Tammes problem
{{Short description|Circle packing 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.{{cite web|first1=Pieter Merkus Lambertus|last1=Tammes|year=1930|title=On the number and arrangements of the places of exit on the surface of pollen-grains|url=https://www.hars.us/Papers/Numerical_Tammes.pdf}}
{{unsolved|mathematics|What is the optimal packing of circles on the surface of a sphere for every possible amount of circles?}}
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.{{cite news |last1=Batagelj |first1=Vladimir |last2=Plestenjak |first2=Bor |title=Optimal arrangements of n points on a sphere and in a circle |url=https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf |archive-date=25 June 2018 |archive-url=https://web.archive.org/web/20180625050324/https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf |publisher=IMFM/TCS}} Thus far, solutions have been proven only for small numbers of circles: 3 through 14, and 24.{{cite journal |last1=Musin |first1=Oleg R. |last2=Tarasov |first2=Alexey S. |title=The Tammes Problem for N = 14 |journal=Experimental Mathematics |date=2015 |volume=24 |issue=4 |pages=460–468 |doi=10.1080/10586458.2015.1022842|s2cid=39429109 }} There are conjectured solutions for many other cases, including those in higher dimensions.{{cite web |last1=Sloane |first1=N. J. A. |title=Spherical Codes: Nice arrangements of points on a sphere in various dimensions |url=http://neilsloane.com/packings/}}
See also
References
=Bibliography=
; Journal articles
- {{cite journal |author1=Tarnai T |author2=((Gáspár Zs)) | year = 1987 | title = Multi-symmetric close packings of equal spheres on the spherical surface | journal = Acta Crystallographica | volume = A43 |issue=5 | pages = 612–616 | doi = 10.1107/S0108767387098842 | doi-access = |bibcode=1987AcCrA..43..612T }}
- {{cite journal | vauthors = Erber T, Hockney GM | year = 1991 | title = Equilibrium configurations of N equal charges on a sphere | journal = Journal of Physics A: Mathematical and General | volume = 24 | issue = 23 | url = http://www.iop.org/EJ/article/0305-4470/24/23/008/ja912308.pdf | pages = Ll369–Ll377 | doi = 10.1088/0305-4470/24/23/008 | bibcode = 1991JPhA...24L1369E | s2cid = 122561279 }}
- {{cite journal | author = Melissen JBM | year = 1998 | title = How Different Can Colours Be? Maximum Separation of Points on a Spherical Octant | journal = Proceedings of the Royal Society A | volume = 454 | pages = 1499–1508 | doi = 10.1098/rspa.1998.0218 | issue = 1973|bibcode = 1998RSPSA.454.1499M | s2cid = 122700006 }}
- {{cite journal | vauthors = Bruinsma RF, Gelbart WM, Reguera D, Rudnick J, Zandi R | year = 2003 | title = Viral Self-Assembly as a Thermodynamic Process | journal = Physical Review Letters | volume = 90 | pages = 248101–1–248101–4 | url = http://personnel.physics.ucla.edu/directory/faculty/fac_files/bruinsma/viral_self-assembly.pdf | doi = 10.1103/PhysRevLett.90.248101 | issue = 24 | pmid = 12857229 | bibcode = 2003PhRvL..90x8101B | arxiv = cond-mat/0211390 | hdl = 2445/13275 | s2cid = 1353095 | url-status = dead | archiveurl = https://web.archive.org/web/20070915043650/http://personnel.physics.ucla.edu/directory/faculty/fac_files/bruinsma/viral_self-assembly.pdf | archivedate = 2007-09-15 }}
- {{cite journal|first=Xiangjing|last1=Lai|first2=Dong|last2=Yue|first3=Jin-Kao|last3=Hao|title =Iterated dynamic neighborhood search for packing equal circles on a sphere|doi=10.1016/j.cor.2022.106121|journal = Comput. Operat. Res.|volume=151|year=2023|url=https://github.com/XiangjingLai/Tammes-problem}}
; Books
- {{cite book | vauthors = Aste T, Weaire DL | year = 2000 | title = The Pursuit of Perfect Packing | title-link=The Pursuit of Perfect Packing | publisher = Taylor and Francis | isbn = 978-0-7503-0648-5 | pages = 108–110}}
- {{cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = [https://archive.org/details/penguindictionar0000well/page/31 31] | url = https://archive.org/details/penguindictionar0000well/page/31 }}
External links
- {{cite journal|first1=Bhaskar|last1=Bagchi|journal=Reconance J. Sci. Edu.|year=1997|volume=2|number=9|pages=18-26|url=https://www.ias.ac.in/article/fulltext/reso/002/09/0018-0026|title=How to Stay Away from Each Other in a Spherical Universe}}.
- [http://www.mi.sanu.ac.rs/vismath/visbook/sugimoto/index.html Packing and Covering of Congruent Spherical Caps on a Sphere].
- [http://users.ipfw.edu/dragnevp/Sigma_Xi1.ppt Science of Spherical Arrangements] (PPT).
- [https://web.archive.org/web/20080806042822/http://wwwmaths.anu.edu.au/events/sy2005/odatalks/womersley.pdf General discussion of packing points on surfaces], with focus on tori (PDF).
{{Packing problem}}