Tammes problem

In geometry, the Tammes problem is a problem in packing a given number of points on the surface of a sphere such that the minimum distance between points is maximized. It is named after the Dutch botanist Pieter Merkus Lambertus Tammes (the nephew of pioneering botanist Jantina Tammes) who posed the problem in his 1930 doctoral dissertation on the distribution of pores on pollen grains.^[1]

It can be viewed as a particular special case of the generalized Thomson problem of minimizing the total Coulomb force of electrons in a spherical arrangement.^[2] Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24.^[3] There are conjectured solutions for many other cases, including those in higher dimensions.^{[4] [5]}

• Spherical code
• Kissing number problem
• Cylinder sphere packings

Journal articles

• Tarnai T; Gáspár Zs (1987). "Multi-symmetric close packings of equal spheres on the spherical surface". Acta Crystallographica. A43 (5): 612–616. Bibcode:1987AcCrA..43..612T. doi:10.1107/S0108767387098842.
• Erber T, Hockney GM (1991). "Equilibrium configurations of N equal charges on a sphere" (PDF). Journal of Physics A: Mathematical and General. 24 (23): Ll369 – Ll377. Bibcode:1991JPhA...24L1369E. doi:10.1088/0305-4470/24/23/008. S2CID 122561279.
• Melissen JBM (1998). "How Different Can Colours Be? Maximum Separation of Points on a Spherical Octant". Proceedings of the Royal Society A. 454 (1973): 1499–1508. Bibcode:1998RSPSA.454.1499M. doi:10.1098/rspa.1998.0218. S2CID 122700006.
• Bruinsma RF, Gelbart WM, Reguera D, Rudnick J, Zandi R (2003). "Viral Self-Assembly as a Thermodynamic Process" (PDF). Physical Review Letters. 90 (24): 248101–1–248101–4. arXiv:cond-mat/0211390. Bibcode:2003PhRvL..90x8101B. doi:10.1103/PhysRevLett.90.248101. hdl:2445/13275. PMID 12857229. S2CID 1353095. Archived from the original (PDF) on 2007-09-15.
• Lai, Xiangjing; Yue, Dong; Hao, Jin-Kao (2023). "Iterated dynamic neighborhood search for packing equal circles on a sphere". Comput. Operat. Res. 151. doi:10.1016/j.cor.2022.106121.

Books

• Aste T, Weaire DL (2000). The Pursuit of Perfect Packing. Taylor and Francis. pp. 108–110. ISBN 978-0-7503-0648-5.
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 31. ISBN 0-14-011813-6.

• Bagchi, Bhaskar (1997). "How to Stay Away from Each Other in a Spherical Universe". Reconance J. Sci. Edu. 2 (9): 18–26..
• Packing and Covering of Congruent Spherical Caps on a Sphere.
• Science of Spherical Arrangements (PPT).
• General discussion of packing points on surfaces, with focus on tori (PDF).