Edit Mesh / Add Divisions .. will cut a polygonal mesh into uniform squares.
Exponential tessellates by triangular divisions. For example, I’ve tessellated this icosahedral sphere into a round version of the same by going to Edit Mesh / Add Divisions ..
After that, you can triangulate, using Edit Mesh / Add Divisions ..
You can experiment with the settings, but Exponential is the setting you need to add divisions to this type of mesh.
See also Mesh / Smooth.