Tammes problem

12 points in an optimal arrangement around a sphere, which also are the vertices of a regular icosahedron.

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] If circles with diameter equal to the minimum distance are drawn around each point, they will not cross. Another equivalent way of phrasing the problem is to ask for the largest radius such that the given number of circles of that radius can be packed disjointly on the sphere.

Unsolved problem in mathematics

What is the optimal packing of circles on the surface of a sphere for every possible amount of circles?

More unsolved problems in mathematics

It can be viewed as a particular special case of the generalized Thomson problem of minimizing the total Coulomb energy 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]}

Some natural systems such as this coral require approximate solutions to problems similar to the Tammes problem

See also

• Spherical code
• Kissing number problem
• Cylinder sphere packings

References

1. Tammes, Pieter Merkus Lambertus (1930). "On the number and arrangements of the places of exit on the surface of pollen-grains" (PDF).
2. Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle" (PDF). IMFM/TCS. Archived from the original (PDF) on 25 June 2018.
3. Musin, Oleg R.; Tarasov, Alexey S. (2015). "The Tammes Problem for N = 14". Experimental Mathematics. 24 (4): 460–468. doi:10.1080/10586458.2015.1022842. S2CID   39429109.
4. Sloane, N. J. A. "Spherical Codes: Nice arrangements of points on a sphere in various dimensions".
5. Hars, Laslo (2020). "Numerical solutions of the Tammes problem for up to 60 points" (PDF).

Bibliography

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.

External links

• 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).