# Given number of regions, find maximum number of boundaries

Problem: Supoose we want to divide $\mathbb{R^2}$ into $N$ regions separated by straight lines. What is the maximum number of boundary lines we could have? I also pose the restriction that between any two regions, there can be only one boundary line (so zigzagging a boundary does not increase the count). For example, here I attempt to find the maximum number of boundaries for $N = 4$ regions in $\mathbb{R}^2$, and I count 6 boundary lines. But is there a general formula for general integer $N$? Can someone point me to a reference on this problem?

• When you say "straight lines", do you mean infinite lines, rays and segments? Can those lines intersect? – ajotatxe Sep 4 '18 at 18:32
• These lines cannot intersect (except at the vertices of course). They can be segments, rays (these two are shown in my example) and infinite lines (sure, we can split $\mathbb{R}^2$ in half). I just thought this seems to be an interesting problem, so please excuse my problem statement not being fully rigorous... – Longti Sep 4 '18 at 18:42
• – lhf Sep 5 '18 at 0:15
• @lhf It seems that I can bound the number of edges $e$ with the number of vertices $v$, but unless I pose more restriction on the faces, I can't seem to find a bound of $e$ by the number of faces – Longti Sep 5 '18 at 1:23