Is there a relationship with how many lines are necessary to divide a circle?

Question

I want to use the example of a circle being divided increasing by 1 each time ( 1/1 , 1/2 , 1/3 , 1/4 , etc.). In order it would take this amount of lines: 0, 1, 3, 2, 5, 3, 7, and so on. Is it that an odd division would require an inverse amount of lines (i.e. dividing by 9 correlates to 9 lines) while even division would require half of it’s inverse (i.e. dividing by 16 would require 8 lines)? And if that is the case is there any equation to express this relationship? Would the series diverge or converge?

Answer 1

I’m assuming that you need to find the number of line segments required to divide a circle into $n$ sectors of equal area. In that case each sector has a central angle $\frac{2\pi}n$. Let $D(n)$ denote the number of line segments you need to perform the required division.

You can think of it this way: draw the radii for the first sector inclined to each other at the angle $\frac{2\pi}{n}$ calling them $r_1$ and $r_2$, then draw the next radius $r_3$ inclined to $r_2$ at the same angle, and continue on until you have drawn the $n$th radius. It’s clear from here that we have drawn $n$ radii, so if none of them form a straight line segment then $D(n)=n$.

Denote by $k = \frac{n}2$. Clearly if $k \in \mathbb Z$, then $r_m$ and $r_{m+k}$ for $m \le k$ would form a straight line segment since $k\frac{2\pi}n = \pi$. That would mean exactly $k$ line segments would be required for the division since $k$ radii get added up with the remaining $k$ radii in pairs to form line segments.

So we conclude that $$D(n) = \begin{cases} n : \text{odd n} \\ \frac{n}2 : \text{even n} \end{cases}$$