Ratio of Largest / Smallest triangle area on Octahedron subdivided into Sphere

Question

I wrote a simple Python code that draws Sphere in Blender 3D program. My approach was:

1. Start with a Platonic seed solid (Octahedron).
2. For each triangle, get the middle points of 3 sides and raise their coordinates to sphere radius.
3. Connect these points in order and formulate 4 new sub-triangles.
4. Repeat step 1 to 3 for the number of subdivision level.

Subdivision Level 0 (Seed)

Subdivision Level 1

Subdivision Level 2

Subdivision Level 3

Now if we look at the area of triangles on the subdivided sphere, they become noticeably different. On the seed octahedron, divided faces that were close to a vertex with 4 converging edges become smaller compared to the ones divided in the middle triangle(furthest from 4-edge converging corners).

So I tried to calculate the ratio of Maximum Area / Minimum Area of triangles as it gets divided closer into sphere. The result seems to suggest this ratio is not an ever increasing number but converges into a certain constant.

There is a limit I can calculate this with brute-force approach. Can anyone help me calculate limit value of Max / Min ratio as number of subdivision becomes infinity? I would like to see a mathematical approach, not another brute-force like I tried.