# Minimum radius of N congruent circles on a sphere, placed optimally, such that the sphere is covered by the circles?

What is the minimum (radius/radii/range of such) of N congruent circles that are placed (optimally) on a sphere in such a way that they cover the entire surface of the sphere?

For 2 circles, a easier visualization would be having two points, growing them into circles and to keep growing them until they touch each other... Then keep growing them until the... 'tunnel-thing' of area covered by the circles is as... wide as the circumference of the sphere that you want, or something...

• How do you place $5$ equally-sized circle on a sphere in such a way they are "equidistantly placed"? Two on the poles and three on the equator? :D – Jack D'Aurizio Aug 6 '15 at 0:08
• Moreover, do you want to deal with two circles or with $N$ circles? And what have you attempted? – Jack D'Aurizio Aug 6 '15 at 0:13
• @JackD'Aurizio - Equidistant from each other, $N$ circles... ... Oh, yeah, you need research for this SE... Shouldn't be that hard to do... But, hmm... I'll put down some thoughts and see what comes up... – Malady Aug 6 '15 at 0:19
• @JoelReyesNoche - Yes, I just adding information saying that I was looking for a minimum, and also tried to allow the possiblity of ranges. ... If you can think of a better title, please, edit it in! – Malady Aug 6 '15 at 1:46
• This is how I interpreted this question in light of your other post. Assuming that you have a fixed number of emitters, can place them intelligently, and want to minimize their transmission power withou compromising full coverage of the sphere. – Jyrki Lahtonen Aug 6 '15 at 21:50

Tarnai and Gaspar (1991) state that

How must a sphere be covered by $n$ equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for $n=15$ to $20$ are given and some sporadic results for $n>20$ $(n=22,26,38,42,50)$ are presented.

and

Solutions have been established for $n=2$ to $7$ and for $n=10,12,14$; conjectured solutions exist for $n=8,9,11,13,16,20,32,72,122,132$

and they provide the references.

Reference:

T. Tarnai and Zs. Gáspár, Covering a sphere by equal circles, and the rigidity of its graph, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 110, Issue 01, July 1991, pp. 71-89.

Your question has the serious shortcoming of leaving the meaning of equidistant undefined. Anyway, there is an entire problem area devoted to this, called spherical codes.

We call any arrangement of $N$ points on the (unit) sphere a spherical code. Two important parameters of such a code are its minimum distance (= the distance between closest neighbors), and its covering radius (= the maximum of the minimum distance from any point on the sphere to a point of the code). Your question asks to minimize the covering radius given the number of points.

Neil Sloane (the author of OEIS) also maintains databases of best known spherical codes. For you the relevant is the database on coverings, but Sloane also has a database of best available minimum distances. He gives the covering radius and/or minimum distance as angular separation (when looking at the relevant points from the center of the sphere). The database covers all the values up to $N=130$ (and some sporadic larger values of $N$).

Relevant literature (the telecommunication applications do mean that they soon generalize to higher dimensions):

It is hopefully clear by now that you should not expect a formula for that radius as a function on $N$. The problem is exactly that it is unknown what is the optimal arrangement of centers of those circles.

• My modest study of the use of the reflection group of type E_8 in constructing good spherical codes (as well as other reflection groups) once earned me a pleasant trip to the French countryside :-) – Jyrki Lahtonen Aug 6 '15 at 21:27
• Hmmm.... so how would you consisely say: "a set of congruent circles, where the distance between every pair of different circles in the set is the same distance."? Or, would you leave it like that and not try to condense it? [Seriously asking here...] ... How did that study result in a trip to France? A conference? Something else? – Malady Aug 6 '15 at 21:27
• @Malandy: Every pair? No can do (unless N=4, and the centers form a tetrahedron). Look at the pics in Joel's answer! Some pairs of circles will be diametrically opposite and have a larger distance. Some other pairs of circles will overlap each other, and be much closer. IMO the meaningful parameters are the minimum distance and the covering radius. I got from your other question that you can place the transmitters anywhere you want, and are looking for the arrangement that gives you full coverage with minimum transmission power. – Jyrki Lahtonen Aug 6 '15 at 21:34
• Well, edit the question to fit your interpretation! If I think it doesn't match the nature of the unedited version, then I'll revert, and if I accept and it invalidates too many answers or something, I'll expect the community to fix it or something... – Malady Aug 6 '15 at 21:44

For generalizing the case of $N$ equally sized circles equally distant on the sphere, orientation of the circles with respect to each other must be known i.e. number of circles touching one circle.

This orientation of the identical circles on the sphere is very important to find out the radius of the circles.

There are many spherical polyhedrons analogous to Archimedean solids

For different configurations/orientations of the identical circles touching each other on the sphere is explained here mathematical analysis of identical circles touching one another on a sphere analogous to Archimedean solid (you may also download PDF from here)

When two equally sized circles at the poles are grown at the same rate then the radius of both the circles will be equal to the radius of globe or the Earth.

• While I'd love to be able to read that pdf... I don't want to register to any more sites... – Malady Aug 6 '15 at 0:35
• Kindly, take it as a hint not otherwise because the explanation of your question is much lengthy involving different cases for generalization. I really appreciate your nice question of the spherical geometry. – Harish Chandra Rajpoot Aug 6 '15 at 0:38
• You may also download PDF from works.bepress.com/harishchandrarajpoot_hcrajpoot/47 without any registration – Harish Chandra Rajpoot Aug 6 '15 at 0:45

Start with in-circles of faces of the only possible 5 Platonic solids. What you say "Circles" are spherical caps. Unless the regular polygons grow at their corners the sphere cannot be fully covered.