Tammes problem explained

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]

See also

• Spherical code
• Kissing number problem
• Cylinder sphere packings

References

1. Pieter Merkus Lambertus Tammes (1930): On the number and arrangements of the places of exit on the surface of pollen-grains, University of Groningen
2. News: Batagelj . Vladimir . Plestenjak . Bor . Optimal arrangements of n points on a sphere and in a circle . 25 June 2018 . https://web.archive.org/web/20180625050324/https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf . IMFM/TCS.
3. Musin . Oleg R. . Tarasov . Alexey S. . The Tammes Problem for N = 14 . Experimental Mathematics . 2015 . 24 . 4 . 460–468 . 10.1080/10586458.2015.1022842. 39429109 .
4. Web site: Sloane . N. J. A. . Spherical Codes: Nice arrangements of points on a sphere in various dimensions .

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 . 10.1107/S0108767387098842 .
• Erber T, Hockney GM . 1991 . Equilibrium configurations of N equal charges on a sphere . Journal of Physics A: Mathematical and General . 24 . 23 . Ll369 - Ll377 . 10.1088/0305-4470/24/23/008 . 1991JPhA...24L1369E . 122561279 .
• Melissen JBM . 1998 . How Different Can Colours Be? Maximum Separation of Points on a Spherical Octant . . 454 . 1499 - 1508 . 10.1098/rspa.1998.0218 . 1973. 1998RSPSA.454.1499M . 122700006 .
• . 2003 . Viral Self-Assembly as a Thermodynamic Process . Physical Review Letters . 90 . 248101–1 - 248101–4 . 10.1103/PhysRevLett.90.248101 . 24 . 12857229 . 2003PhRvL..90x8101B . cond-mat/0211390 . 2445/13275 . 1353095 . dead . https://web.archive.org/web/20070915043650/http://personnel.physics.ucla.edu/directory/faculty/fac_files/bruinsma/viral_self-assembly.pdf . 2007-09-15 .

Books

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

External links

• How to Stay Away from Each Other in a Spherical Universe (PDF).
• 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).