Dividing a circle with $n$ points

April 8, 2018

A couple of months ago, I came upon a problem from 3Blue1Brown if $n$ points are placed on a circle, and there is a line connecting every point on the circle, how many regions would be inside the shape. Overall, I found this problem to be quite fascinating, because while the solution was initially not the expected one, a beautiful and intuitive one emerged.

Initially, I thought this problem had to do with sequences. This is because, based on the value of $n$, we can see some type of pattern emerge when it comes to the number of regions in the circle. If we define $R(n)$ to be the number of regions in the circle when $n$ points have been placed on the circle, we can see the following pattern emerge: \begin{align*} R(2) &= 1\\ R(3) &= 4\\ R(4) &= 8\\ R(5) &= 16\\ \end{align*} Based on evaluating $R(2)$ until $R(5)$, it can be observed that $R(n) = 2^{n-1}$. However, if $n=6$, and there is a line from every point to every other point, there are only 31 regions, and when $n=7$, $R(7)=57$. Therefore, $R(n)=2^{n-1}$ does not hold true for every value of $n$ meaning that the formula cannot be used to find the number of regions in the circle for any arbitrary value of $n$.

Instead of trying to observe a sequence, let us try to represent the circle in another way. A cirlce with intersecting lines can also be represented as a graph. In the branch of mathematics called graph theory, a structure consisting of points, called vertices (also known as nodes), are connected by lines called edges.

A circle, with $n$ points on it, where a line is connected between every point, can be represented as a graph. The $n$ points on the circumference of the circle and the points of intersections are the vertices, and the lines which connect all the vertices together are the edges.

Calculating the number of regions in the circle can be solved in the same way as finding the number of faces the circle has. Using the number of edges and vertices in the circle (graph), we can find the number of faces in the circle. For example, consider a cube. A cube has 8 vertices and 12 edges. These 8 vertices and 12 edges are connected in such a way that there are 6 faces. While it may seem that the relationship between the faces and edges and vertices changes for each different shape, it is not the case.

Euler's Characteristic Formula states that $$\chi = V-E+F$$, where $F$ represents the number of faces, $V$ represents the number of vertices, and $E$ represents the number of edges.