Earlier I wrote a post showing what happens when you start with an equilateral triangle, then repeatedly subdivide it into smaller and smaller triangles by drawing lines from the centroid (barycenter) to each of the vertices.
I mentioned in that post that I moved the code for finding the center to its own function because in the future I might want to see what happens when you look at different choices of center. There are thousands of ways to define the center of a triangle.
This post will look at 4 levels of recursive division, using the barycenter, incenter, and circumcenter.
The barycenter of a set of points is the point that would be the center of mass if each point had the same weight. (The name comes from the Greek baros for weight. Think barium or bariatric surgery.)
This is the method used in the earlier post.
The incenter of a triangle is the center of the largest circle that can be drawn inside the triangle. When we use this definition of center and repeatedly divide our triangle, we get a substantially different image.
The circumcenter of a triangle is the center of the unique circle that passes through each of the three vertices. This results in a very different image because the circumcenter of a triangle may be outside of the triangle.
By recursively dividing our triangle, we get a hexagon!