex Polytopes.” In Polytopes—Combinatorics and Computation (Oberwolfach, 1997), pp. 43–73, DMV Sem. 29. Basel: Birkhäuser, 2000.
37. L. Glasser and A. G. Every. “Energies and Spacings of Point Charges on a Sphere.” J. Phys. A: Math. Gen. 25 (1992), 2473–2482.
38. R. Griess. “Few-Cosine Spherical Codes and Barnes–Wall Lattices.” J. Combin. Theory Ser. A 115 (2008), 1211–1234.
39. D. P. Hardin and E. B. Saff. “Discretizing Manifolds via Minimum Energy Points.” Notices Amer. Math. Soc. 51 (2004), 1186–1194.
40. D. P. Hardin and E. B. Saff. “Minimal Riesz Energy Point Configurations for Rectifiable d- Dimensional Manifolds. Adv. Math. 193 (2005), 174–204.
41. R. H. Hardin and N. J. A. Sloane. “GOSSET: A General-Purpose Program for Designing Experiments.” Available online (http://www.research.att.com/∼njas/gosset/), 2003.
42. S. H. Hovinga. “Polytopes and Optimal Packing of p Points in n Dimensional Spheres.” Available online (http://presh.com/hovinga/), 2004.
43. W. Kantor. “Some Generalized Quadrangles with Parameters q2, q.” Math. Z. 192 (1986), 45–50.
44. A. Katanforoush and M. Shahshahani. “Distributing Points on the Sphere I.” Experiment. Math. 12 (2003), 199–209.
45. A. V. Kolushov and V. A. Yudin. “On the Korkin–Zolotarev Construction” (Russian). Diskret. Mat. 6 (1994), 155–157
46. translation in Discrete Math. Appl. 4 (1994), 143–146.
47. A. V. Kolushov and V. A. Yudin. “Extremal Dispositions of Points on the Sphere.” Anal. Math. 23 (1997), 25–34.
48. D. A. Kottwitz. “The Densest Packing of Equal Circles on a Sphere.” Acta Cryst. A47 (1991), 158–165.
49. A. B. J. Kuijlaars and E. B. Saff. “Asymptotics for Minimal Discrete Energy on the Sphere.” Trans. Amer. Math. Soc. 350 (1998), 523–538.
50. J. Leech. “Equilibrium of Sets of Particles on a Sphere.” Math. Gaz. 41 (1957), 81–90.
51. A. M. Livshits and Yu. E. Lozovik. “Coulomb Clusters on a Sphere: Topological Classification.” Chemical Physics Letters 314 (1999), 577–583.
52. F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes, North- Holland Mathematical Library 16. Amsterdam: North- Holland Publishing Company, 1977.
53. A. Martínez-Finkelshtein, V. Maymeskul, E. A. Rakhmanov, and E. B. Saff. “Asymptotics for Minimal Discrete Riesz Energy on Curves in Rd.” Canad. J. Math. 56 (2004), 529–552.
54. B. D. McKay. “Practical Graph Isomorphism” (10th Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, 1980)). Congressus Numerantium 30 (1981), 45–87. Software available online (http://cs.anu.edu.au/∼bdm/nauty), 1981.
55. T. W. Melnyk, O. Knop, and W. R. Smith. “Extremal Arrangements of Points and Unit Charges on a Sphere: Equilibrium Configurations Revisited.” Canad. J. Chem. 55 (1977), 1745–1761.
56. J. R. Morris, D. M. Deaven, and K. M. Ho. “Genetic-Algorithm Energy Minimization for Point Charges on a Sphere.” Phys. Rev. B 53 (1996), R1740–R1743.
57. A. W. Nordstrom and J. P. Robinson. “An Optimum Nonlinear Code.” Information and Control 11 (1967), 613–616.
58. A. Pérez-Garrido and M. Moore. “Symmetric Patterns of Dislocations in Thomson’s Problem.” Phys. Rev. B 60 (1999), 15628–15631.
59. A. Pérez-Garrido, M. J. W. Dodgson, M. A. Moore, M. Ortuño, and A. Díaz-Sánchez. “Comment on ‘Possible Global Minimum Lattice Configurations for Thomson’s Problem of Charges on a Sphere.” Phys. Rev. Lett. 79 (1997), 1417.
60. A. Pérez-Garrido, M. J. W. Dodgson, and M. Moore. “Influence of Dislocations in Thomson’s Problem.” Phys. Rev. B 56 (1997), 3640–3643.
61. E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou. “Minimal Discrete Energy on the Sphere.” Math. Res. Lett. 1 (1994), 647–662.
62. E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou. “Electrons on the Sphere.” In Computational Methods and Function Theory, edited by R. M. Ali, S. Ruscheweyh, and E. B. Saff, pp. 111–127. River Edge, NJ: World Scientific, 1995.
63. E. B. Saff and A. B. J. Kuijlaars. “Distributing Many Points on a Sphere.” Math. Intelligencer 19 (1997), 5–11.
64. P. H. Schoutte. “On the Relation between the Vertices of a Definite Six-Dimensional Polytope and the Lines of a Cubic Surface.” Proc. Roy. Acad. Amsterdam 13 (1910), 375–383.
65. N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others. “Tables of Spherical Codes.” Available online (http://www.research.att.com/∼njas/packings/), 2000.
66. N. J. A. Sloane, R. H. Hardin, and P. Cara. “Spherical Designs in Four Dimensions.” In Proceedings of the 2003 IEEE Information Theory Workshop, pp. 253–258. Los Alamitos: IEEE, 2003.
67. J. J. Thomson. “Cathode Rays.” Phil. Mag. 44 (1897), 293–316.
68. L. L. Whyte. “Unique Arrangements of Points on a Sphere.” Amer. Math. Monthly 59 (1952), 606–611.
69. D. V. Widder. The Laplace Transform. Princeton: Princeton University Press, 1941.
70. L. T. Willie. “Searching Potential Energy Surfaces by Simulated Annealing.” Nature 324 (1986), 46–48.
71. V. A. Yudin. “Minimum Potential Energy of a Point System of Charges” (Russian). Diskret. Mat. 4 (1992), 115–121
72. translation in Discrete Math. Appl. 3 (1993), 75–81.