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] Mathematicians independent of Tammes began studying circle packing on the sphere in the early 1940s; it was not until twenty years later that the problem became associated with his name.[citation needed]
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
- ↑ Pieter Merkus Lambertus Tammes (1930): On the number and arrangements of the places of exit on the surface of pollen-grains, University of Groningen
- ↑ Batagelj, Vladimir; Plestenjak, Bor. "Optimal arrangements of n points on a sphere and in a circle". IMFM/TCS. https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf.
- ↑ 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.
- ↑ Sloane, N. J. A.. "Spherical Codes: Nice arrangements of points on a sphere in various dimensions". http://neilsloane.com/packings/.
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. doi:10.1107/S0108767387098842.
- "Equilibrium configurations of N equal charges on a sphere". Journal of Physics A: Mathematical and General 24 (23): Ll369–Ll377. 1991. doi:10.1088/0305-4470/24/23/008. Bibcode: 1991JPhA...24L1369E. http://www.iop.org/EJ/article/0305-4470/24/23/008/ja912308.pdf.
- 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. doi:10.1098/rspa.1998.0218. Bibcode: 1998RSPSA.454.1499M.
- "Viral Self-Assembly as a Thermodynamic Process". Physical Review Letters 90 (24): 248101–1–248101–4. 2003. doi:10.1103/PhysRevLett.90.248101. PMID 12857229. Bibcode: 2003PhRvL..90x8101B. Archived from the original on 2007-09-15. https://web.archive.org/web/20070915043650/http://personnel.physics.ucla.edu/directory/faculty/fac_files/bruinsma/viral_self-assembly.pdf.
- Books
- The Pursuit of Perfect Packing. Taylor and Francis. 2000. 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. https://archive.org/details/penguindictionar0000well/page/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).
 |  |
