Convex and discrete geometry Packing and covering Problem

Tammes problem

Problem

Let $N \ge 2$ be an integer and let $S^2 \subset \R^3$ be the unit sphere. Find the largest number $d_N$ such that there are $N$ points on $S^2$ with pairwise distances at least $d_N$ . Find also the configurations of $N$ points that attain it.

Remarks

Equivalently, find the largest radius of $N$ equal non-overlapping circular caps on a sphere, or the best spherical code of size $N$ in $\R^3$ . The botanist Tammes posed it in 1930 while studying the distribution of pores on pollen grains. For $N = 4$ , $6$ and $12$ the answers are the vertices of the regular tetrahedron, octahedron and icosahedron. The problem is solved for $N \le 14$ and for $N = 24$ . Oleg Musin and Alexey Tarasov solved the case $N = 13$ . It is linked with the kissing number problem in three dimensions: thirteen unit balls cannot touch a central one, so the optimal angle for $13$ points is below $60^\circ$ .

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Spherical code concept
Tammes problem problem

Tag	`9WD2TZ4ZB5`
URI	`m2://9WD2TZ4ZB5`
Kind	problem
Writing	W0 stub
MSC	52C17, 05B40
Status	This copy of the site has no build data. Run `m2 status 9WD2TZ4ZB5`.
About	Spherical code
Uses	Spherical code
Same as	Wikidata Q3406239 Wikipedia Tammes problem

Tammes problem

Problem

Remarks

Reading path

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