# Great circles on a sphere

• What is the maximal number of disjoint regions obtained on the sphere by dividing it with $n$ great circles?

For $n= 1$ we have $2$ regions, for $n=2$ we have $2^2$, for $n=3$ the number is $2^3$,... what next? - what would be the best approach to this problem? How to use geometrical constraints solving it?

... and ...

• Is it possible to find always $n$ great circles (in the case when the number of regions is maximal) for such division that areas of regions would be equal ? (I suppose not but how to prove it?)

For the count: We use the fact that the Euler characteristic of the sphere is $2$. A generic configuration of great circles will have $2\times \frac {n(n-1)}2=n(n-1)$ vertices and $n\times 2(n-1)$ edges. Thus we have $$2=\#F_n -2n^2+2n+n^2-n=\#F_n-n^2+n\implies \boxed {\#F_n=n^2-n+2}$$

As a sanity check we remark that $\#F_1=2,\#F_2=4,\#F_3=9-3+2=8$ as desired.

• So in this case a good approach is linked with graph theory, without even defining equations for circles? – Widawensen Dec 16 '16 at 14:33
• I wouldn't even know how to use the defining equations for circles in this case....so, yes. I'm just using basic notions about triangulations. – lulu Dec 16 '16 at 14:34
• Fine, so it's simply a topology, however the second question is probably impossible to solve without geometry... but who knows.. "equal areas" is also a very general condition.. – Widawensen Dec 16 '16 at 14:43
• I didn't think about the area question. Like you, my gut reaction is that it isn't possible...but i really don't know. – lulu Dec 16 '16 at 14:45